Chapter 10: Resampling Methods

Foldable textbook-to-Lean result crosswalk for Chapter 10.

This page is generated from the canonical Chapter 10 inventory and the compiled Lean environment. It contains 68 textbook result groups and 338 selected Lean endpoints. When an inventory row links a large proof surface, this page shows at most six theorem-facing endpoints. The inventory remains the source of truth for all supporting links, qualifications, and open gaps.

Definition 10.11 endpoint

Bootstrap convergence in probability means P^*(\lVert Z_n^*-Z\rVert\gt \varepsilon)\xrightarrow{p}0 for every \varepsilon\gt 0.

def HansenEconometrics.TendstoInBootstrapProbability

Hansen Definition 10.1: convergence in bootstrap probability.

Zstar n ->p* Z means that for every positive tolerance η, the conditional tail probability Pstar[dist (Zstar n) Z ≥ η] converges to zero in ordinary probability under the original-sample law μ.

The inequality is written with ≤ to match Mathlib’s TendstoInMeasure convention; this is the usual harmless closed-tail version of convergence in probability.

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {E : Type u_3} →
      {mΩ : MeasurableSpace Ω} →
        {mΩs : MeasurableSpace Ωs} →
          [PseudoMetricSpace E] →
            MeasureTheory.Measure Ω → (Nat → Ω → MeasureTheory.Measure Ωs) → (Nat → Ω → Ωs → E) → (Ω → E) → Prop

HansenEconometrics/BootstrapUtils.lean:65

Equation 10.12 endpoints

Tukey’s jackknife covariance is \hat V_J=\frac{n-1}{n}\sum_{i=1}^n(\hat\theta_{(-i)}-\bar\theta_J)(\hat\theta_{(-i)}-\bar\theta_J)'.

def HansenEconometrics.jackknifeMean

Mean of jackknife pseudo-sample estimators.

Formal statement

{E : Type u_4} → {ι : Type u_7} → [Fintype ι] → [inst : NormedAddCommGroup E] → [NormedSpace Real E] → (ι → E) → E

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:73

def HansenEconometrics.jackknifeCovariance

Hansen equation (10.1): Tukey’s jackknife covariance estimator.

Formal statement

{ι : Type u_7} → [Fintype ι] → {k : Type u_8} → [Fintype k] → (ι → k → Real) → Matrix k k Real

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:79

Equation 10.25 endpoints

For the sample mean, \bar Y_{(-i)}-\bar Y=(\bar Y-Y_i)/(n-1) and n^{-1}\sum_i\bar Y_{(-i)}=\bar Y.

def HansenEconometrics.jackknifeLeaveOneOutMean

Leave-one-out sample mean for Hansen’s jackknife discussion.

For observation i, this is the empirical mean of the sample with i deleted, matching equation (10.2).

Formal statement

{E : Type u_4} →
  {ι : Type u_7} →
    [Fintype ι] → [DecidableEq ι] → [inst : NormedAddCommGroup E] → [NormedSpace Real E] → (ι → E) → ι → E

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:66

theorem HansenEconometrics.card_leaveOneOutIndex

Cardinality of the row-deleted jackknife index.

Formal statement

∀ {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] (i : ι),
  Eq (Fintype.card (HansenEconometrics.LeaveOneOutIndex i)) (instHSub.hSub (Fintype.card ι) 1)

Direct statement dependencies (1)

HansenEconometrics.LeaveOneOutIndex

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:88

theorem HansenEconometrics.jackknifeLeaveOneOutMean_eq

Hansen equation (10.2): the leave-one-out mean written in terms of the full-sample mean and the deleted observation.

Formal statement

∀ {E : Type u_4} {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nontrivial ι]
  [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace Real E] (Y : ι → E) (i : ι),
  Eq (HansenEconometrics.jackknifeLeaveOneOutMean Y i)
    (instHSub.hSub
      (instHSMul.hSMul (instHDiv.hDiv (Fintype.card ι).cast (instHSub.hSub (Fintype.card ι).cast 1))
        (HansenEconometrics.empiricalMean Y))
      (instHSMul.hSMul (Real.instInv.inv (instHSub.hSub (Fintype.card ι).cast 1)) (Y i)))

Direct statement dependencies (2)

HansenEconometrics.empiricalMean
HansenEconometrics.jackknifeLeaveOneOutMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:95

theorem HansenEconometrics.jackknifeMean_leaveOneOutMean_eq_empiricalMean

The average of the leave-one-out sample means is the full-sample mean.

Formal statement

∀ {E : Type u_4} {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nontrivial ι]
  [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace Real E] (Y : ι → E),
  Eq (HansenEconometrics.jackknifeMean fun i => HansenEconometrics.jackknifeLeaveOneOutMean Y i)
    (HansenEconometrics.empiricalMean Y)

Direct statement dependencies (3)

HansenEconometrics.empiricalMean
HansenEconometrics.jackknifeLeaveOneOutMean
HansenEconometrics.jackknifeMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:130

theorem HansenEconometrics.jackknifeLeaveOneOutMean_sub_empiricalMean_eq

Hansen’s displayed identity after (10.2): each leave-one-out mean differs from the full-sample mean by (Ybar - Yᵢ)/(n-1).

Formal statement

∀ {E : Type u_4} {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nontrivial ι]
  [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace Real E] (Y : ι → E) (i : ι),
  Eq (instHSub.hSub (HansenEconometrics.jackknifeLeaveOneOutMean Y i) (HansenEconometrics.empiricalMean Y))
    (instHSMul.hSMul (Real.instInv.inv (instHSub.hSub (Fintype.card ι).cast 1))
      (instHSub.hSub (HansenEconometrics.empiricalMean Y) (Y i)))

Direct statement dependencies (2)

HansenEconometrics.empiricalMean
HansenEconometrics.jackknifeLeaveOneOutMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:169

Equation 10.31 endpoint

For the sample mean, the jackknife covariance equals the usual estimated variance: \hat V_J(\bar Y)=s_Y^2/n.

theorem HansenEconometrics.jackknifeCovariance_leaveOneOutMean_eq_sampleMeanCovariance

Hansen equation (10.3): for the sample mean, Tukey’s jackknife covariance equals the conventional covariance estimator for the variance of the mean.

Formal statement

∀ {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nontrivial ι] {k : Type u_8} [inst_3 : Fintype k]
  (Y : ι → k → Real),
  Eq (HansenEconometrics.jackknifeCovariance fun i => HansenEconometrics.jackknifeLeaveOneOutMean Y i) fun a b =>
    instHMul.hMul
      (instHMul.hMul (Real.instInv.inv (Fintype.card ι).cast)
        (Real.instInv.inv (instHSub.hSub (Fintype.card ι).cast 1)))
      (Finset.univ.sum fun i =>
        instHMul.hMul (instHSub.hSub (HansenEconometrics.empiricalMean Y a) (Y i a))
          (instHSub.hSub (HansenEconometrics.empiricalMean Y b) (Y i b)))

Direct statement dependencies (3)

HansenEconometrics.empiricalMean
HansenEconometrics.jackknifeCovariance
HansenEconometrics.jackknifeLeaveOneOutMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:203

Equation 10.45 endpoints

Leave-one-out OLS obeys \hat\beta_{(-i)}=\hat\beta-(X'X)^{-1}X_i\hat e_i/(1-h_{ii}) and \hat e_{i,(-i)}=\hat e_i/(1-h_{ii}).

def HansenEconometrics.LeaveOneOutIndex

Row index type for the sample with observation i deleted.

Formal statement

{n : Type u_1} → n → Type (max 0 u_1)

HansenEconometrics/Chapter3Projections.lean:225

def HansenEconometrics.leaveOneOutBeta

Hansen equation (3.42): leave-one-out coefficient written with the reduced Gram matrix.

Formal statement

{n : Type u_1} →
  {k : Type u_2} →
    [inst : Fintype n] →
      [inst_1 : Fintype k] →
        [inst_2 : DecidableEq k] →
          (X : Matrix n k Real) →
            (n → Real) → (i : n) → [Invertible (HansenEconometrics.leaveOneOutGram X i)] → k → Real

Direct statement dependencies (1)

HansenEconometrics.leaveOneOutGram

HansenEconometrics/Chapter3Projections.lean:265

theorem HansenEconometrics.leaveOneOutBeta_eq_olsBeta_sub_invGram_mulVec

Hansen Theorem 3.7 / equation (3.43): leave-one-out coefficients can be computed from the full-sample coefficient and prediction error.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] (X : Matrix n k Real)
  (y : n → Real) (i : n) [inst_3 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  [inst_4 : Invertible (HansenEconometrics.leaveOneOutGram X i)],
  Eq (HansenEconometrics.leaveOneOutBeta X y i)
    (instHSub.hSub (HansenEconometrics.olsBeta X y)
      (instHSMul.hSMul (HansenEconometrics.leaveOneOutResidual X y i)
        ((inst_3.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)).mulVec (X i))))

Direct statement dependencies (4)

HansenEconometrics.leaveOneOutBeta
HansenEconometrics.leaveOneOutGram
HansenEconometrics.leaveOneOutResidual
HansenEconometrics.olsBeta

HansenEconometrics/Chapter3Projections.lean:308

theorem HansenEconometrics.leaveOneOutResidual_eq_inv_one_sub_leverage_mul_residual

Hansen Theorem 3.7 / equation (3.44): leave-one-out prediction errors are full-sample residuals scaled by (1 - hᵢᵢ)⁻¹.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] (X : Matrix n k Real)
  (y : n → Real) (i : n) [inst_3 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  [inst_4 : Invertible (HansenEconometrics.leaveOneOutGram X i)],
  Ne (instHSub.hSub 1 (HansenEconometrics.leverageValue X i)) 0 →
    Eq (HansenEconometrics.leaveOneOutResidual X y i)
      (instHMul.hMul (Real.instInv.inv (instHSub.hSub 1 (HansenEconometrics.leverageValue X i)))
        (HansenEconometrics.residual X y i))

Direct statement dependencies (4)

HansenEconometrics.leaveOneOutGram
HansenEconometrics.leaveOneOutResidual
HansenEconometrics.leverageValue
HansenEconometrics.residual

HansenEconometrics/Chapter3Projections.lean:364

theorem HansenEconometrics.olsBeta_sub_leaveOneOutBeta_eq_invGram_mulVec

Hansen equation (3.48): the change in coefficient estimates after dropping observation i.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] (X : Matrix n k Real)
  (y : n → Real) (i : n) [inst_3 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  [inst_4 : Invertible (HansenEconometrics.leaveOneOutGram X i)],
  Eq (instHSub.hSub (HansenEconometrics.olsBeta X y) (HansenEconometrics.leaveOneOutBeta X y i))
    (instHSMul.hSMul (HansenEconometrics.leaveOneOutResidual X y i)
      ((inst_3.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)).mulVec (X i)))

Direct statement dependencies (4)

HansenEconometrics.leaveOneOutBeta
HansenEconometrics.leaveOneOutGram
HansenEconometrics.leaveOneOutResidual
HansenEconometrics.olsBeta

HansenEconometrics/Chapter3Projections.lean:375

Equation 10.56 of 7 linked endpoints

The OLS jackknife covariance is the HC3 covariance minus its finite-sample mean-adjustment outer product: \hat V_J=\hat V_{\mathrm{HC3}}-aa'.

theorem HansenEconometrics.jackknifeCovariance_leaveOneOutBeta_eq_HC3_sub_meanAdjustment

Hansen equation (10.5): the OLS leave-one-out jackknife covariance equals the HC3 covariance estimator less the finite-sample mean-adjustment outer product.

Formal statement

∀ {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nontrivial ι] {k : Type u_8} [inst_3 : Fintype k]
  [inst_4 : DecidableEq k] (X : Matrix ι k Real) (y : ι → Real)
  [inst_5 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hloo : (i : ι) → Invertible (HansenEconometrics.leaveOneOutGram X i)),
  (∀ (i : ι), Ne (instHSub.hSub 1 (HansenEconometrics.leverageValue X i)) 0) →
    Eq (HansenEconometrics.jackknifeCovariance (HansenEconometrics.olsLeaveOneOutBetaFamily X y hloo))
      (instHSub.hSub
        (instHSMul.hSMul (instHDiv.hDiv (instHSub.hSub (Fintype.card ι).cast 1) (Fintype.card ι).cast)
          (HansenEconometrics.olsHuberWhiteHC3VarianceEstimator X y))
        (instHSMul.hSMul (instHSub.hSub (Fintype.card ι).cast 1)
          (Matrix.vecMulVec (HansenEconometrics.empiricalMean (HansenEconometrics.olsLeaveOneOutInfluence X y hloo))
            (HansenEconometrics.empiricalMean (HansenEconometrics.olsLeaveOneOutInfluence X y hloo)))))

Direct statement dependencies (7)

HansenEconometrics.empiricalMean
HansenEconometrics.jackknifeCovariance
HansenEconometrics.leaveOneOutGram
HansenEconometrics.leverageValue
HansenEconometrics.olsHuberWhiteHC3VarianceEstimator
HansenEconometrics.olsLeaveOneOutBetaFamily
HansenEconometrics.olsLeaveOneOutInfluence

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:421

def HansenEconometrics.olsLeaveOneOutScoreMean

Hansen equation (10.5) score mean = n^{-1}_i X_i e_i, stated with the leave-one-out prediction errors from Chapter 3.

Formal statement

{ι : Type u_7} →
  [inst : Fintype ι] →
    [DecidableEq ι] →
      {k : Type u_8} →
        [inst_2 : Fintype k] →
          [inst_3 : DecidableEq k] →
            (X : Matrix ι k Real) →
              (ι → Real) →
                [Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)] →
                  ((i : ι) → Invertible (HansenEconometrics.leaveOneOutGram X i)) → k → Real

Direct statement dependencies (1)

HansenEconometrics.leaveOneOutGram

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:294

theorem HansenEconometrics.leaveOneOutBeta_sub_jackknifeMean_eq_influenceMean_sub

Leave-one-out coefficient deviations equal the centered influence deviations used in Hansen equation (10.5).

Formal statement

∀ {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] [Nontrivial ι] {k : Type u_8} [inst_3 : Fintype k]
  [inst_4 : DecidableEq k] (X : Matrix ι k Real) (y : ι → Real)
  [inst_5 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hloo : (i : ι) → Invertible (HansenEconometrics.leaveOneOutGram X i)) (i : ι),
  Eq
    (instHSub.hSub (HansenEconometrics.olsLeaveOneOutBetaFamily X y hloo i)
      (HansenEconometrics.jackknifeMean (HansenEconometrics.olsLeaveOneOutBetaFamily X y hloo)))
    (instHSub.hSub (HansenEconometrics.empiricalMean (HansenEconometrics.olsLeaveOneOutInfluence X y hloo))
      (HansenEconometrics.olsLeaveOneOutInfluence X y hloo i))

Direct statement dependencies (5)

HansenEconometrics.empiricalMean
HansenEconometrics.jackknifeMean
HansenEconometrics.leaveOneOutGram
HansenEconometrics.olsLeaveOneOutBetaFamily
HansenEconometrics.olsLeaveOneOutInfluence

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:346

theorem HansenEconometrics.sum_vecMulVec_olsLeaveOneOutInfluence_eq_HC3

The uncentered OLS leave-one-out influence outer-product sum is the HC3 covariance estimator when the leave-one-out residuals are written as prediction errors.

Formal statement

∀ {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] {k : Type u_8} [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] (X : Matrix ι k Real) (y : ι → Real)
  [inst_4 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hloo : (i : ι) → Invertible (HansenEconometrics.leaveOneOutGram X i)),
  (∀ (i : ι), Ne (instHSub.hSub 1 (HansenEconometrics.leverageValue X i)) 0) →
    Eq
      (Finset.univ.sum fun i =>
        Matrix.vecMulVec (HansenEconometrics.olsLeaveOneOutInfluence X y hloo i)
          (HansenEconometrics.olsLeaveOneOutInfluence X y hloo i))
      (HansenEconometrics.olsHuberWhiteHC3VarianceEstimator X y)

Direct statement dependencies (4)

HansenEconometrics.leaveOneOutGram
HansenEconometrics.leverageValue
HansenEconometrics.olsHuberWhiteHC3VarianceEstimator
HansenEconometrics.olsLeaveOneOutInfluence

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:391

theorem HansenEconometrics.empiricalMean_olsLeaveOneOutInfluence_eq_invGram_mulVec_scoreMean

The mean of the OLS leave-one-out influence vectors is (X’X)^{-1}.

Formal statement

∀ {ι : Type u_7} [inst : Fintype ι] [inst_1 : DecidableEq ι] {k : Type u_8} [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] (X : Matrix ι k Real) (y : ι → Real)
  [inst_4 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hloo : (i : ι) → Invertible (HansenEconometrics.leaveOneOutGram X i)),
  Eq (HansenEconometrics.empiricalMean (HansenEconometrics.olsLeaveOneOutInfluence X y hloo))
    ((inst_4.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)).mulVec
      (HansenEconometrics.olsLeaveOneOutScoreMean X y hloo))

Direct statement dependencies (4)

HansenEconometrics.empiricalMean
HansenEconometrics.leaveOneOutGram
HansenEconometrics.olsLeaveOneOutInfluence
HansenEconometrics.olsLeaveOneOutScoreMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:327

def HansenEconometrics.olsLeaveOneOutInfluence

Influence vector (X’X)^{-1}X_ie_i in Hansen’s OLS jackknife calculation.

Formal statement

{ι : Type u_7} →
  [inst : Fintype ι] →
    [DecidableEq ι] →
      {k : Type u_8} →
        [inst_2 : Fintype k] →
          [inst_3 : DecidableEq k] →
            (X : Matrix ι k Real) →
              (ι → Real) →
                [Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)] →
                  ((i : ι) → Invertible (HansenEconometrics.leaveOneOutGram X i)) → ι → k → Real

Direct statement dependencies (1)

HansenEconometrics.leaveOneOutGram

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:316

Section 10.56 of 9 linked endpoints

The delete-cluster estimator uses \hat\beta_{(-g)} and the CR3 adjusted cluster score X_g'(I-H_{gg})^{-1}\hat e_g.

theorem HansenEconometrics.clusterLeaveOutBeta_eq_olsBeta_sub_invGram_mulVec_cr3Residual

Hansen equation (4.53), reduced-Gram form.

The cluster-deleted coefficient equals the full-sample coefficient minus the cluster leverage correction based on the CR3-style prediction errors.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] {G : Type u_3}
  [inst_3 : DecidableEq G] (X : Matrix n k Real) (y : n → Real) (cluster : n → G) (g : G) [inst_4 : DecidableEq n]
  [inst_5 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  [inst_6 : Invertible (HansenEconometrics.clusterLeaveOutAdjustmentMatrix X cluster g)]
  [inst_7 : Invertible (HansenEconometrics.clusterLeaveOutGram X cluster g)],
  Eq (HansenEconometrics.clusterLeaveOutBeta X y cluster g)
    (instHSub.hSub (HansenEconometrics.olsBeta X y)
      ((inst_5.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)).mulVec
        ((HansenEconometrics.clusterDesign X cluster g).transpose.mulVec
          (HansenEconometrics.clusterCR3Residual X y cluster g))))

Direct statement dependencies (7)

HansenEconometrics.ClusterIndex
HansenEconometrics.clusterCR3Residual
HansenEconometrics.clusterDesign
HansenEconometrics.clusterLeaveOutAdjustmentMatrix
HansenEconometrics.clusterLeaveOutBeta
HansenEconometrics.clusterLeaveOutGram
HansenEconometrics.olsBeta

HansenEconometrics/Chapter4LeastSquaresRegression.lean:953

theorem HansenEconometrics.olsClusteredCR3VarianceEstimator_eq_clusterCovarianceMiddle_cr3Outer

Hansen equation (4.54) with the CR3 residual block outer-product expression for the clustered middle matrix.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] {G : Type u_3}
  [inst_3 : Fintype G] [inst_4 : DecidableEq G] (X : Matrix n k Real) (y : n → Real) (cluster : n → G)
  [inst_5 : DecidableEq n] [inst_6 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hInv : (g : G) → Invertible (HansenEconometrics.clusterLeaveOutAdjustmentMatrix X cluster g)),
  Eq (HansenEconometrics.olsClusteredCR3VarianceEstimator X y cluster hInv)
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
        (inst_6.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X))
        (HansenEconometrics.clusterCovarianceMiddle X cluster fun g =>
          Matrix.vecMulVec (HansenEconometrics.clusterCR3Residual X y cluster g)
            (HansenEconometrics.clusterCR3Residual X y cluster g)))
      (inst_6.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)))

Direct statement dependencies (5)

HansenEconometrics.ClusterIndex
HansenEconometrics.clusterCR3Residual
HansenEconometrics.clusterCovarianceMiddle
HansenEconometrics.clusterLeaveOutAdjustmentMatrix
HansenEconometrics.olsClusteredCR3VarianceEstimator

HansenEconometrics/Chapter4LeastSquaresRegression.lean:1116

def HansenEconometrics.clusterCR3Residual

Hansen equation (4.52): CR3-style cluster prediction errors tilde e_g = (I_g - X_g (X’X)^{-1} X_g’)^{-1} e_hat_g.

Formal statement

{n : Type u_1} →
  {k : Type u_2} →
    [inst : Fintype n] →
      [inst_1 : Fintype k] →
        [inst_2 : DecidableEq k] →
          {G : Type u_3} →
            [inst_3 : DecidableEq G] →
              (X : Matrix n k Real) →
                (n → Real) →
                  (cluster : n → G) →
                    (g : G) →
                      [inst_4 : DecidableEq n] →
                        [inst_5 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)] →
                          [Invertible (HansenEconometrics.clusterLeaveOutAdjustmentMatrix X cluster g)] →
                            HansenEconometrics.ClusterIndex cluster g → Real

Direct statement dependencies (2)

HansenEconometrics.ClusterIndex
HansenEconometrics.clusterLeaveOutAdjustmentMatrix

HansenEconometrics/Chapter4LeastSquaresRegression.lean:593

def HansenEconometrics.olsClusteredCR3VarianceEstimator

Hansen equation (4.54): CR3-style cluster-robust covariance estimator using leave-cluster-out prediction errors.

Formal statement

{n : Type u_1} →
  {k : Type u_2} →
    [inst : Fintype n] →
      [inst_1 : Fintype k] →
        [inst_2 : DecidableEq k] →
          {G : Type u_3} →
            [Fintype G] →
              [inst_4 : DecidableEq G] →
                (X : Matrix n k Real) →
                  (n → Real) →
                    (cluster : n → G) →
                      [inst_5 : DecidableEq n] →
                        [inst_6 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)] →
                          ((g : G) → Invertible (HansenEconometrics.clusterLeaveOutAdjustmentMatrix X cluster g)) →
                            Matrix k k Real

Direct statement dependencies (2)

HansenEconometrics.ClusterIndex
HansenEconometrics.clusterLeaveOutAdjustmentMatrix

HansenEconometrics/Chapter4LeastSquaresRegression.lean:1029

theorem HansenEconometrics.olsClusteredCR3VarianceEstimator_eq_clusterCR3Score

The CR3 clustered sandwich written with the named CR3 cluster-score API.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] {G : Type u_3}
  [inst_3 : Fintype G] [inst_4 : DecidableEq G] (X : Matrix n k Real) (y : n → Real) (cluster : n → G)
  [inst_5 : DecidableEq n] [inst_6 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hInv : (g : G) → Invertible (HansenEconometrics.clusterLeaveOutAdjustmentMatrix X cluster g)),
  Eq (HansenEconometrics.olsClusteredCR3VarianceEstimator X y cluster hInv)
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
        (inst_6.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X))
        (Finset.univ.sum fun g =>
          Matrix.vecMulVec (HansenEconometrics.clusterCR3Score X y cluster g)
            (HansenEconometrics.clusterCR3Score X y cluster g)))
      (inst_6.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)))

Direct statement dependencies (4)

HansenEconometrics.ClusterIndex
HansenEconometrics.clusterCR3Score
HansenEconometrics.clusterLeaveOutAdjustmentMatrix
HansenEconometrics.olsClusteredCR3VarianceEstimator

HansenEconometrics/Chapter4LeastSquaresRegression.lean:1055

theorem HansenEconometrics.olsClusteredCR3VarianceEstimator_eq_clusterCR3ScoreMiddle

The CR3 clustered sandwich written through the named CR3 score-middle matrix.

Formal statement

∀ {n : Type u_1} {k : Type u_2} [inst : Fintype n] [inst_1 : Fintype k] [inst_2 : DecidableEq k] {G : Type u_3}
  [inst_3 : Fintype G] [inst_4 : DecidableEq G] (X : Matrix n k Real) (y : n → Real) (cluster : n → G)
  [inst_5 : DecidableEq n] [inst_6 : Invertible (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)]
  (hInv : (g : G) → Invertible (HansenEconometrics.clusterLeaveOutAdjustmentMatrix X cluster g)),
  Eq (HansenEconometrics.olsClusteredCR3VarianceEstimator X y cluster hInv)
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
        (inst_6.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X))
        (HansenEconometrics.clusterCR3ScoreMiddle X y cluster hInv))
      (inst_6.invOf (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul X.transpose X)))

Direct statement dependencies (4)

HansenEconometrics.ClusterIndex
HansenEconometrics.clusterCR3ScoreMiddle
HansenEconometrics.clusterLeaveOutAdjustmentMatrix
HansenEconometrics.olsClusteredCR3VarianceEstimator

HansenEconometrics/Chapter4LeastSquaresRegression.lean:1081

Equation 10.63 endpoints

A fixed observation appears in an n-draw bootstrap sample with probability 1-(1-1/n)^n\to1-e^{-1}.

def HansenEconometrics.bootstrapObservationInclusionProbability

Hansen equation (10.6): displayed probability that one fixed observation is included at least once in an n-draw bootstrap sample from n observations.

The expression is totalized at n = 0; the asymptotic result below is the textbook statement.

Formal statement

Nat → Real

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:527

theorem HansenEconometrics.bootstrapObservationInclusionProbability_eq

No plain-language docstring is available.

Formal statement

∀ (n : Nat),
  Eq (HansenEconometrics.bootstrapObservationInclusionProbability n)
    (instHSub.hSub 1 (instHPow.hPow (instHSub.hSub 1 (instHDiv.hDiv 1 n.cast)) n))

Direct statement dependencies (1)

HansenEconometrics.bootstrapObservationInclusionProbability

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:530

theorem HansenEconometrics.bootstrapObservationInclusionProbability_tendsto

Hansen equation (10.6): 1 - (1 - 1/n)^n → 1 - e^{-1}.

Formal statement

Filter.Tendsto HansenEconometrics.bootstrapObservationInclusionProbability Filter.atTop
  (nhds (instHSub.hSub 1 (Real.exp (-1))))

Direct statement dependencies (1)

HansenEconometrics.bootstrapObservationInclusionProbability

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:536

Equation 10.75 endpoints

With B draws, \hat V_B^*=\frac{1}{B-1}\sum_{b=1}^B(\hat\theta_b^*-\bar\theta^*)(\hat\theta_b^*-\bar\theta^*)'.

def HansenEconometrics.finiteReplicationMeanReal

Mean across B finite bootstrap replications of a real statistic.

Formal statement

{Ω : Type u_1} → (Nat → Nat → Ω → Real) → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:25

def HansenEconometrics.finiteReplicationMeanVec

Mean vector across B finite bootstrap replications.

Formal statement

{Ω : Type u_1} → {k : Type u_6} → (Nat → Nat → Ω → k → Real) → Nat → Ω → k → Real

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:74

def HansenEconometrics.finiteReplicationCovarianceCenteredReal

Centered finite-replication covariance estimator for two real statistics.

Formal statement

{Ω : Type u_1} → (Nat → Nat → Ω → Real) → (Nat → Nat → Ω → Real) → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:58

def HansenEconometrics.finiteReplicationVarianceCenteredReal

Centered finite-replication variance estimator for a real statistic.

This is the scalar X = Y notation for Hansen’s centered finite-replication covariance estimator.

Formal statement

{Ω : Type u_1} → (Nat → Nat → Ω → Real) → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:69

def HansenEconometrics.finiteReplicationCovarianceCenteredMat

Centered finite-replication covariance matrix estimator.

Formal statement

{Ω : Type u_1} → {k : Type u_6} → (Nat → Nat → Ω → k → Real) → Nat → Ω → Matrix k k Real

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:92

Equation 10.85 endpoints

The percentile interval is C_{\mathrm{pct}}=[q_{\alpha/2}^*,q_{1-\alpha/2}^*].

def HansenEconometrics.percentileCIEvent

Hansen percentile confidence interval event, qLower <= θ <= qUpper.

Formal statement

Real → Real → Real → Prop

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:26

def HansenEconometrics.lowerCDFQuantile

Lower generalized inverse of a real CDF-like function.

Formal statement

(Real → Real) → Real → Real

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:38

def HansenEconometrics.bootstrapScalarCDF

Scalar conditional bootstrap CDF P*[Zₙ* ≤ x].

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {mΩs : MeasurableSpace Ωs} → (Nat → Ω → MeasureTheory.Measure Ωs) → (Nat → Ω → Ωs → Real) → Real → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:576

theorem HansenEconometrics.bootstrapScalarLowerQuantile_tendsto_of_strictMono_cdf

Bootstrap scalar lower-quantile convergence with a strictly increasing limiting CDF.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real} {G : Real → Real} {p q : Real},
  (∀ (n : Nat) (ω : Ω), Monotone fun x => HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω) →
    (∀ (n : Nat) (ω : Ω),
        (setOf fun x => Real.instLE.le p (HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω)).Nonempty) →
      (∀ (n : Nat) (ω : Ω),
          BddBelow (setOf fun x => Real.instLE.le p (HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω))) →
        (∀ (n : Nat) (ω : Ω) (x : Real),
            Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω) p →
              Exists fun δ =>
                And (Real.instLT.lt 0 δ)
                  (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Zstar (instHAdd.hAdd x δ) n ω) p)) →
          StrictMono G →
            Eq (G q) p →
              (∀ (x : Real),
                  MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω)
                    Filter.atTop fun x_1 => G x) →
                MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Zstar p)
                  Filter.atTop fun x => q

Direct statement dependencies (2)

HansenEconometrics.bootstrapScalarCDF
HansenEconometrics.bootstrapScalarLowerQuantile

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:932

theorem HansenEconometrics.chapter10_percentileCI_coverage_tendsto_one_sub_alpha_of_bootstrap_lowerQuantiles

Symmetric percentile-interval coverage from bootstrap lower quantiles.

The bootstrap lower quantiles identify the scaled endpoint deviations aₙ(q* - θhatₙ). Dividing by aₙ and adding θhatₙ puts the endpoints on the original parameter scale, after which the symmetric percentile-coverage wrapper applies.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν]
  {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → Real} {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θ : Real} {θhat : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
      MeasureTheory.TendstoInDistribution (fun n ω => instHMul.hMul (a n) (instHSub.hSub (θhat n ω) θ)) Filter.atTop ξ
          (fun x => μ) ν →
        (∀ (n : Nat) (ω : Ω), Monotone fun x => HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) →
          (∀ (n : Nat) (ω : Ω),
              (setOf fun x =>
                  Real.instLE.le (instHDiv.hDiv α 2)
                    (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
            (∀ (n : Nat) (ω : Ω),
                BddBelow
                  (setOf fun x =>
                    Real.instLE.le (instHDiv.hDiv α 2) (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
              (∀ (n : Nat) (ω : Ω) (x : Real),
                  Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) (instHDiv.hDiv α 2) →
                    Exists fun δ =>
                      And (Real.instLT.lt 0 δ)
                        (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω)
                          (instHDiv.hDiv α 2))) →
                (∀ (n : Nat) (ω : Ω),
                    (setOf fun x =>
                        Real.instLE.le (instHSub.hSub 1 (instHDiv.hDiv α 2))
                          (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
                  (∀ (n : Nat) (ω : Ω),
                      BddBelow
                        (setOf fun x =>
                          Real.instLE.le (instHSub.hSub 1 (instHDiv.hDiv α 2))
                            (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
                    (∀ (n : Nat) (ω : Ω) (x : Real),
                        Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)
                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                          Exists fun δ =>
                            And (Real.instLT.lt 0 δ)
                              (Real.instLT.lt
                                (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω)
                                (instHSub.hSub 1 (instHDiv.hDiv α 2)))) →
                      (StrictMono fun x => (ProbabilityTheory.cdf η).toFun x) →
                        (∀ (x : Real),
                            MeasureTheory.TendstoInMeasure μ
                              (fun n ω => HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) Filter.atTop
                              fun x_1 => (ProbabilityTheory.cdf η).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar (instHDiv.hDiv α 2) n) μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              ProbabilityTheory.HasLaw ξ η ν →
                                Real.instLE.le 0 q →
                                  Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                    Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                      Filter.Tendsto
                                        (fun n =>
                                          MeasureTheory.Measure.instFunLike.coe μ
                                            (setOf fun ω =>
                                              HansenEconometrics.percentileCIEvent θ
                                                (instHAdd.hAdd (θhat n ω)
                                                  (instHDiv.hDiv
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (a n)))
                                                (instHAdd.hAdd (θhat n ω)
                                                  (instHDiv.hDiv
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                                    (a n)))))
                                        Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (3)

HansenEconometrics.bootstrapScalarCDF
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.percentileCIEvent

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:948

Equation 10.96 endpoints

The ideal bootstrap CDF is F_n^*(x)=P^*(Z_n^*\le x).

def HansenEconometrics.bootstrapVectorCDF

Conditional bootstrap CDF Gₙ(x) = P[Zₙ* ≤ x] for a finite-dimensional random vector.

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {k : Type u_6} →
      {mΩs : MeasurableSpace Ωs} →
        (Nat → Ω → MeasureTheory.Measure Ωs) → (Nat → Ω → Ωs → k → Real) → (k → Real) → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:22

def HansenEconometrics.bootstrapScalarCDF

Scalar conditional bootstrap CDF P*[Zₙ* ≤ x].

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {mΩs : MeasurableSpace Ωs} → (Nat → Ω → MeasureTheory.Measure Ωs) → (Nat → Ω → Ωs → Real) → Real → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:576

def HansenEconometrics.TendstoInBootstrapDistribution

Hansen Definition 10.2: convergence in bootstrap distribution.

The conditional CDF of Zstar n converges in ordinary probability, under the original-sample law μ, to the limit CDF at every continuity point of the limit CDF.

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {Ωlim : Type u_3} →
      {k : Type u_6} →
        {mΩ : MeasurableSpace Ω} →
          {mΩs : MeasurableSpace Ωs} →
            {mΩlim : MeasurableSpace Ωlim} →
              MeasureTheory.Measure Ω →
                (Nat → Ω → MeasureTheory.Measure Ωs) →
                  (Nat → Ω → Ωs → k → Real) → MeasureTheory.Measure Ωlim → (Ωlim → k → Real) → Prop

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:37

def HansenEconometrics.bootstrapVectorCDFIndexed

Indexed conditional bootstrap CDF for sample-size-dependent bootstrap spaces.

Formal statement

{Ω : Type u_1} →
  {k : Type u_6} →
    {Ωboot : Nat → Type u_7} →
      [inst : (n : Nat) → MeasurableSpace (Ωboot n)] →
        ((n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)) →
          ((n : Nat) → Ω → Ωboot n → k → Real) → (k → Real) → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:126

def HansenEconometrics.bootstrapScalarCDFIndexed

Scalar conditional bootstrap CDF for sample-size-dependent bootstrap spaces.

Formal statement

{Ω : Type u_1} →
  {Ωboot : Nat → Type u_7} →
    [inst : (n : Nat) → MeasurableSpace (Ωboot n)] →
      ((n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)) → ((n : Nat) → Ω → Ωboot n → Real) → Real → Nat → Ω → Real

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:1310

def HansenEconometrics.TendstoInBootstrapDistributionIndexed

Indexed-space Hansen Definition 10.2.

This is the distributional counterpart of TendstoInBootstrapProbabilityIndexed; it is needed for ordinary nonparametric bootstrap constructions whose resampling space varies with sample size, such as Fin (n + 1) → Fin (n + 1).

Formal statement

{Ω : Type u_1} →
  {Ωlim : Type u_3} →
    {k : Type u_6} →
      {mΩ : MeasurableSpace Ω} →
        {mΩlim : MeasurableSpace Ωlim} →
          {Ωboot : Nat → Type u_7} →
            [inst : (n : Nat) → MeasurableSpace (Ωboot n)] →
              MeasureTheory.Measure Ω →
                ((n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)) →
                  ((n : Nat) → Ω → Ωboot n → k → Real) → MeasureTheory.Measure Ωlim → (Ωlim → k → Real) → Prop

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:138

Equations 10.10/10.126 of 22 linked endpoints

One empirical bootstrap draw satisfies \mathbb E^*[Y_i^*]=\bar Y.

theorem HansenEconometrics.iIndepFun_uniformOn_fun_eval

Coordinate projections of the finite ordinary-bootstrap resampling space are independent under the uniform law.

This exposes the iid coordinate fact used implicitly in the finite covariance proofs and needed by the ordinary-bootstrap CLT route.

Formal statement

∀ {κ : Type u_8} {ι : Type u_9} [inst : MeasurableSpace ι] [Finite κ] [Finite ι] [Nonempty ι]
  [MeasurableSingletonClass ι] [MeasurableSingletonClass (κ → ι)],
  ProbabilityTheory.iIndepFun (fun t ωs => ωs t) (ProbabilityTheory.uniformOn Set.univ)

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:558

theorem HansenEconometrics.identDistrib_uniformOn_fun_eval

Each coordinate projection of the finite ordinary-bootstrap resampling space has the empirical uniform law.

Formal statement

∀ {κ : Type u_8} {ι : Type u_9} [inst : MeasurableSpace ι] [Finite κ] [Finite ι] [Nonempty ι]
  [MeasurableSingletonClass ι] [MeasurableSingletonClass (κ → ι)] (t : κ),
  ProbabilityTheory.IdentDistrib (fun ωs => ωs t) (fun i => i) (ProbabilityTheory.uniformOn Set.univ)
    (ProbabilityTheory.uniformOn Set.univ)

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:582

theorem HansenEconometrics.iIndepFun_uniformOn_fun_eval_comp

Transformed ordinary-bootstrap draws are independent coordinates under the finite uniform resampling law.

Formal statement

∀ {κ : Type u_8} {ι : Type u_9} {E : Type u_10} [inst : MeasurableSpace ι] [inst_1 : MeasurableSpace E] [Finite κ]
  [Finite ι] [Nonempty ι] [MeasurableSingletonClass ι] [MeasurableSingletonClass (κ → ι)] (g : ι → E),
  Measurable g → ProbabilityTheory.iIndepFun (fun t ωs => g (ωs t)) (ProbabilityTheory.uniformOn Set.univ)

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:614

theorem HansenEconometrics.identDistrib_uniformOn_fun_eval_comp

Transformed ordinary-bootstrap draws are identically distributed with the same transform under the empirical uniform law.

Formal statement

∀ {κ : Type u_8} {ι : Type u_9} {E : Type u_10} [inst : MeasurableSpace ι] [inst_1 : MeasurableSpace E] [Finite κ]
  [Finite ι] [Nonempty ι] [MeasurableSingletonClass ι] [MeasurableSingletonClass (κ → ι)] (g : ι → E),
  Measurable g →
    ∀ (t : κ),
      ProbabilityTheory.IdentDistrib (fun ωs => g (ωs t)) g (ProbabilityTheory.uniformOn Set.univ)
        (ProbabilityTheory.uniformOn Set.univ)

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:628

theorem HansenEconometrics.uniformOn_fun_univ_eq_pi_uniformOn_univ

Uniform law on finite resampling functions is the product of empirical uniform laws.

This is the measure-level iid structure behind ordinary finite nonparametric-bootstrap resampling.

Formal statement

∀ {κ : Type u_8} {ι : Type u_9} [inst : MeasurableSpace ι] [inst_1 : Fintype κ] [Finite ι] [Nonempty ι]
  [MeasurableSingletonClass ι] [MeasurableSingletonClass (κ → ι)],
  Eq (ProbabilityTheory.uniformOn Set.univ) (MeasureTheory.Measure.pi fun x => ProbabilityTheory.uniformOn Set.univ)

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:509

theorem HansenEconometrics.iIndepFun_uniformOn_fun_eval_sub_empiricalMean

Centered transformed ordinary-bootstrap draws are independent coordinates under the finite uniform resampling law.

This is the iid summand shape used in Hansen’s ordinary-bootstrap CLT proof: each draw is centered at the finite empirical mean.

Formal statement

∀ {κ : Type u_8} {ι : Type u_9} {E : Type u_10} [inst : MeasurableSpace ι] [inst_1 : MeasurableSpace E] [Finite κ]
  [inst_3 : Fintype ι] [Nonempty ι] [MeasurableSingletonClass ι] [MeasurableSingletonClass (κ → ι)]
  [inst_7 : NormedAddCommGroup E] [inst_8 : NormedSpace Real E] (Y : ι → E),
  ProbabilityTheory.iIndepFun (fun t ωs => instHSub.hSub (Y (ωs t)) (HansenEconometrics.empiricalMean Y))
    (ProbabilityTheory.uniformOn Set.univ)

Direct statement dependencies (1)

HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:646

Equation 10.116 of 7 linked endpoints

One empirical bootstrap draw satisfies \operatorname{Var}^*(Y_i^*)=n^{-1}\sum_{i=1}^n(Y_i-\bar Y)(Y_i-\bar Y)'.

theorem HansenEconometrics.covMat_uniformOn_univ_eq_card_inv_smul_sum_centered

Finite-dimensional empirical covariance identity for one bootstrap draw.

This is the matrix form of Hansen’s exact bootstrap covariance formula (10.11): under uniform resampling from a finite empirical support, the covariance matrix is the average outer product of deviations from the empirical mean.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {k : Type u_8}
  (Y : ι → k → Real),
  Eq (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) Y) fun a b =>
    instHSMul.hSMul (ENNReal.instInv.inv (Fintype.card ι).cast).toReal
      (Finset.univ.sum fun i =>
        instHMul.hMul
          (instHSub.hSub (Y i a)
            (instHSMul.hSMul (ENNReal.instInv.inv (Fintype.card ι).cast).toReal (Finset.univ.sum fun j => Y j a)))
          (instHSub.hSub (Y i b)
            (instHSMul.hSMul (ENNReal.instInv.inv (Fintype.card ι).cast).toReal (Finset.univ.sum fun j => Y j b))))

Direct statement dependencies (1)

HansenEconometrics.covMat

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:6147

theorem HansenEconometrics.variance_uniformOn_univ_eq_card_inv_smul_sum_sq_centered

Scalar empirical variance identity for one bootstrap draw.

This is the scalar version of Hansen’s exact bootstrap covariance formula (10.11): under uniform resampling from a finite empirical support, the variance is the average squared deviation from the empirical mean.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] (Y : ι → Real),
  Eq (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ))
    (instHSMul.hSMul (ENNReal.instInv.inv (Fintype.card ι).cast).toReal
      (Finset.univ.sum fun i =>
        instHPow.hPow
          (instHSub.hSub (Y i)
            (instHSMul.hSMul (ENNReal.instInv.inv (Fintype.card ι).cast).toReal (Finset.univ.sum fun j => Y j)))
          2))

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4570

theorem HansenEconometrics.taylor_charFun_centered_standardized_uniformOn_univ

Taylor expansion at zero for the standardized centered empirical one-draw characteristic function.

This packages Mathlib’s second-order characteristic-function Taylor lemma with the finite empirical mean-zero and variance identities. It is the local analytic input used by the characteristic-function proof of Hansen Theorem 10.4.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] [Nonempty ι]
  (Y : ι → Real),
  Ne (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)) 0 →
    Asymptotics.IsLittleO (nhds 0)
      (fun t =>
        instHSub.hSub
          (MeasureTheory.charFun
            (MeasureTheory.Measure.map
              (fun i =>
                instHDiv.hDiv (instHSub.hSub (Y i) (HansenEconometrics.empiricalMean Y))
                  (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)).sqrt)
              (ProbabilityTheory.uniformOn Set.univ))
            t)
          (instHSub.hSub 1 (instHDiv.hDiv (instHPow.hPow (Complex.ofReal t) 2) 2)))
      fun t => instHPow.hPow t 2

Direct statement dependencies (1)

HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4602

theorem HansenEconometrics.integral_sq_sub_empiricalMean_uniformOn_univ_eq_variance

Raw second moment of a centered empirical one-draw statistic.

Since the centered empirical one-draw statistic has exact mean zero, its raw second moment is the empirical one-draw variance.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun i =>
      instHPow.hPow (instHSub.hSub (Y i) (HansenEconometrics.empiricalMean Y)) 2)
    (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ))

Direct statement dependencies (1)

HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4588

theorem HansenEconometrics.charFun_centered_uniformOn_univ_inv_sqrt_succ_pow_tendsto

Gaussian characteristic-function power limit for a centered empirical one-draw law with fixed support.

This removes the standardization from charFun_centered_standardized_uniformOn_univ_inv_sqrt_succ_pow_tendsto: the fixed-support limit has the empirical one-draw variance as its Gaussian scale.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] [Nonempty ι]
  (Y : ι → Real),
  Ne (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)) 0 →
    ∀ (t : Real),
      Filter.Tendsto
        (fun n =>
          instHPow.hPow
            (MeasureTheory.charFun
              (MeasureTheory.Measure.map (fun i => instHSub.hSub (Y i) (HansenEconometrics.empiricalMean Y))
                (ProbabilityTheory.uniformOn Set.univ))
              (instHMul.hMul (Real.instInv.inv (instHAdd.hAdd n.cast 1).sqrt) t))
            n.succ)
        Filter.atTop
        (nhds
          (Complex.exp
            (instHDiv.hDiv
              (Complex.instNeg.neg
                (instHPow.hPow
                  (Complex.ofReal
                    (instHMul.hMul (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)).sqrt t))
                  2))
              2)))

Direct statement dependencies (1)

HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4793

theorem HansenEconometrics.charFun_centered_standardized_uniformOn_univ_inv_sqrt_succ_pow_tendsto

Gaussian characteristic-function power limit for a standardized centered empirical one-draw law with fixed support.

This is the fixed-support analytic bridge behind the characteristic-function proof of Hansen Theorem 10.4: once the empirical one-draw statistic is centered and divided by its empirical standard deviation, the n+1 iid bootstrap draws have the standard-normal characteristic-function limit.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] [Nonempty ι]
  (Y : ι → Real),
  Ne (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)) 0 →
    ∀ (t : Real),
      Filter.Tendsto
        (fun n =>
          instHPow.hPow
            (MeasureTheory.charFun
              (MeasureTheory.Measure.map
                (fun i =>
                  instHDiv.hDiv (instHSub.hSub (Y i) (HansenEconometrics.empiricalMean Y))
                    (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)).sqrt)
                (ProbabilityTheory.uniformOn Set.univ))
              (instHMul.hMul (Real.instInv.inv (instHAdd.hAdd n.cast 1).sqrt) t))
            n.succ)
        Filter.atTop (nhds (Complex.exp (instHDiv.hDiv (Complex.instNeg.neg (instHPow.hPow (Complex.ofReal t) 2)) 2)))

Direct statement dependencies (1)

HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4767

Equation 10.136 of 10 linked endpoints

The bootstrap mean satisfies \operatorname{Var}^*(\bar Y^*)=m^{-1}\operatorname{Var}^*(Y_i^*) for resample size m.

theorem HansenEconometrics.trace_covMat_resampleMean_eq_inv_card_mul

Trace of the finite-dimensional nonparametric-bootstrap sample-mean covariance matrix.

This is the trace form of Hansen equation (10.13).

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [MeasurableSingletonClass ι] {κ : Type u_8} {k : Type u_9}
  [inst_2 : Fintype κ] [Nonempty κ] [inst_4 : Fintype k] [Finite ι] [Nonempty ι] [MeasurableSingletonClass (κ → ι)]
  (Y : ι → k → Real),
  Eq
    (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) fun ωs a =>
        HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs a).trace
    (instHMul.hMul (Real.instInv.inv (Fintype.card κ).cast)
      (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) Y).trace)

Direct statement dependencies (2)

HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:6484

theorem HansenEconometrics.integral_sq_resampleMean_sub_empiricalMean_eq_inv_card_mul_variance

Centered second moment of the ordinary finite nonparametric bootstrap sample mean.

This is Hansen equation (10.13) in the exact second-moment form used by the bootstrap WLLN proof.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
          (HansenEconometrics.empiricalMean Y))
        2)
    (instHMul.hMul (Real.instInv.inv (Fintype.card κ).cast)
      (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)))

Direct statement dependencies (2)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:6066

theorem HansenEconometrics.covMat_empiricalBootstrapResampleMean_uniformOn_fun_eq_inv_card_smul

Covariance matrix of the ordinary finite nonparametric bootstrap sample mean.

This is the finite-dimensional form of Hansen equation (10.13): the conditional covariance matrix of the bootstrap sample mean is the empirical one-draw covariance matrix divided by the number of bootstrap draws.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [MeasurableSingletonClass ι] {κ : Type u_8} {k : Type u_9}
  [inst_2 : Fintype κ] [Nonempty κ] [inst_4 : Fintype k] [Finite ι] [Nonempty ι] [MeasurableSingletonClass (κ → ι)]
  (Y : ι → k → Real),
  Eq
    (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) fun ωs a =>
      HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs a)
    (instHSMul.hSMul (Real.instInv.inv (Fintype.card κ).cast)
      (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) Y))

Direct statement dependencies (2)

HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:6174

theorem HansenEconometrics.variance_empiricalBootstrapResampleMean_uniformOn_fun_eq_inv_card_mul

Scalar variance of the ordinary finite nonparametric bootstrap sample mean.

This is the scalar form of Hansen equation (10.13): the conditional variance of the bootstrap sample mean is the empirical one-draw variance divided by the number of bootstrap draws.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [MeasurableSingletonClass ι] {κ : Type u_8} [inst_2 : Fintype κ]
  [Nonempty κ] [Finite ι] [Nonempty ι] (Y : ι → Real),
  Eq
    (ProbabilityTheory.variance (fun ωs => HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
      (ProbabilityTheory.uniformOn Set.univ))
    (instHMul.hMul (Real.instInv.inv (Fintype.card κ).cast)
      (ProbabilityTheory.variance Y (ProbabilityTheory.uniformOn Set.univ)))

Direct statement dependencies (1)

HansenEconometrics.empiricalBootstrapResampleMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:5908

theorem HansenEconometrics.integral_norm_sq_resampleMean_sub_empiricalMean_eq_trace_covMat

Expected squared Euclidean norm of the centered nonparametric-bootstrap sample mean as a covariance trace.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] {κ : Type u_8} {k : Type u_9} [inst_2 : Fintype κ]
  [Nonempty κ] [inst_4 : Fintype k] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → EuclideanSpace Real k),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        ((PiLp.instNorm 2 fun x => Real).norm
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        2)
    (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) fun ωs a =>
        (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs).ofLp a).trace

Direct statement dependencies (3)

HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:6590

theorem HansenEconometrics.integral_sq_finSucc_resampleMean_sub_empiricalMean_le_marcinkiewicz

Scalar sample-size-indexed finite-resample second-moment bound in Hansen’s Theorem 10.2 scale.

For sample size n+1, the expected squared centered ordinary nonparametric-bootstrap mean is bounded by Hansen’s (n+1)^{-2} sum_{i<n+1} |Y_i|^2 Marcinkiewicz statistic.

Formal statement

∀ {Ω : Type u_1} (Y : Nat → Ω → Real) (n : Nat) (ω : Ω),
  Real.instLE.le
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
          (HansenEconometrics.empiricalMean fun i => Y i.val ω))
        2)
    (HansenEconometrics.marcinkiewiczWLLNStatisticNat Y 2 (instHAdd.hAdd n 1) ω)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.marcinkiewiczWLLNStatisticNat

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1718

Theorem 10.12 endpoints

If Z_n\xrightarrow{p}Z, then the nonresampled statistic also satisfies Z_n\xrightarrow{p^*}Z.

theorem HansenEconometrics.chapter10_bootstrap_convergence_in_probability_of_convergence_in_probability

Hansen Theorem 10.1, chapter-facing name.

Ordinary convergence in probability implies bootstrap convergence in probability when the sequence is non-random under the bootstrap resampling law.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : PseudoMetricSpace E] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Zseq : Nat → Ω → E} {Z : Ω → E},
      MeasureTheory.TendstoInMeasure μ Zseq Filter.atTop Z →
        HansenEconometrics.TendstoInBootstrapProbability μ Pstar (fun n ω x => Zseq n ω) Z

Direct statement dependencies (1)

HansenEconometrics.TendstoInBootstrapProbability

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:27

theorem HansenEconometrics.chapter10_indexed_bootstrap_convergence_in_probability_of_convergence_in_probability

Indexed Hansen Theorem 10.1, chapter-facing name.

Ordinary convergence in probability implies indexed bootstrap convergence in probability when the sequence is non-random under the bootstrap resampling law and the bootstrap sample space may vary with n.

Formal statement

∀ {Ω : Type u_1} {E : Type u_4} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : PseudoMetricSpace E]
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Zseq : Nat → Ω → E} {Z : Ω → E},
      MeasureTheory.TendstoInMeasure μ Zseq Filter.atTop Z →
        HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar (fun n ω x => Zseq n ω) Z

Direct statement dependencies (1)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1304

Theorem 10.26 of 26 linked endpoints

If Y_i are independent and uniformly integrable, then \bar Y^*-\bar Y\xrightarrow{p^*}0 and \bar Y^*\xrightarrow{p^*}\mu.

theorem HansenEconometrics.chapter10_bootstrap_wlln_level_from_centered

Hansen Theorem 10.2, second conclusion from the centered bootstrap WLLN.

Once the centered bootstrap sample mean satisfies Ybar* - Ybar ->p* 0, and the ordinary sample mean satisfies Ybar ->p μY, the bootstrap sample mean itself satisfies Ybar* ->p* μY. This is the bootstrap Slutsky/addition step used in the textbook proof after the centered WLLN is established by the conditional variance bound.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : SeminormedAddCommGroup E] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {YbarStar : Nat → Ω → Ωs → E} {Ybar : Nat → Ω → E} {μY : E},
      (HansenEconometrics.TendstoInBootstrapProbability μ Pstar
          (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) fun x => 0) →
        (MeasureTheory.TendstoInMeasure μ Ybar Filter.atTop fun x => μY) →
          HansenEconometrics.TendstoInBootstrapProbability μ Pstar YbarStar fun x => μY

Direct statement dependencies (1)

HansenEconometrics.TendstoInBootstrapProbability

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:113

theorem HansenEconometrics.chapter10_bootstrap_wlln_centered_of_tail_bound

Hansen Theorem 10.2, centered WLLN from the conditional tail bound.

This is the reusable form of the textbook proof: Markov’s inequality and the conditional variance calculation supply hle; the Marcinkiewicz/WLLN argument supplies hbound.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : SeminormedAddCommGroup E] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {YbarStar : Nat → Ω → Ωs → E} {Ybar : Nat → Ω → E} {bound : Real → Nat → Ω → Real},
  (∀ (η : Real),
      Real.instLT.lt 0 η → MeasureTheory.TendstoInMeasure μ (fun n ω => bound η n ω) Filter.atTop fun x => 0) →
    (∀ (η : Real),
        Real.instLT.lt 0 η →
          ∀ (n : Nat) (ω : Ω),
            Real.instLE.le
              (HansenEconometrics.bootstrapTailProb Pstar (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω))
                (fun x => 0) η n ω)
              (bound η n ω)) →
      HansenEconometrics.TendstoInBootstrapProbability μ Pstar
        (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) fun x => 0

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.bootstrapTailProb

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:88

theorem HansenEconometrics.chapter10_bootstrap_wlln_level_of_l2_eLpNorm_bound

Hansen Theorem 10.2, vector level WLLN from a bootstrap L² seminorm bound.

This packages the vector conditional-Markov centered result with the ordinary-sample WLLN for Ybar, giving the textbook level conclusion Ybar* ->p* μY.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E] [MeasureTheory.IsFiniteMeasure μ]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {YbarStar : Nat → Ω → Ωs → E} {Ybar : Nat → Ω → E} {μY : E} {u : Nat → Ω → Real},
      MeasureTheory.UniformIntegrable u 1 μ →
        (∀ (n : Nat) (ω : Ω),
            MeasureTheory.MemLp (fun ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) 2 (Pstar n ω)) →
          (∀ (η : Real),
              Real.instLT.lt 0 η →
                ∀ (n : Nat) (ω : Ω),
                  Real.instLE.le
                    (HansenEconometrics.bootstrapL2ENNTailBound Pstar
                      (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) η n ω)
                    (HansenEconometrics.bootstrapWLLNSecondMomentBound u η n ω)) →
            (MeasureTheory.TendstoInMeasure μ Ybar Filter.atTop fun x => μY) →
              HansenEconometrics.TendstoInBootstrapProbability μ Pstar YbarStar fun x => μY

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.bootstrapL2ENNTailBound
HansenEconometrics.bootstrapWLLNSecondMomentBound

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:936

theorem HansenEconometrics.chapter10_bootstrap_wlln_centered_of_l2_eLpNorm_bound

Hansen Theorem 10.2, vector centered WLLN from a bootstrap L² seminorm bound.

This is the vector-valued conditional Markov constructor. The remaining empirical-bootstrap specialization identifies the displayed L² seminorm through the finite empirical covariance/norm calculation.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E] [MeasureTheory.IsFiniteMeasure μ]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {YbarStar : Nat → Ω → Ωs → E} {Ybar : Nat → Ω → E} {u : Nat → Ω → Real},
  MeasureTheory.UniformIntegrable u 1 μ →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) 2 (Pstar n ω)) →
      (∀ (η : Real),
          Real.instLT.lt 0 η →
            ∀ (n : Nat) (ω : Ω),
              Real.instLE.le
                (HansenEconometrics.bootstrapL2ENNTailBound Pstar
                  (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) η n ω)
                (HansenEconometrics.bootstrapWLLNSecondMomentBound u η n ω)) →
        HansenEconometrics.TendstoInBootstrapProbability μ Pstar
          (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) fun x => 0

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.bootstrapL2ENNTailBound
HansenEconometrics.bootstrapWLLNSecondMomentBound

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:868

theorem HansenEconometrics.chapter10_bootstrap_wlln_level_of_second_moment_bound

Hansen Theorem 10.2, level WLLN from the textbook second-moment bound.

This packages the centered second-moment/Marcinkiewicz proof with the ordinary-sample WLLN for Ybar, giving the textbook conclusion Ybar* ->p* μY.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : SeminormedAddCommGroup E] [MeasureTheory.IsFiniteMeasure μ]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {YbarStar : Nat → Ω → Ωs → E} {Ybar : Nat → Ω → E} {μY : E} {u : Nat → Ω → Real},
      MeasureTheory.UniformIntegrable u 1 μ →
        (∀ (η : Real),
            Real.instLT.lt 0 η →
              ∀ (n : Nat) (ω : Ω),
                Real.instLE.le
                  (HansenEconometrics.bootstrapTailProb Pstar (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω))
                    (fun x => 0) η n ω)
                  (HansenEconometrics.bootstrapWLLNSecondMomentBound u η n ω)) →
          (MeasureTheory.TendstoInMeasure μ Ybar Filter.atTop fun x => μY) →
            HansenEconometrics.TendstoInBootstrapProbability μ Pstar YbarStar fun x => μY

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.bootstrapTailProb
HansenEconometrics.bootstrapWLLNSecondMomentBound

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:990

theorem HansenEconometrics.chapter10_bootstrap_wlln_centered_of_second_moment_bound

Hansen Theorem 10.2, centered WLLN from the textbook second-moment bound.

Once Chebyshev/Markov and the empirical variance calculation give the conditional tail bound hle, the Marcinkiewicz WLLN proves that bound is oₚ(1), hence the centered bootstrap sample mean converges in bootstrap probability to zero.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : SeminormedAddCommGroup E] [MeasureTheory.IsFiniteMeasure μ]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {YbarStar : Nat → Ω → Ωs → E} {Ybar : Nat → Ω → E} {u : Nat → Ω → Real},
  MeasureTheory.UniformIntegrable u 1 μ →
    (∀ (η : Real),
        Real.instLT.lt 0 η →
          ∀ (n : Nat) (ω : Ω),
            Real.instLE.le
              (HansenEconometrics.bootstrapTailProb Pstar (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω))
                (fun x => 0) η n ω)
              (HansenEconometrics.bootstrapWLLNSecondMomentBound u η n ω)) →
      HansenEconometrics.TendstoInBootstrapProbability μ Pstar
        (fun n ω ωs => instHSub.hSub (YbarStar n ω ωs) (Ybar n ω)) fun x => 0

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.bootstrapTailProb
HansenEconometrics.bootstrapWLLNSecondMomentBound

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:776

Theorem 10.32 endpoints

If Z_n^*\xrightarrow{p^*}c and g is continuous at c, then g(Z_n^*)\xrightarrow{p^*}g(c).

theorem HansenEconometrics.chapter10_bootstrap_continuous_mapping_probability

Hansen Theorem 10.3, chapter-facing name.

If Zₙ* ->p* c and g is continuous at c, then g(Zₙ) ->p g(c).

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {E : Type u_4} {F : Type u_5} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : PseudoMetricSpace E] [inst_1 : PseudoMetricSpace F]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Zstar : Nat → Ω → Ωs → E} {c : E} {g : E → F},
      (HansenEconometrics.TendstoInBootstrapProbability μ Pstar Zstar fun x => c) →
        ContinuousAt g c →
          HansenEconometrics.TendstoInBootstrapProbability μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) fun x => g c

Direct statement dependencies (1)

HansenEconometrics.TendstoInBootstrapProbability

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:40

theorem HansenEconometrics.chapter10_indexed_bootstrap_continuous_mapping_probability

Indexed-space Hansen Theorem 10.3, chapter-facing name.

If Zₙ* ->p* c on sample-size-dependent bootstrap spaces and g is continuous at c, then g(Zₙ) ->p g(c).

Formal statement

∀ {Ω : Type u_1} {E : Type u_4} {F : Type u_5} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω}
  {Ωboot : Nat → Type u_7} [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : PseudoMetricSpace E]
  [inst_2 : PseudoMetricSpace F] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Zstar : (n : Nat) → Ω → Ωboot n → E} {c : E} {g : E → F},
      (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar Zstar fun x => c) →
        ContinuousAt g c →
          HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) fun x => g c

Direct statement dependencies (1)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1586

Definition 10.22 endpoints

Bootstrap convergence in distribution means P^*(Z_n^*\le x)\xrightarrow{p}P(Z\le x) at every continuity point x.

def HansenEconometrics.TendstoInBootstrapDistribution

Hansen Definition 10.2: convergence in bootstrap distribution.

The conditional CDF of Zstar n converges in ordinary probability, under the original-sample law μ, to the limit CDF at every continuity point of the limit CDF.

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {Ωlim : Type u_3} →
      {k : Type u_6} →
        {mΩ : MeasurableSpace Ω} →
          {mΩs : MeasurableSpace Ωs} →
            {mΩlim : MeasurableSpace Ωlim} →
              MeasureTheory.Measure Ω →
                (Nat → Ω → MeasureTheory.Measure Ωs) →
                  (Nat → Ω → Ωs → k → Real) → MeasureTheory.Measure Ωlim → (Ωlim → k → Real) → Prop

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:37

def HansenEconometrics.TendstoInBootstrapDistributionIndexed

Indexed-space Hansen Definition 10.2.

Formal statement

{Ω : Type u_1} →
  {Ωlim : Type u_3} →
    {k : Type u_6} →
      {mΩ : MeasurableSpace Ω} →
        {mΩlim : MeasurableSpace Ωlim} →
          {Ωboot : Nat → Type u_7} →
            [inst : (n : Nat) → MeasurableSpace (Ωboot n)] →
              MeasureTheory.Measure Ω →
                ((n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)) →
                  ((n : Nat) → Ω → Ωboot n → k → Real) → MeasureTheory.Measure Ωlim → (Ωlim → k → Real) → Prop

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:138

Theorem 10.46 of 49 linked endpoints

If \mathbb E\lVert Y\rVert^2\lt \infty and \Sigma=\operatorname{Var}(Y)\succ0, then \sqrt n(\bar Y^*-\bar Y)\xrightarrow{d^*}N(0,\Sigma).

theorem HansenEconometrics.chapter10_bootstrap_clt_gaussian_of_tendsto_cdf

Hansen Theorem 10.4, Gaussian bootstrap CLT CDF wrapper.

If the conditional CDFs of a normalized bootstrap statistic converge in probability to the CDF of N(0, Σ) at every continuity point, then the statistic converges in bootstrap distribution to that Gaussian law. Later ordinary-bootstrap wrappers discharge this premise through pathwise weak convergence, scalar projection/characteristic-function routes, and the iid covariance-tail route.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : Fintype k] [inst_1 : DecidableEq k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → k → Real} {S : Matrix k k Real},
  (∀ (x : k → Real),
      ContinuousAt
          (fun y => HansenEconometrics.vectorCDF (ProbabilityTheory.multivariateGaussian 0 S) (fun z => z.ofLp) y) x →
        MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.bootstrapVectorCDF Pstar Zstar x n ω)
          Filter.atTop fun x_1 =>
          HansenEconometrics.vectorCDF (ProbabilityTheory.multivariateGaussian 0 S) (fun z => z.ofLp) x) →
    HansenEconometrics.TendstoInBootstrapDistribution μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
      fun z => z.ofLp

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapVectorCDF
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:234

theorem HansenEconometrics.chapter10_bootstrap_clt_gaussian_of_weakDistribution

Hansen Theorem 10.4, Gaussian bootstrap CLT from weak bootstrap convergence.

If a normalized bootstrap statistic converges weakly, in the bounded-continuous-test-function bootstrap sense, to N(0, S), then the coordinate-CDF version of Hansen Definition 10.2 follows at all continuity points whose lower-orthant frontiers are null under that Gaussian law.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : Fintype k] [inst_1 : DecidableEq k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → k → Real} {S : Matrix k k Real},
  (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
      fun z => z.ofLp) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
      (∀ (n : Nat) (ω : Ω), Measurable (Zstar n ω)) →
        (∀ (x : k → Real),
            ContinuousAt
                (fun y => HansenEconometrics.vectorCDF (ProbabilityTheory.multivariateGaussian 0 S) (fun z => z.ofLp) y)
                x →
              Eq
                (MeasureTheory.Measure.instFunLike.coe
                  (MeasureTheory.Measure.map (fun z => z.ofLp) (ProbabilityTheory.multivariateGaussian 0 S))
                  (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                0) →
          HansenEconometrics.TendstoInBootstrapDistribution μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
            fun z => z.ofLp

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:2863

theorem HansenEconometrics.chapter10_bootstrap_clt_gaussian_of_weakDistribution_posDef

Hansen Theorem 10.4 Gaussian bootstrap CLT from weak bootstrap convergence with positive definite covariance.

This is the theorem-facing finite-dimensional route: positive definiteness of Σ makes every Gaussian lower-orthant frontier null, so a bounded-continuous bootstrap weak limit to N(0,Σ) directly yields Hansen Definition 10.2.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : Fintype k] [inst_1 : DecidableEq k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → k → Real} {S : Matrix k k Real},
  S.PosDef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
        fun z => z.ofLp) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (Zstar n ω)) →
          HansenEconometrics.TendstoInBootstrapDistribution μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
            fun z => z.ofLp

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:2903

theorem HansenEconometrics.chapter10_bootstrap_clt_gaussian_of_ae_tendsto_integrals

Hansen Theorem 10.4 Gaussian bootstrap CLT from pathwise conditional bounded-continuous integral convergence.

This bridge converts sample-path conditional weak convergence, supplied as bounded-continuous test-function integral convergence for almost every original sample path, into Hansen Definition 10.2.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : Fintype k] [inst_1 : DecidableEq k] [MeasureTheory.IsFiniteMeasure μ]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real} {S : Matrix k k Real},
  (∀ (f : BoundedContinuousFunction (k → Real) Real) (n : Nat),
      MeasureTheory.AEStronglyMeasurable
        (fun ω => HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar Zstar f n ω) μ) →
    (∀ (f : BoundedContinuousFunction (k → Real) Real),
        Filter.Eventually
          (fun ω =>
            Filter.Tendsto (fun n => HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar Zstar f n ω)
              Filter.atTop
              (nhds
                (MeasureTheory.integral (ProbabilityTheory.multivariateGaussian 0 S) fun z =>
                  BoundedContinuousFunction.instFunLike.coe f z.ofLp)))
          (MeasureTheory.ae μ)) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (Zstar n ω)) →
          (∀ (x : k → Real),
              ContinuousAt
                  (fun y =>
                    HansenEconometrics.vectorCDF (ProbabilityTheory.multivariateGaussian 0 S) (fun z => z.ofLp) y)
                  x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe
                    (MeasureTheory.Measure.map (fun z => z.ofLp) (ProbabilityTheory.multivariateGaussian 0 S))
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
              fun z => z.ofLp

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapBoundedContinuousIntegral
HansenEconometrics.coordinateLE
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:2929

theorem HansenEconometrics.chapter10_indexed_bootstrap_clt_gaussian_of_tendsto_cdf

Indexed-space Hansen Theorem 10.4 Gaussian bootstrap CLT CDF wrapper.

This is the sample-size-dependent counterpart of chapter10_bootstrap_clt_gaussian_of_tendsto_cdf, for ordinary finite nonparametric bootstrap constructions whose resampling type varies with n.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : Fintype k] [inst_2 : DecidableEq k]
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {S : Matrix k k Real},
  (∀ (x : k → Real),
      ContinuousAt
          (fun y => HansenEconometrics.vectorCDF (ProbabilityTheory.multivariateGaussian 0 S) (fun z => z.ofLp) y) x →
        MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.bootstrapVectorCDFIndexed Pstar Zstar x n ω)
          Filter.atTop fun x_1 =>
          HansenEconometrics.vectorCDF (ProbabilityTheory.multivariateGaussian 0 S) (fun z => z.ofLp) x) →
    HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar Zstar (ProbabilityTheory.multivariateGaussian 0 S)
      fun z => z.ofLp

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.bootstrapVectorCDFIndexed
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:262

theorem HansenEconometrics.covMat_normalized_finSucc_resampleMean_sub_empiricalMean_eq

Matrix Fin (n+1) CLT-scale covariance identity for the ordinary nonparametric bootstrap.

This is the sample-size-indexed finite-resample covariance normalization used by the concrete Theorem 10.4 path.

Formal statement

∀ {Ω : Type u_1} {k : Type u_8} [inst : Fintype k] (Y : Nat → Ω → k → Real) (n : Nat) (ω : Ω),
  Eq
    (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) fun ωs a =>
      instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
        (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs a)
          (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
    (HansenEconometrics.covMat (ProbabilityTheory.uniformOn Set.univ) fun i a => Y i.val ω a)

Direct statement dependencies (3)

HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:2227

Theorem 10.56 of 25 linked endpoints

If Z_n^*\xrightarrow{d^*}Z and g is continuous on the support of Z, then g(Z_n^*)\xrightarrow{d^*}g(Z).

theorem HansenEconometrics.chapter10_bootstrap_continuous_mapping_event_probability

Hansen Theorem 10.5, globally continuous event-probability face.

After a continuous transformation g, bounded-continuous lower/upper sandwiches for an event A imply convergence in probability of the conditional bootstrap event probabilities. The remaining textbook discontinuity-set-null case supplies these sandwiches from the null-boundary hypothesis.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {E : Type u_4} {F : Type u_5} {mΩ : MeasurableSpace Ω}
  {mΩs : MeasurableSpace Ωs} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : TopologicalSpace E] [inst_1 : TopologicalSpace F]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → E} {Z : Ωlim → E} {g : E → F} {A : Set F}
  {c : Real},
  HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
    Continuous g →
      (∀ (ε : Real),
          Real.instLT.lt 0 ε →
            Exists fun lower =>
              Exists fun upper =>
                And
                  (Real.instLE.le
                    (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe lower (g (Z ωlim)))
                    c)
                  (And
                    (Real.instLE.le c
                      (MeasureTheory.integral ν fun ωlim =>
                        BoundedContinuousFunction.instFunLike.coe upper (g (Z ωlim))))
                    (And
                      (Real.instLE.le
                        (instHSub.hSub
                          (MeasureTheory.integral ν fun ωlim =>
                            BoundedContinuousFunction.instFunLike.coe upper (g (Z ωlim)))
                          (MeasureTheory.integral ν fun ωlim =>
                            BoundedContinuousFunction.instFunLike.coe lower (g (Z ωlim))))
                        ε)
                      (And
                        (∀ (n : Nat) (ω : Ω),
                          Real.instLE.le
                            (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar
                              (fun n ω ωs => g (Zstar n ω ωs)) lower n ω)
                            (HansenEconometrics.bootstrapEventProbability Pstar (fun n ω ωs => g (Zstar n ω ωs)) A n ω))
                        (∀ (n : Nat) (ω : Ω),
                          Real.instLE.le
                            (HansenEconometrics.bootstrapEventProbability Pstar (fun n ω ωs => g (Zstar n ω ωs)) A n ω)
                            (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar
                              (fun n ω ωs => g (Zstar n ω ωs)) upper n ω)))))) →
        MeasureTheory.TendstoInMeasure μ
          (HansenEconometrics.bootstrapEventProbability Pstar (fun n ω ωs => g (Zstar n ω ωs)) A) Filter.atTop fun x =>
          c

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapBoundedContinuousIntegral
HansenEconometrics.bootstrapEventProbability

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:5253

theorem HansenEconometrics.chapter10_bootstrap_mapping_distribution_of_sandwich_null_frontiers

Hansen Theorem 10.5, sandwich-mapped finite-dimensional CDF face.

Bounded-continuous sandwich approximations give mapped weak convergence; null frontiers for transformed lower orthants then recover Hansen Definition 10.2.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {E : Type u_4} {k : Type u_6} {mΩ : MeasurableSpace Ω}
  {mΩs : MeasurableSpace Ωs} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} [inst : TopologicalSpace E]
  [Finite k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → E} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsProbabilityMeasure ν] {Z : Ωlim → E} {g : E → k → Real},
  HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
    (∀ (f : BoundedContinuousFunction (k → Real) Real) (ε : Real),
        Real.instLT.lt 0 ε →
          Exists fun lower =>
            Exists fun upper =>
              And
                (Real.instLE.le
                  (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe lower (Z ωlim))
                  (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe f (g (Z ωlim))))
                (And
                  (Real.instLE.le
                    (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe f (g (Z ωlim)))
                    (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe upper (Z ωlim)))
                  (And
                    (Real.instLE.le
                      (instHSub.hSub
                        (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe upper (Z ωlim))
                        (MeasureTheory.integral ν fun ωlim => BoundedContinuousFunction.instFunLike.coe lower (Z ωlim)))
                      ε)
                    (And
                      (∀ (n : Nat) (ω : Ω),
                        Real.instLE.le (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar Zstar lower n ω)
                          (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar (fun n ω ωs => g (Zstar n ω ωs))
                            f n ω))
                      (∀ (n : Nat) (ω : Ω),
                        Real.instLE.le
                          (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar (fun n ω ωs => g (Zstar n ω ωs))
                            f n ω)
                          (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar Zstar upper n ω)))))) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable fun ωs => g (Zstar n ω ωs)) →
          AEMeasurable (fun ωlim => g (Z ωlim)) ν →
            (∀ (x : k → Real),
                ContinuousAt (fun y => HansenEconometrics.vectorCDF ν (fun ωlim => g (Z ωlim)) y) x →
                  Eq
                    (MeasureTheory.Measure.instFunLike.coe (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν)
                      (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                    0) →
              HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) ν fun ωlim =>
                g (Z ωlim)

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapBoundedContinuousIntegral
HansenEconometrics.coordinateLE
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:5492

theorem HansenEconometrics.chapter10_bootstrap_continuous_mapping_distribution_of_null_frontiers

Hansen Theorem 10.5, globally continuous finite-dimensional CDF face.

After a continuous transformation into k → ℝ, the bounded-continuous bootstrap CMT implies Hansen Definition 10.2 whenever the transformed limiting lower orthants have null frontier at the relevant continuity points. The measurability premises are stated for the transformed statistics so this wrapper can also be used when measurability is supplied by a model-specific layer.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {E : Type u_4} {k : Type u_6} {mΩ : MeasurableSpace Ω}
  {mΩs : MeasurableSpace Ωs} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} [inst : TopologicalSpace E]
  [Finite k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → E} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsProbabilityMeasure ν] {Z : Ωlim → E} {g : E → k → Real},
  HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
    Continuous g →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable fun ωs => g (Zstar n ω ωs)) →
          AEMeasurable (fun ωlim => g (Z ωlim)) ν →
            (∀ (x : k → Real),
                ContinuousAt (fun y => HansenEconometrics.vectorCDF ν (fun ωlim => g (Z ωlim)) y) x →
                  Eq
                    (MeasureTheory.Measure.instFunLike.coe (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν)
                      (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                    0) →
              HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) ν fun ωlim =>
                g (Z ωlim)

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:5026

theorem HansenEconometrics.chapter10_bootstrap_ae_continuous_mapping_distribution_of_null_frontiers

Hansen Theorem 10.5, a.e.-continuous finite-dimensional CDF face.

This is the Definition 10.2 counterpart of chapter10_bootstrap_ae_continuous_mapping_event_probability_of_null_frontier. The a.e.-continuity package records Hansen’s mapping premise, while the transformed weak-convergence hypothesis is explicit; null frontiers for transformed lower orthants then give conditional-CDF convergence.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {E : Type u_4} {k : Type u_6} {mΩ : MeasurableSpace Ω}
  {mΩs : MeasurableSpace Ωs} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} [inst : TopologicalSpace E]
  [Finite k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → E} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsProbabilityMeasure ν] {Z : Ωlim → E} {g : E → k → Real},
  HansenEconometrics.BootstrapAEMappingPremise ν Z g →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) ν fun ωlim =>
        g (Z ωlim)) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable fun ωs => g (Zstar n ω ωs)) →
          (∀ (x : k → Real),
              ContinuousAt (fun y => HansenEconometrics.vectorCDF ν (fun ωlim => g (Z ωlim)) y) x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν)
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) ν fun ωlim =>
              g (Z ωlim)

Direct statement dependencies (5)

HansenEconometrics.BootstrapAEMappingPremise
HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:5724

theorem HansenEconometrics.chapter10_bootstrap_continuous_mapping_event_probability_of_null_frontier

Hansen Theorem 10.5, globally continuous event-probability face with a null-frontier event.

If Zₙ* ->d* Z, g is continuous, the conditional bootstrap laws are finite, and the transformed limit law gives zero mass to the frontier of A, then the conditional probabilities P*[g(Zₙ*) ∈ A] converge in probability to P[g(Z) ∈ A].

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {E : Type u_4} {F : Type u_5} {mΩ : MeasurableSpace Ω}
  {mΩs : MeasurableSpace Ωs} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} [inst : TopologicalSpace E]
  [inst_1 : MeasurableSpace E] [OpensMeasurableSpace E] [inst_3 : PseudoEMetricSpace F] [inst_4 : MeasurableSpace F]
  [BorelSpace F] [OpensMeasurableSpace F] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → E}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsProbabilityMeasure ν] {Z : Ωlim → E} {g : E → F} {A : Set F},
  HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
    Continuous g →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (Zstar n ω)) →
          AEMeasurable Z ν →
            MeasurableSet A →
              Eq
                  (MeasureTheory.Measure.instFunLike.coe (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν)
                    (frontier A))
                  0 →
                MeasureTheory.TendstoInMeasure μ
                  (HansenEconometrics.bootstrapEventProbability Pstar (fun n ω ωs => g (Zstar n ω ωs)) A) Filter.atTop
                  fun x => (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν).real A

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapEventProbability

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:5326

theorem HansenEconometrics.chapter10_bootstrap_ae_continuous_mapping_event_probability_of_null_frontier

Hansen Theorem 10.5, a.e.-continuous transformed-event face.

The a.e.-continuity package records the textbook mapping premise, while the transformed weak-convergence hypothesis is explicit. This gives the null-frontier event-probability conclusion without assuming that g is globally continuous.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {E : Type u_4} {F : Type u_5} {mΩ : MeasurableSpace Ω}
  {mΩs : MeasurableSpace Ωs} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} [inst : TopologicalSpace E]
  [inst_1 : PseudoEMetricSpace F] [inst_2 : MeasurableSpace F] [BorelSpace F] [OpensMeasurableSpace F]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → E} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsProbabilityMeasure ν] {Z : Ωlim → E} {g : E → F} {A : Set F},
  HansenEconometrics.BootstrapAEMappingPremise ν Z g →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar (fun n ω ωs => g (Zstar n ω ωs)) ν fun ωlim =>
        g (Z ωlim)) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable fun ωs => g (Zstar n ω ωs)) →
          MeasurableSet A →
            Eq
                (MeasureTheory.Measure.instFunLike.coe (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν)
                  (frontier A))
                0 →
              MeasureTheory.TendstoInMeasure μ
                (HansenEconometrics.bootstrapEventProbability Pstar (fun n ω ωs => g (Zstar n ω ωs)) A) Filter.atTop
                fun x => (MeasureTheory.Measure.map (fun ωlim => g (Z ωlim)) ν).real A

Direct statement dependencies (3)

HansenEconometrics.BootstrapAEMappingPremise
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapEventProbability

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:5664

Theorem 10.66 of 20 linked endpoints

If a_n(Z_n^*-Z_n)\xrightarrow{d^*}Z and g is differentiable, then a_n(g(Z_n^*)-g(Z_n))\xrightarrow{d^*}G'Z.

theorem HansenEconometrics.chapter10_bootstrap_delta_method_gaussian

Hansen Theorem 10.6, Gaussian covariance specialization.

If the bootstrap linearized statistic converges weakly to N(0, V), then its matrix-derivative image converges weakly to N(0, G V G’), matching the textbook covariance formula.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {V : Matrix d d Real} (G : Matrix r d Real),
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
        (fun n ω ωs => ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))
        (ProbabilityTheory.multivariateGaussian 0
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
            G.transpose))
        fun z => z

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:59

theorem HansenEconometrics.chapter10_bootstrap_delta_method_gaussian_event_probability

Hansen Theorem 10.6, Gaussian event-probability specialization.

The matrix-linear bootstrap Delta method implies convergence of conditional bootstrap probabilities for events whose transformed Gaussian limit-law frontier has zero mass.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {V : Matrix d d Real} (G : Matrix r d Real) {A : Set (EuclideanSpace Real r)},
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
          MeasurableSet A →
            Eq
                (MeasureTheory.Measure.instFunLike.coe
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                  (frontier A))
                0 →
              MeasureTheory.TendstoInMeasure μ
                (HansenEconometrics.bootstrapEventProbability Pstar
                  (fun n ω ωs =>
                    ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))
                  A)
                Filter.atTop fun x =>
                (ProbabilityTheory.multivariateGaussian 0
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)).real
                  A

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapEventProbability
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:101

theorem HansenEconometrics.chapter10_bootstrap_delta_method_gaussian_distribution_posDef

Hansen Theorem 10.6, Gaussian CDF specialization with positive definite transformed covariance.

When G V G’ is positive definite, the Gaussian lower-orthant null-frontier premise in chapter10_bootstrap_delta_method_gaussian_distribution is automatic.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {V : Matrix d d Real} (G : Matrix r d Real),
  V.PosSemidef →
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose).PosDef →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
              fun z => z.ofLp

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:230

theorem HansenEconometrics.chapter10_bootstrap_delta_method_gaussian_distribution_of_compact_range_remainder_bound

Hansen Theorem 10.6 Gaussian Delta-method CDF wrapper from a compact-range pointwise remainder bound.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {R : Nat → Ω → Ωs → Real} {V : Matrix d d Real}
  (G : Matrix r d Real) {K : Set (EuclideanSpace Real r)},
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      IsCompact K →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (thetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                  Set.instMembership.mem K
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
                (∀ (n : Nat) (ω : Ω) (ωs : Ωs), Set.instMembership.mem K (thetaStar n ω ωs)) →
                  (∀ (δ : Real),
                      Real.instLT.lt 0 δ →
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω => (Pstar n ω).real (setOf fun ωs => Real.instLE.le δ (R n ω ωs))) Filter.atTop
                          fun x => 0) →
                    (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                        Real.instLE.le
                          ((PiLp.instDist 2 fun x => Real).dist (thetaStar n ω ωs)
                            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                              (Tstar n ω ωs)))
                          (R n ω ωs)) →
                      (∀ (x : r → Real),
                          ContinuousAt
                              (fun y =>
                                HansenEconometrics.vectorCDF
                                  (ProbabilityTheory.multivariateGaussian 0
                                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                                  (fun z => z.ofLp) y)
                              x →
                            Eq
                              (MeasureTheory.Measure.instFunLike.coe
                                (MeasureTheory.Measure.map (fun z => z.ofLp)
                                  (ProbabilityTheory.multivariateGaussian 0
                                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)))
                                (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                              0) →
                        HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
                          (fun n ω ωs => (thetaStar n ω ωs).ofLp)
                          (ProbabilityTheory.multivariateGaussian 0
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                          fun z => z.ofLp

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:1041

theorem HansenEconometrics.chapter10_bootstrap_delta_method_gaussian_distribution_of_compact_range_remainder_posDef

Hansen Theorem 10.6 Gaussian Delta-method CDF wrapper from a compact-range pointwise remainder bound with positive definite transformed covariance.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {R : Nat → Ω → Ωs → Real} {V : Matrix d d Real}
  (G : Matrix r d Real) {K : Set (EuclideanSpace Real r)},
  V.PosSemidef →
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose).PosDef →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        IsCompact K →
          (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (thetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                    Set.instMembership.mem K
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                        (Tstar n ω ωs))) →
                  (∀ (n : Nat) (ω : Ω) (ωs : Ωs), Set.instMembership.mem K (thetaStar n ω ωs)) →
                    (∀ (δ : Real),
                        Real.instLT.lt 0 δ →
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω => (Pstar n ω).real (setOf fun ωs => Real.instLE.le δ (R n ω ωs))) Filter.atTop
                            fun x => 0) →
                      (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                          Real.instLE.le
                            ((PiLp.instDist 2 fun x => Real).dist (thetaStar n ω ωs)
                              (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)))
                            (R n ω ωs)) →
                        HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
                          (fun n ω ωs => (thetaStar n ω ωs).ofLp)
                          (ProbabilityTheory.multivariateGaussian 0
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                          fun z => z.ofLp

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:1095

theorem HansenEconometrics.chapter10_bootstrap_delta_method_gaussian_event_probability_of_compact_range_remainder_bound

Hansen Theorem 10.6 Gaussian Delta-method event-probability wrapper from a compact-range pointwise remainder bound.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {R : Nat → Ω → Ωs → Real} {V : Matrix d d Real}
  (G : Matrix r d Real) {A K : Set (EuclideanSpace Real r)},
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      IsCompact K →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (thetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                  Set.instMembership.mem K
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
                (∀ (n : Nat) (ω : Ω) (ωs : Ωs), Set.instMembership.mem K (thetaStar n ω ωs)) →
                  (∀ (δ : Real),
                      Real.instLT.lt 0 δ →
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω => (Pstar n ω).real (setOf fun ωs => Real.instLE.le δ (R n ω ωs))) Filter.atTop
                          fun x => 0) →
                    (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                        Real.instLE.le
                          ((PiLp.instDist 2 fun x => Real).dist (thetaStar n ω ωs)
                            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                              (Tstar n ω ωs)))
                          (R n ω ωs)) →
                      MeasurableSet A →
                        Eq
                            (MeasureTheory.Measure.instFunLike.coe
                              (ProbabilityTheory.multivariateGaussian 0
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                              (frontier A))
                            0 →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.bootstrapEventProbability Pstar thetaStar A) Filter.atTop fun x =>
                            (ProbabilityTheory.multivariateGaussian 0
                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)).real
                              A

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapEventProbability
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:992

Theorem 10.76 of 24 linked endpoints

In the smooth-function model, p_n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}N(0,V_\theta), the same limit as p_n(\hat\theta-\theta).

theorem HansenEconometrics.chapter10_bootstrap_smooth_function_gaussian_of_linearization

Hansen Theorem 10.7, smooth-function Gaussian bootstrap wrapper.

If the bootstrap moment/statistic has Gaussian bootstrap limit N(0,V) and the centered bootstrap smooth-function estimator has already been reduced to the derivative-linearized statistic G T, then it has bootstrap limit N(0, G V G’). The remaining theorem-specific work is the nonlinear differentiability/oₚ constructor that supplies this linearization for Hansen’s smooth-function estimator.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real),
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
          Eq (thetaStar n ω ωs)
            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar thetaStar
          (ProbabilityTheory.multivariateGaussian 0
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
              G.transpose))
          fun z => z

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:2239

theorem HansenEconometrics.chapter10_bootstrap_smooth_function_gaussian_of_integral_linearization

Hansen Theorem 10.7 smooth-function Gaussian wrapper from bounded-continuous test-function linearization.

This is the weak-distribution transfer form used when differentiability gives an oₚ* nonlinear remainder strong enough to make every bounded-continuous test-function conditional expectation agree asymptotically with the derivative-linearized statistic.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real),
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      (∀ (f : BoundedContinuousFunction (EuclideanSpace Real r) Real),
          MeasureTheory.TendstoInMeasure μ
            (fun n ω =>
              instHSub.hSub (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar thetaStar f n ω)
                (HansenEconometrics.bootstrapBoundedContinuousIntegral Pstar
                  (fun n ω ωs =>
                    ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))
                  f n ω))
            Filter.atTop fun x => 0) →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar thetaStar
          (ProbabilityTheory.multivariateGaussian 0
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
              G.transpose))
          fun z => z

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapBoundedContinuousIntegral
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:2407

theorem HansenEconometrics.chapter10_bootstrap_smooth_function_gaussian_distribution_of_linearization

Hansen Theorem 10.7, smooth-function Gaussian CDF wrapper from exact linearization.

This is the Hansen Definition 10.2 face of chapter10_bootstrap_smooth_function_gaussian_of_linearization: the matrix-linear Gaussian Delta-method CDF theorem supplies the coordinate-CDF conclusion, and the supplied pointwise linearization identifies the smooth-function bootstrap statistic with the derivative-linearized statistic.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real),
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
          (∀ (x : r → Real),
              ContinuousAt
                  (fun y =>
                    HansenEconometrics.vectorCDF
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                      (fun z => z.ofLp) y)
                  x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe
                    (MeasureTheory.Measure.map (fun z => z.ofLp)
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)))
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs => (thetaStar n ω ωs).ofLp)
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                fun z => z.ofLp

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:2324

theorem HansenEconometrics.chapter10_bootstrap_smooth_function_gaussian_of_compact_range_remainder_bound

Hansen Theorem 10.7 smooth-function Gaussian wrapper from a compact-range remainder bound.

This is the o_p* remainder form of the compact-range route: a smooth-model Taylor argument may supply the pointwise distance bound, while the bootstrap tail premise states that the bound is negligible in conditional bootstrap probability.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {R : Nat → Ω → Ωs → Real} {V : Matrix d d Real}
  (G : Matrix r d Real) {K : Set (EuclideanSpace Real r)},
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      IsCompact K →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (thetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                  Set.instMembership.mem K
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
                (∀ (n : Nat) (ω : Ω) (ωs : Ωs), Set.instMembership.mem K (thetaStar n ω ωs)) →
                  (∀ (δ : Real),
                      Real.instLT.lt 0 δ →
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω => (Pstar n ω).real (setOf fun ωs => Real.instLE.le δ (R n ω ωs))) Filter.atTop
                          fun x => 0) →
                    (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                        Real.instLE.le
                          ((PiLp.instDist 2 fun x => Real).dist (thetaStar n ω ωs)
                            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                              (Tstar n ω ωs)))
                          (R n ω ωs)) →
                      HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar thetaStar
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                        fun z => z

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:2838

theorem HansenEconometrics.chapter10_bootstrap_smooth_function_gaussian_event_probability_of_linearization

Hansen Theorem 10.7 smooth-function Gaussian event-probability wrapper from exact derivative linearization.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  {A : Set (EuclideanSpace Real r)},
  V.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
        fun z => z) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
          MeasurableSet A →
            Eq
                (MeasureTheory.Measure.instFunLike.coe
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                  (frontier A))
                0 →
              (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                  Eq (thetaStar n ω ωs)
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
                MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapEventProbability Pstar thetaStar A)
                  Filter.atTop fun x =>
                  (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)).real
                    A

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapEventProbability
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:2262

theorem HansenEconometrics.chapter10_bootstrap_smooth_function_gaussian_distribution_of_linearization_posDef

Hansen Theorem 10.7 smooth-function Gaussian CDF wrapper from exact linearization with positive definite transformed covariance.

This is the positive-definite covariance specialization of chapter10_bootstrap_smooth_function_gaussian_distribution_of_linearization; the Gaussian lower-orthant null-frontier premise is discharged automatically.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real),
  V.PosSemidef →
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose).PosDef →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (Tstar n ω)) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs => (thetaStar n ω ωs).ofLp)
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                fun z => z.ofLp

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/DeltaMethod.lean:2373

Theorem 10.86 of 14 linked endpoints

The smooth plug-in covariance satisfies V_\theta^*=G^{*'}V^*G^*\xrightarrow{p^*}V_\theta.

theorem HansenEconometrics.chapter10_bootstrap_smooth_variance_consistency

Hansen Theorem 10.8, plug-in covariance continuous-mapping bridge.

If the bootstrap Jacobian/covariance pair converges in bootstrap probability to the population pair, then the smooth-function covariance plug-in Gstarᵀ Vstar Gstar converges in bootstrap probability to Gᵀ V G. The concrete Theorem 10.8 constructors provide the joint bootstrap-probability premise from the smooth-function model and the bootstrap WLLN/CLT layer.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Gstar : Nat → Ω → Ωs → Matrix d r Real} {Vstar : Nat → Ω → Ωs → Matrix d d Real} {G : Matrix d r Real}
      {V : Matrix d d Real},
      (HansenEconometrics.TendstoInBootstrapProbability μ Pstar
          (fun n ω ωs => { fst := Gstar n ω ωs, snd := Vstar n ω ωs }) fun x => { fst := G, snd := V }) →
        HansenEconometrics.TendstoInBootstrapProbability μ Pstar
          (fun n ω ωs => HansenEconometrics.smoothFunctionVarianceFunctional (Gstar n ω ωs) (Vstar n ω ωs)) fun x =>
          HansenEconometrics.smoothFunctionVarianceFunctional G V

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Variance.lean:52

theorem HansenEconometrics.chapter10_bootstrap_smooth_variance_consistency_of_components

Hansen Theorem 10.8, componentwise plug-in covariance bridge.

This wrapper packages the usual proof shape: establish separate bootstrap convergence of the plug-in Jacobian and covariance inputs, combine them into a joint convergence statement, then apply the smooth covariance CMT.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Gstar : Nat → Ω → Ωs → Matrix d r Real} {Vstar : Nat → Ω → Ωs → Matrix d d Real} {G : Matrix d r Real}
      {V : Matrix d d Real},
      (HansenEconometrics.TendstoInBootstrapProbability μ Pstar Gstar fun x => G) →
        (HansenEconometrics.TendstoInBootstrapProbability μ Pstar Vstar fun x => V) →
          HansenEconometrics.TendstoInBootstrapProbability μ Pstar
            (fun n ω ωs => HansenEconometrics.smoothFunctionVarianceFunctional (Gstar n ω ωs) (Vstar n ω ωs)) fun x =>
            HansenEconometrics.smoothFunctionVarianceFunctional G V

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Variance.lean:81

theorem HansenEconometrics.chapter10_bootstrap_smooth_variance_consistency_of_continuous_plugins

Hansen Theorem 10.8, plug-in covariance bridge from continuous stochastic component maps.

This is the CMT-shaped constructor for stochastic plug-in Jacobian/covariance inputs: if a bootstrap statistic Ustar converges to a constant u and the Jacobian and covariance plug-ins are continuous at u, then G(Ustar)ᵀ V(Ustar) G(Ustar) converges to G(u)ᵀ V(u) G(u).

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} {A : Type u_9} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Ustar : Nat → Ω → Ωs → A} {u : A} {Gfun : A → Matrix d r Real} {Vfun : A → Matrix d d Real},
      (HansenEconometrics.TendstoInBootstrapProbability μ Pstar Ustar fun x => u) →
        ContinuousAt Gfun u →
          ContinuousAt Vfun u →
            HansenEconometrics.TendstoInBootstrapProbability μ Pstar
              (fun n ω ωs =>
                HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Ustar n ω ωs)) (Vfun (Ustar n ω ωs)))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (Vfun u)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Variance.lean:131

theorem HansenEconometrics.chapter10_bootstrap_smooth_variance_consistency_of_continuous_jacobian

Hansen Theorem 10.8, mixed stochastic Jacobian plug-in bridge.

This covers the common case where the Jacobian is a continuous function of a bootstrap plug-in statistic, while the covariance input has its own convergence proof, such as a conditional covariance or finite-replication covariance route.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} {A : Type u_9} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Ustar : Nat → Ω → Ωs → A} {u : A} {Gfun : A → Matrix d r Real} {Vstar : Nat → Ω → Ωs → Matrix d d Real}
      {V : Matrix d d Real},
      (HansenEconometrics.TendstoInBootstrapProbability μ Pstar Ustar fun x => u) →
        ContinuousAt Gfun u →
          (HansenEconometrics.TendstoInBootstrapProbability μ Pstar Vstar fun x => V) →
            HansenEconometrics.TendstoInBootstrapProbability μ Pstar
              (fun n ω ωs => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Ustar n ω ωs)) (Vstar n ω ωs))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) V

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Variance.lean:157

theorem HansenEconometrics.chapter10_bootstrap_smooth_variance_consistency_of_deterministic_plugins

Hansen Theorem 10.8, deterministic continuous plug-in covariance bridge.

This wrapper covers the common smooth-function case where the plug-in source statistic is non-random under the bootstrap law: ordinary convergence in probability of U_n to u, plus continuity of the Jacobian and covariance plug-in maps at u, implies bootstrap convergence of G(U_n)’ V(U_n) G(U_n).

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} {A : Type u_9} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Useq : Nat → Ω → A} {u : A} {Gfun : A → Matrix d r Real} {Vfun : A → Matrix d d Real},
      (MeasureTheory.TendstoInMeasure μ Useq Filter.atTop fun x => u) →
        ContinuousAt Gfun u →
          ContinuousAt Vfun u →
            HansenEconometrics.TendstoInBootstrapProbability μ Pstar
              (fun n ω x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Useq n ω)) (Vfun (Useq n ω)))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (Vfun u)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Variance.lean:183

theorem HansenEconometrics.chapter10_bootstrap_smooth_variance_consistency_of_deterministic_jacobian

Hansen Theorem 10.8, mixed deterministic Jacobian plug-in bridge.

Ordinary convergence in probability of a non-bootstrap plug-in statistic supplies the continuous Jacobian input, while a separate ordinary convergence proof supplies the covariance input; Theorem 10.1 lifts both to bootstrap probability before the smooth covariance CMT is applied.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} {A : Type u_9} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    ∀ {Useq : Nat → Ω → A} {u : A} {Gfun : A → Matrix d r Real} {Vseq : Nat → Ω → Matrix d d Real}
      {V : Matrix d d Real},
      (MeasureTheory.TendstoInMeasure μ Useq Filter.atTop fun x => u) →
        ContinuousAt Gfun u →
          (MeasureTheory.TendstoInMeasure μ Vseq Filter.atTop fun x => V) →
            HansenEconometrics.TendstoInBootstrapProbability μ Pstar
              (fun n ω x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Useq n ω)) (Vseq n ω)) fun x =>
              HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) V

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Variance.lean:210

Theorem 10.8 ordinary-bootstrap covariance input4 endpoints

The smooth plug-in covariance satisfies V_\theta^*=G^{*'}V^*G^*\xrightarrow{p^*}V_\theta. The ordinary Fin (n+1) nonparametric-bootstrap covariance matrix can feed the smooth plug-in covariance estimator when the Jacobian source is deterministic or stochastic and continuous

theorem HansenEconometrics.chapter10_indexed_smoothVariance_detJacobian_finSuccCovariance_iid

Theorem 10.8 ordinary-bootstrap covariance-input route with a deterministic Jacobian plug-in and iid finite-dimensional observations.

The Fin (n+1) ordinary nonparametric-bootstrap covariance matrix supplies the covariance input V; ordinary convergence of the deterministic Jacobian source supplies G. The smooth plug-in covariance CMT then gives convergence of G’ V G* to G’ V G.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Useq : Nat → Ω → A} {u : A} {Gfun : A → Matrix d r Real} (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (MeasureTheory.TendstoInMeasure μ Useq Filter.atTop fun x => u) →
          ContinuousAt Gfun u →
            HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
              (fun n ω x =>
                HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Useq n ω))
                  (HansenEconometrics.bootstrapCovarianceMatIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                    (fun n ω ωs a =>
                      instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                        (instHSub.hSub
                          (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs
                            a)
                          (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                    n ω))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (6)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2663

theorem HansenEconometrics.chapter10_indexed_smoothVariance_detJacobian_finSuccCovariance_iIndep

Theorem 10.8 ordinary-bootstrap covariance-input route with a deterministic Jacobian plug-in and the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Useq : Nat → Ω → A} {u : A} {Gfun : A → Matrix d r Real} (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (MeasureTheory.TendstoInMeasure μ Useq Filter.atTop fun x => u) →
          ContinuousAt Gfun u →
            HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
              (fun n ω x =>
                HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Useq n ω))
                  (HansenEconometrics.bootstrapCovarianceMatIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                    (fun n ω ωs a =>
                      instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                        (instHSub.hSub
                          (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs
                            a)
                          (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                    n ω))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (6)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2727

theorem HansenEconometrics.chapter10_indexed_smoothVariance_contJacobian_finSuccCovariance_iid

Theorem 10.8 ordinary-bootstrap covariance-input route with a stochastic continuous Jacobian plug-in and iid finite-dimensional observations.

The Fin (n+1) ordinary nonparametric-bootstrap covariance matrix supplies the covariance input V; the bootstrap-probability convergence premise for U_n supplies the stochastic continuous Jacobian G(U*_n).

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Ustar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → A} {u : A} {Gfun : A → Matrix d r Real}
  (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
            Ustar fun x => u) →
          ContinuousAt Gfun u →
            HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
              (fun n ω ωs =>
                HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Ustar n ω ωs))
                  (HansenEconometrics.bootstrapCovarianceMatIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                    (fun n ω ωs a =>
                      instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                        (instHSub.hSub
                          (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs
                            a)
                          (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                    n ω))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (6)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2767

theorem HansenEconometrics.chapter10_indexed_smoothVariance_contJacobian_finSuccCovariance_iIndep

Theorem 10.8 ordinary-bootstrap covariance-input route with a stochastic continuous Jacobian plug-in and the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Ustar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → A} {u : A} {Gfun : A → Matrix d r Real}
  (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
            Ustar fun x => u) →
          ContinuousAt Gfun u →
            HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
              (fun n ω ωs =>
                HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Ustar n ω ωs))
                  (HansenEconometrics.bootstrapCovarianceMatIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                    (fun n ω ωs a =>
                      instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                        (instHSub.hSub
                          (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs
                            a)
                          (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                    n ω))
              fun x => HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (6)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2845

Theorem 10.8/10.11 ordinary-bootstrap finite-replication covariance input6 of 8 linked endpoints

The smooth plug-in covariance satisfies V_\theta^*=G^{*'}V^*G^*\xrightarrow{p^*}V_\theta. The ordinary Fin (n+1) nonparametric-bootstrap covariance target can also feed the moment-form and Hansen-centered finite-replication covariance estimators, and then the smooth plug-in covariance estimator

theorem HansenEconometrics.chapter10_indexed_smoothVariance_detJacobian_finSuccFiniteReplicationCovariance_l2_iid

Theorem 10.8/10.11 ordinary-bootstrap finite-replication smooth plug-in route with a deterministic Jacobian source and iid observations.

The normalized Fin (n+1) ordinary bootstrap covariance supplies the target conditional covariance; coordinatewise O(n⁻¹) simulation error transfers Hansen’s centered finite-replication covariance estimator to that target before the smooth plug-in covariance CMT is applied.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Useq : Nat → Ω → A} {u : A} {Gfun : A → Matrix d r Real} {Zsim : Nat → Nat → Ω → d → Real} {Cfinite : d → d → Real}
  (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (MeasureTheory.TendstoInMeasure μ Useq Filter.atTop fun x => u) →
          ContinuousAt Gfun u →
            (∀ (a c : d) (n : Nat),
                MeasureTheory.Integrable
                  (fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                          (HansenEconometrics.bootstrapCovarianceMatIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs a =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs a)
                                  (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                            n ω)
                          a c))
                      2)
                  μ) →
              (∀ (a c : d),
                  Filter.Eventually
                    (fun n =>
                      Real.instLE.le
                        (MeasureTheory.integral μ fun ω =>
                          instHPow.hPow
                            (Real.norm.norm
                              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMatIndexed
                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                  (fun n ω ωs a =>
                                    instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                      (instHSub.hSub
                                        (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                          (fun ωs t => ωs t) ωs a)
                                        (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                                  n ω)
                                a c))
                            2)
                        (instHDiv.hDiv (Cfinite a c) n.cast))
                    Filter.atTop) →
                HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ
                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                  (fun n ω x =>
                    HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Useq n ω))
                      (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω))
                  fun x =>
                  HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (7)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:15818

theorem HansenEconometrics.chapter10_indexed_smoothVariance_contJacobian_finSuccFiniteReplicationCovariance_l2_iid

Theorem 10.8/10.11 ordinary-bootstrap finite-replication smooth plug-in route with a stochastic continuous Jacobian source and iid observations.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Ustar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → A} {u : A} {Gfun : A → Matrix d r Real}
  {Zsim : Nat → Nat → Ω → d → Real} {Cfinite : d → d → Real} (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
            Ustar fun x => u) →
          ContinuousAt Gfun u →
            (∀ (a c : d) (n : Nat),
                MeasureTheory.Integrable
                  (fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                          (HansenEconometrics.bootstrapCovarianceMatIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs a =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs a)
                                  (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                            n ω)
                          a c))
                      2)
                  μ) →
              (∀ (a c : d),
                  Filter.Eventually
                    (fun n =>
                      Real.instLE.le
                        (MeasureTheory.integral μ fun ω =>
                          instHPow.hPow
                            (Real.norm.norm
                              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMatIndexed
                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                  (fun n ω ωs a =>
                                    instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                      (instHSub.hSub
                                        (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                          (fun ωs t => ωs t) ωs a)
                                        (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                                  n ω)
                                a c))
                            2)
                        (instHDiv.hDiv (Cfinite a c) n.cast))
                    Filter.atTop) →
                HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ
                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                  (fun n ω ωs =>
                    HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Ustar n ω ωs))
                      (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω))
                  fun x =>
                  HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (7)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:15958

theorem HansenEconometrics.chapter10_indexed_smoothVariance_detJacobian_finSuccFiniteReplicationCovariance_l2_iIndep

Theorem 10.8/10.11 ordinary-bootstrap finite-replication smooth plug-in route with a deterministic Jacobian source and the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Useq : Nat → Ω → A} {u : A} {Gfun : A → Matrix d r Real} {Zsim : Nat → Nat → Ω → d → Real} {Cfinite : d → d → Real}
  (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (MeasureTheory.TendstoInMeasure μ Useq Filter.atTop fun x => u) →
          ContinuousAt Gfun u →
            (∀ (a c : d) (n : Nat),
                MeasureTheory.Integrable
                  (fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                          (HansenEconometrics.bootstrapCovarianceMatIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs a =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs a)
                                  (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                            n ω)
                          a c))
                      2)
                  μ) →
              (∀ (a c : d),
                  Filter.Eventually
                    (fun n =>
                      Real.instLE.le
                        (MeasureTheory.integral μ fun ω =>
                          instHPow.hPow
                            (Real.norm.norm
                              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMatIndexed
                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                  (fun n ω ωs a =>
                                    instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                      (instHSub.hSub
                                        (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                          (fun ωs t => ωs t) ωs a)
                                        (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                                  n ω)
                                a c))
                            2)
                        (instHDiv.hDiv (Cfinite a c) n.cast))
                    Filter.atTop) →
                HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ
                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                  (fun n ω x =>
                    HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Useq n ω))
                      (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω))
                  fun x =>
                  HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (7)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:15896

theorem HansenEconometrics.chapter10_indexed_smoothVariance_contJacobian_finSuccFiniteReplicationCovariance_l2_iIndep

Theorem 10.8/10.11 ordinary-bootstrap finite-replication smooth plug-in route with a stochastic continuous Jacobian source and the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} {A : Type u_9}
  [MeasureTheory.IsProbabilityMeasure μ] [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : PseudoMetricSpace A]
  {Ustar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → A} {u : A} {Gfun : A → Matrix d r Real}
  {Zsim : Nat → Nat → Ω → d → Real} {Cfinite : d → d → Real} (Y : Nat → Ω → d → Real),
  (∀ (a : d), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
            Ustar fun x => u) →
          ContinuousAt Gfun u →
            (∀ (a c : d) (n : Nat),
                MeasureTheory.Integrable
                  (fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                          (HansenEconometrics.bootstrapCovarianceMatIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs a =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs a)
                                  (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                            n ω)
                          a c))
                      2)
                  μ) →
              (∀ (a c : d),
                  Filter.Eventually
                    (fun n =>
                      Real.instLE.le
                        (MeasureTheory.integral μ fun ω =>
                          instHPow.hPow
                            (Real.norm.norm
                              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMatIndexed
                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                  (fun n ω ωs a =>
                                    instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                      (instHSub.hSub
                                        (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                          (fun ωs t => ωs t) ωs a)
                                        (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                                  n ω)
                                a c))
                            2)
                        (instHDiv.hDiv (Cfinite a c) n.cast))
                    Filter.atTop) →
                HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ
                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                  (fun n ω ωs =>
                    HansenEconometrics.smoothFunctionVarianceFunctional (Gfun (Ustar n ω ωs))
                      (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω))
                  fun x =>
                  HansenEconometrics.smoothFunctionVarianceFunctional (Gfun u) (HansenEconometrics.covMat μ (Y 0))

Direct statement dependencies (7)

HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.smoothFunctionVarianceFunctional

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:16050

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_finSucc_l2_iid

Theorem 10.11 ordinary nonparametric-bootstrap finite-replication moment-form covariance route for iid finite-dimensional observations.

The conditional covariance target is the normalized Fin (n+1) ordinary bootstrap covariance from Theorem 10.4; coordinatewise O(n⁻¹) mean-square simulation error transfers the moment-form finite-replication covariance estimator to the same population covariance target.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst : Fintype k] {Zsim : Nat → Nat → Ω → k → Real} {Cfinite : k → k → Real} (Y : Nat → Ω → k → Real),
  (∀ (a : k), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (∀ (a c : k) (n : Nat),
            MeasureTheory.Integrable
              (fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                      (HansenEconometrics.bootstrapCovarianceMatIndexed
                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                        (fun n ω ωs a =>
                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                            (instHSub.hSub
                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t)
                                ωs a)
                              (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                        n ω)
                      a c))
                  2)
              μ) →
          (∀ (a c : k),
              Filter.Eventually
                (fun n =>
                  Real.instLE.le
                    (MeasureTheory.integral μ fun ω =>
                      instHPow.hPow
                        (Real.norm.norm
                          (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                            (HansenEconometrics.bootstrapCovarianceMatIndexed
                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                              (fun n ω ωs a =>
                                instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                  (instHSub.hSub
                                    (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                      (fun ωs t => ωs t) ωs a)
                                    (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                              n ω)
                            a c))
                        2)
                    (instHDiv.hDiv (Cfinite a c) n.cast))
                Filter.atTop) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop
              fun x => HansenEconometrics.covMat μ (Y 0)

Direct statement dependencies (5)

HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationCovarianceMomentMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:7128

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_finSucc_l2_iIndep

Theorem 10.11 ordinary nonparametric-bootstrap finite-replication moment-form covariance route with the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst : Fintype k] {Zsim : Nat → Nat → Ω → k → Real} {Cfinite : k → k → Real} (Y : Nat → Ω → k → Real),
  (∀ (a : k), MeasureTheory.MemLp (fun ω => Y 0 ω a) 2 μ) →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (∀ (a c : k) (n : Nat),
            MeasureTheory.Integrable
              (fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                      (HansenEconometrics.bootstrapCovarianceMatIndexed
                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                        (fun n ω ωs a =>
                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                            (instHSub.hSub
                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t)
                                ωs a)
                              (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                        n ω)
                      a c))
                  2)
              μ) →
          (∀ (a c : k),
              Filter.Eventually
                (fun n =>
                  Real.instLE.le
                    (MeasureTheory.integral μ fun ω =>
                      instHPow.hPow
                        (Real.norm.norm
                          (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                            (HansenEconometrics.bootstrapCovarianceMatIndexed
                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                              (fun n ω ωs a =>
                                instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                  (instHSub.hSub
                                    (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                      (fun ωs t => ωs t) ωs a)
                                    (HansenEconometrics.empiricalMean (fun i => Y i.val ω) a)))
                              n ω)
                            a c))
                        2)
                    (instHDiv.hDiv (Cfinite a c) n.cast))
                Filter.atTop) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop
              fun x => HansenEconometrics.covMat μ (Y 0)

Direct statement dependencies (5)

HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationCovarianceMomentMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:7191

Theorem 10.96 of 22 linked endpoints

Under a common limit law and uniform square integrability, \operatorname{Var}^*(Z_n^*)\xrightarrow{p}V and the finite-replication variance is consistent for V.

theorem HansenEconometrics.chapter10_bootstrap_mean_tendsto_of_weak_distribution_of_uniformSquareTail

Hansen Theorem 10.9 conditional mean convergence from the named uniform-square-tail condition package.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real}
  {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTail μ Pstar Zstar ν Z →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanReal Pstar Zstar) Filter.atTop fun x =>
              MeasureTheory.integral ν fun ωlim => Z ωlim

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapMeanReal

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4264

theorem HansenEconometrics.chapter10_bootstrap_meanVec_tendsto_of_weak_distribution_of_uniformSquareTail

Hansen Theorem 10.9 finite-dimensional mean-vector wrapper.

Bootstrap weak convergence of the vector statistic plus the named uniform-square-tail condition on each coordinate implies convergence in probability of the conditional bootstrap mean vector. This is the coordinatewise vector surface used by the covariance and trimmed-variance layers, where the textbook proofs first establish scalar uniform square-integrability for every coordinate.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim} [inst : Fintype k]
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real}
  {Z : Ωlim → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (∀ (a : k), MeasureTheory.MemLp (fun ωlim => Z ωlim a) 2 ν) →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          (∀ (a : k),
              HansenEconometrics.BootstrapUniformSquareTail μ Pstar (fun n ω ωs => Zstar n ω ωs a) ν fun ωlim =>
                Z ωlim a) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanVec Pstar Zstar) Filter.atTop fun x a =>
              MeasureTheory.integral ν fun ωlim => Z ωlim a

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapMeanVec

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2893

theorem HansenEconometrics.chapter10_finiteReplicationVariance_tendsto_of_weak_distribution_uniformSquareTail

Hansen Theorem 10.9/10.11 finite-replication variance from bootstrap weak convergence and a uniform-square-tail condition.

This packages the two variance layers used in the theorem: a finite-replication simulation-error premise estimates the conditional bootstrap variance, while bootstrap weak convergence plus the named uniform-square-tail condition sends that conditional variance to the limiting variance functional.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Zsim : Nat → Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTail μ Pstar Zstar ν Z →
            (MeasureTheory.TendstoInMeasure μ
                (fun n ω =>
                  instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                    (HansenEconometrics.bootstrapVarianceReal Pstar Zstar n ω))
                Filter.atTop fun x => 0) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim)
                Filter.atTop fun x =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                  (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (4)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:802

theorem HansenEconometrics.chapter10_bootstrap_secondMoment_tendsto_of_weak_distribution_of_uniformSquareTail

Hansen Theorem 10.9 conditional second-moment convergence from the named uniform-square-tail condition package.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real}
  {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTail μ Pstar Zstar ν Z →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentReal Pstar Zstar) Filter.atTop
              fun x => MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapSecondMomentReal

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4280

theorem HansenEconometrics.chapter10_bootstrap_variance_consistency_of_weak_distribution_of_uniformSquareTail

Hansen Theorem 10.9 from a named uniform-square-tail condition.

This is the public theorem-facing wrapper: bootstrap weak convergence plus BootstrapUniformSquareTail gives conditional bootstrap variance consistency.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real}
  {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTail μ Pstar Zstar ν Z →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapVarianceReal Pstar Zstar) Filter.atTop
              fun x =>
              instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4339

theorem HansenEconometrics.chapter10_bootstrap_crossMomentMat_tendsto_of_weak_distribution_of_uniformSquareTail

Hansen Theorem 10.9 finite-dimensional cross-moment wrapper.

Bootstrap weak convergence plus named uniform-square-tail conditions for each coordinate and each coordinate sum imply convergence in probability of the conditional bootstrap cross-moment matrix. The proof uses xy = ((x + y)^2 - x^2 - y^2) / 2, so model-specific layers can verify scalar square-tail conditions rather than developing a separate product-tail API.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim} [inst : Fintype k]
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real}
  {Z : Ωlim → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (∀ (a : k), MeasureTheory.MemLp (fun ωlim => Z ωlim a) 2 ν) →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          (∀ (a : k),
              HansenEconometrics.BootstrapUniformSquareTail μ Pstar (fun n ω ωs => Zstar n ω ωs a) ν fun ωlim =>
                Z ωlim a) →
            (∀ (a c : k),
                HansenEconometrics.BootstrapUniformSquareTail μ Pstar
                  (fun n ω ωs => instHAdd.hAdd (Zstar n ω ωs a) (Zstar n ω ωs c)) ν fun ωlim =>
                  instHAdd.hAdd (Z ωlim a) (Z ωlim c)) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCrossMomentMat Pstar Zstar) Filter.atTop
                fun x a c => MeasureTheory.integral ν fun ωlim => instHMul.hMul (Z ωlim a) (Z ωlim c)

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCrossMomentMat

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2994

Theorem 10.9 indexed tail-integral support5 endpoints

Under a common limit law and uniform square integrability, \operatorname{Var}^*(Z_n^*)\xrightarrow{p}V and the finite-replication variance is consistent for V. Sample-size-dependent bootstrap spaces can use the same concrete first/second tail-integral and squared-tail-integral routes as the fixed bootstrap-space theorem

theorem HansenEconometrics.bootstrapMeanRealIndexed_realClip_tails_of_tail_integrals

Indexed tail-integral constructor for the first-moment clipping premise used in Hansen Theorem 10.9.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
    MeasureTheory.Integrable Z ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.Integrable (Zstar n ω) (Pstar n ω)) →
        (∀ (ε : Real),
            Real.instLT.lt 0 ε →
              Exists fun R =>
                And (Real.instLE.le 0 R)
                  (And
                    (Real.instLE.le
                      (MeasureTheory.integral ν fun ωlim =>
                        (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator (fun ωlim => abs (Z ωlim)) ωlim)
                      ε)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                            (fun ωs => abs (Zstar n ω ωs)) ωs)
                      Filter.atTop fun x => 0))) →
          ∀ (ε : Real),
            Real.instLT.lt 0 ε →
              Exists fun R =>
                And (Real.instLE.le 0 R)
                  (And
                    (Real.instLE.le
                      (abs
                        (instHSub.hSub (MeasureTheory.integral ν fun ωlim => Z ωlim)
                          (MeasureTheory.integral ν fun ωlim => HansenEconometrics.realClip R (Z ωlim))))
                      ε)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        instHSub.hSub (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar n ω)
                          (MeasureTheory.integral (Pstar n ω) fun x =>
                            (fun ωs => HansenEconometrics.realClip R (Zstar n ω ωs)) x))
                      Filter.atTop fun x => 0))

Direct statement dependencies (2)

HansenEconometrics.bootstrapMeanRealIndexed
HansenEconometrics.realClip

HansenEconometrics/Chapter10Bootstrap/Variance.lean:880

theorem HansenEconometrics.bootstrapSecondMomentRealIndexed_realClip_tails_of_tail_integrals

Indexed tail-integral constructor for the second-moment clipping premise used in Hansen Theorem 10.9.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
    MeasureTheory.MemLp Z 2 ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
        (∀ (ε : Real),
            Real.instLT.lt 0 ε →
              Exists fun R =>
                And (Real.instLE.le 0 R)
                  (And
                    (Real.instLE.le
                      (MeasureTheory.integral ν fun ωlim =>
                        (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                          (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                      ε)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                            (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
                      Filter.atTop fun x => 0))) →
          ∀ (ε : Real),
            Real.instLT.lt 0 ε →
              Exists fun R =>
                And (Real.instLE.le 0 R)
                  (And
                    (Real.instLE.le
                      (abs
                        (instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                          (MeasureTheory.integral ν fun ωlim =>
                            instHPow.hPow (HansenEconometrics.realClip R (Z ωlim)) 2)))
                      ε)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        instHSub.hSub (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar n ω)
                          (MeasureTheory.integral (Pstar n ω) fun x =>
                            (fun ωs => instHPow.hPow (HansenEconometrics.realClip R (Zstar n ω ωs)) 2) x))
                      Filter.atTop fun x => 0))

Direct statement dependencies (2)

HansenEconometrics.bootstrapSecondMomentRealIndexed
HansenEconometrics.realClip

HansenEconometrics/Chapter10Bootstrap/Variance.lean:923

theorem HansenEconometrics.bootstrapTailAbsIntegralIndexed_tendsto_zero_of_tailSqIntegral

Indexed conditional absolute-tail integrals vanish in probability when dominated by squared-tail integrals at a threshold at least one.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      ∀ {R : Real},
        Real.instLE.le 1 R →
          (MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                MeasureTheory.integral (Pstar n ω) fun ωs =>
                  (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                    (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
              Filter.atTop fun x => 0) →
            MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                MeasureTheory.integral (Pstar n ω) fun ωs =>
                  (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator (fun ωs => abs (Zstar n ω ωs)) ωs)
              Filter.atTop fun x => 0

HansenEconometrics/Chapter10Bootstrap/Variance.lean:997

theorem HansenEconometrics.chapter10_indexed_bootstrap_variance_consistency_of_weak_distribution_tail_integrals

Indexed Hansen Theorem 10.9, weak-distribution plus concrete tail-integral variance bridge.

This is the sample-size-dependent bootstrap-space version of chapter10_bootstrap_variance_consistency_of_weak_distribution_tail_integrals.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (ε : Real),
              Real.instLT.lt 0 ε →
                Exists fun R =>
                  And (Real.instLE.le 0 R)
                    (And
                      (Real.instLE.le
                        (MeasureTheory.integral ν fun ωlim =>
                          (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator (fun ωlim => abs (Z ωlim)) ωlim)
                        ε)
                      (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                              (fun ωs => abs (Zstar n ω ωs)) ωs)
                        Filter.atTop fun x => 0))) →
            (∀ (ε : Real),
                Real.instLT.lt 0 ε →
                  Exists fun R =>
                    And (Real.instLE.le 0 R)
                      (And
                        (Real.instLE.le
                          (MeasureTheory.integral ν fun ωlim =>
                            (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                              (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                          ε)
                        (MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                                (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
                          Filter.atTop fun x => 0))) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar)
                Filter.atTop fun x =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                  (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapVarianceRealIndexed

HansenEconometrics/Chapter10Bootstrap/Variance.lean:1316

theorem HansenEconometrics.chapter10_indexed_bootstrap_variance_consistency_of_weak_distribution_square_tail_integrals

Indexed Hansen Theorem 10.9, weak-distribution plus squared-tail-integral variance bridge.

For thresholds at least one, squared tails dominate absolute tails, so one indexed squared-tail control supplies the first- and second-tail integral premises needed for conditional bootstrap variance consistency.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (ε : Real),
              Real.instLT.lt 0 ε →
                Exists fun R =>
                  And (Real.instLE.le 1 R)
                    (And
                      (Real.instLE.le
                        (MeasureTheory.integral ν fun ωlim =>
                          (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                            (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                        ε)
                      (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                              (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
                        Filter.atTop fun x => 0))) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar) Filter.atTop
              fun x =>
              instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapVarianceRealIndexed

HansenEconometrics/Chapter10Bootstrap/Variance.lean:1370

Theorem 10.9 indexed uniform-square-tail support6 of 10 linked endpoints

Under a common limit law and uniform square integrability, \operatorname{Var}^*(Z_n^*)\xrightarrow{p}V and the finite-replication variance is consistent for V. Sample-size-dependent bootstrap spaces satisfying the same uniform square-tail premise have conditional indexed mean, second-moment, variance, mean-vector, cross-moment, and covariance convergence

theorem HansenEconometrics.chapter10_indexed_bootstrap_mean_tendsto_of_weak_distribution_uniform_square_tail

Indexed Hansen Theorem 10.9 conditional mean convergence from weak convergence and uniform square-tail control.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (ε : Real),
              Real.instLT.lt 0 ε →
                Exists fun R =>
                  And (Real.instLE.le 1 R)
                    (And
                      (Real.instLE.le
                        (MeasureTheory.integral ν fun ωlim =>
                          (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                            (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                        ε)
                      (Filter.Tendsto
                        (fun n =>
                          MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω =>
                              Real.instLE.le ε
                                (Real.pseudoMetricSpace.dist
                                  (MeasureTheory.integral (Pstar n ω) fun ωs =>
                                    (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                                      (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
                                  0)))
                        Filter.atTop (nhds 0)))) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar) Filter.atTop
              fun x => MeasureTheory.integral ν fun ωlim => Z ωlim

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapMeanRealIndexed

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4800

theorem HansenEconometrics.chapter10_indexed_bootstrap_mean_tendsto_of_weak_distribution_of_uniformSquareTail

Indexed Hansen Theorem 10.9 conditional mean convergence from the named uniform-square-tail condition package.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar Zstar ν Z →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar) Filter.atTop
              fun x => MeasureTheory.integral ν fun ωlim => Z ωlim

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapMeanRealIndexed

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4995

theorem HansenEconometrics.chapter10_indexed_bootstrap_meanVec_tendsto_of_weak_distribution_of_uniformSquareTail

Indexed Hansen Theorem 10.9 finite-dimensional mean-vector wrapper.

This is the sample-size-dependent counterpart of chapter10_bootstrap_meanVec_tendsto_of_weak_distribution_of_uniformSquareTail.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim}
  {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim} [inst : Fintype k] [MeasureTheory.IsFiniteMeasure ν]
  {Ωboot : Nat → Type u_7} [inst_2 : (n : Nat) → MeasurableSpace (Ωboot n)]
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {Z : Ωlim → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (∀ (a : k), MeasureTheory.MemLp (fun ωlim => Z ωlim a) 2 ν) →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (a : k),
              HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar (fun n ω ωs => Zstar n ω ωs a) ν fun ωlim =>
                Z ωlim a) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanVecIndexed Pstar Zstar) Filter.atTop
              fun x a => MeasureTheory.integral ν fun ωlim => Z ωlim a

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapMeanVecIndexed

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:3662

theorem HansenEconometrics.chapter10_indexed_bootstrap_covarianceMat_tendsto_of_weak_distribution_uniformSquareTail

Indexed Hansen Theorem 10.9/10.12 covariance matrix from bootstrap weak convergence and scalar indexed uniform-square-tail controls.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim}
  {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim} [inst : Fintype k] [MeasureTheory.IsFiniteMeasure ν]
  {Ωboot : Nat → Type u_7} [inst_2 : (n : Nat) → MeasurableSpace (Ωboot n)]
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {Z : Ωlim → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (∀ (a : k), MeasureTheory.MemLp (fun ωlim => Z ωlim a) 2 ν) →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (a : k),
              HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar (fun n ω ωs => Zstar n ω ωs a) ν fun ωlim =>
                Z ωlim a) →
            (∀ (a c : k),
                HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar
                  (fun n ω ωs => instHAdd.hAdd (Zstar n ω ωs a) (Zstar n ω ωs c)) ν fun ωlim =>
                  instHAdd.hAdd (Z ωlim a) (Z ωlim c)) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar)
                Filter.atTop fun x a c =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHMul.hMul (Z ωlim a) (Z ωlim c))
                  (instHMul.hMul (MeasureTheory.integral ν fun ωlim => Z ωlim a)
                    (MeasureTheory.integral ν fun ωlim => Z ωlim c))

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:3865

theorem HansenEconometrics.chapter10_indexed_bootstrap_secondMoment_tendsto_of_weak_distribution_uniform_square_tail

Indexed Hansen Theorem 10.9 conditional second-moment convergence from weak convergence and uniform square-tail control.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (ε : Real),
              Real.instLT.lt 0 ε →
                Exists fun R =>
                  And (Real.instLE.le 1 R)
                    (And
                      (Real.instLE.le
                        (MeasureTheory.integral ν fun ωlim =>
                          (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                            (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                        ε)
                      (Filter.Tendsto
                        (fun n =>
                          MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω =>
                              Real.instLE.le ε
                                (Real.pseudoMetricSpace.dist
                                  (MeasureTheory.integral (Pstar n ω) fun ωs =>
                                    (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                                      (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
                                  0)))
                        Filter.atTop (nhds 0)))) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar)
              Filter.atTop fun x => MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapSecondMomentRealIndexed

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4906

theorem HansenEconometrics.chapter10_indexed_bootstrap_secondMoment_tendsto_of_weak_distribution_of_uniformSquareTail

Indexed Hansen Theorem 10.9 conditional second-moment convergence from the named uniform-square-tail condition package.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar Zstar ν Z →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar)
              Filter.atTop fun x => MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapSecondMomentRealIndexed

HansenEconometrics/Chapter10Bootstrap/Variance.lean:5012

Theorem 10.9 ordinary nonparametric scalar variance support6 endpoints

Under a common limit law and uniform square integrability, \operatorname{Var}^*(Z_n^*)\xrightarrow{p}V and the finite-replication variance is consistent for V. The normalized ordinary Fin (n+1) bootstrap mean has conditional variance converging to the population variance once the empirical one-draw variance converges; iid finite-second-moment assumptions discharge that empirical variance premise

theorem HansenEconometrics.variance_uniformOn_finSucc_tendsto_of_mean_second_moments

Empirical scalar variance convergence from empirical first and second moments.

This is the scalar counterpart of covMat_uniformOn_finSucc_tendsto_of_mean_cross_moments: once the empirical mean and raw second moment on Fin (n+1) converge, the finite empirical variance converges to the population variance.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    (MeasureTheory.TendstoInMeasure μ
        (fun n ω => MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun i => Y i.val ω) Filter.atTop
        fun x => MeasureTheory.integral μ fun ω => Y 0 ω) →
      (MeasureTheory.TendstoInMeasure μ
          (fun n ω =>
            MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun i => instHPow.hPow (Y i.val ω) 2)
          Filter.atTop fun x => MeasureTheory.integral μ fun ω => instHPow.hPow (Y 0 ω) 2) →
        MeasureTheory.TendstoInMeasure μ
          (fun n ω => ProbabilityTheory.variance (fun i => Y i.val ω) (ProbabilityTheory.uniformOn Set.univ))
          Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:883

theorem HansenEconometrics.variance_uniformOn_finSucc_tendsto_of_iid

Empirical scalar variance convergence for iid real observations.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        MeasureTheory.TendstoInMeasure μ
          (fun n ω => ProbabilityTheory.variance (fun i => Y i.val ω) (ProbabilityTheory.uniformOn Set.univ))
          Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:1002

theorem HansenEconometrics.variance_uniformOn_finSucc_tendsto_of_iIndep

Empirical scalar variance convergence for iid real observations with the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        MeasureTheory.TendstoInMeasure μ
          (fun n ω => ProbabilityTheory.variance (fun i => Y i.val ω) (ProbabilityTheory.uniformOn Set.univ))
          Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:1056

theorem HansenEconometrics.chapter10_indexed_bootstrap_variance_finSucc_resampleMean_tendsto_of_empirical_variance

Indexed conditional variance convergence for the normalized scalar ordinary nonparametric-bootstrap mean from empirical one-draw variance convergence.

This is the scalar Theorem 10.9 surface for the concrete Fin (n+1) -> Fin (n+1) resampling law: the exact finite identity above reduces the bootstrap conditional variance to the finite empirical variance.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (Y : Nat → Ω → Real) {v : Real},
  (MeasureTheory.TendstoInMeasure μ
      (fun n ω => ProbabilityTheory.variance (fun i => Y i.val ω) (ProbabilityTheory.uniformOn Set.univ)) Filter.atTop
      fun x => v) →
    MeasureTheory.TendstoInMeasure μ
      (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ) fun n ω ωs =>
        instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
      Filter.atTop fun x => v

Direct statement dependencies (3)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Variance.lean:531

theorem HansenEconometrics.chapter10_indexed_bootstrap_variance_finSucc_resampleMean_tendsto_of_iid

Iid-facing scalar ordinary nonparametric-bootstrap variance constructor.

For the normalized bootstrap mean sqrt (n+1) (Ybar* - Ybar), the conditional bootstrap variance converges in probability to the population variance.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        MeasureTheory.TendstoInMeasure μ
          (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
            fun n ω ωs =>
            instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
              (instHSub.hSub
                (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
          Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

Direct statement dependencies (3)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:1075

theorem HansenEconometrics.chapter10_indexed_bootstrap_variance_finSucc_resampleMean_tendsto_of_iIndep

Iid-facing scalar ordinary nonparametric-bootstrap variance constructor with the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        MeasureTheory.TendstoInMeasure μ
          (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
            fun n ω ωs =>
            instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
              (instHSub.hSub
                (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
          Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

Direct statement dependencies (3)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:1101

Theorem 10.9/10.11 ordinary nonparametric finite-replication scalar variance support6 endpoints

Under a common limit law and uniform square integrability, \operatorname{Var}^*(Z_n^*)\xrightarrow{p}V and the finite-replication variance is consistent for V. The normalized ordinary Fin (n+1) bootstrap variance can feed moment-form and Hansen-centered finite-replication scalar variance estimators

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVariance_finSucc_l2_iid

Theorem 10.9/10.11 ordinary nonparametric-bootstrap finite-replication scalar variance route for iid observations.

The conditional variance target is the normalized Fin (n+1) ordinary bootstrap variance. Coordinatewise O(n⁻¹) mean-square simulation error transfers the moment-form finite-replication variance estimator to the population variance target.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {Zsim : Nat → Nat → Ω → Real} {Cfinite : Real} (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (∀ (n : Nat),
            MeasureTheory.Integrable
              (fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                      (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                        (fun n ω ωs =>
                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                            (instHSub.hSub
                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t)
                                ωs)
                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                        n ω)))
                  2)
              μ) →
          Filter.Eventually
              (fun n =>
                Real.instLE.le
                  (MeasureTheory.integral μ fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                          (HansenEconometrics.bootstrapVarianceRealIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs)
                                  (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                            n ω)))
                      2)
                  (instHDiv.hDiv Cfinite n.cast))
              Filter.atTop →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim) Filter.atTop
              fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

Direct statement dependencies (4)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2470

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVariance_finSucc_l2_iIndep

Theorem 10.9/10.11 ordinary nonparametric-bootstrap finite-replication scalar variance route with the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {Zsim : Nat → Nat → Ω → Real} {Cfinite : Real} (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (∀ (n : Nat),
            MeasureTheory.Integrable
              (fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                      (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                        (fun n ω ωs =>
                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                            (instHSub.hSub
                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t)
                                ωs)
                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                        n ω)))
                  2)
              μ) →
          Filter.Eventually
              (fun n =>
                Real.instLE.le
                  (MeasureTheory.integral μ fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                          (HansenEconometrics.bootstrapVarianceRealIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs)
                                  (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                            n ω)))
                      2)
                  (instHDiv.hDiv Cfinite n.cast))
              Filter.atTop →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim) Filter.atTop
              fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

Direct statement dependencies (4)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2531

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVarianceCenteredReal_finSucc_l2_iid

Theorem 10.9/10.11 ordinary nonparametric-bootstrap centered finite-replication scalar variance route for iid observations.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {Zsim : Nat → Nat → Ω → Real} {Cfinite : Real} (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    Pairwise (Function.onFun (fun x1 x2 => ProbabilityTheory.IndepFun x1 x2 μ) Y) →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (∀ (n : Nat),
            MeasureTheory.Integrable
              (fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                      (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                        (fun n ω ωs =>
                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                            (instHSub.hSub
                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t)
                                ωs)
                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                        n ω)))
                  2)
              μ) →
          Filter.Eventually
              (fun n =>
                Real.instLE.le
                  (MeasureTheory.integral μ fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                          (HansenEconometrics.bootstrapVarianceRealIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs)
                                  (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                            n ω)))
                      2)
                  (instHDiv.hDiv Cfinite n.cast))
              Filter.atTop →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim)
              Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

Direct statement dependencies (4)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2579

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVarianceCenteredReal_finSucc_l2_iIndep

Theorem 10.9/10.11 ordinary nonparametric-bootstrap centered finite-replication scalar variance route with the textbook iIndepFun premise.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {Zsim : Nat → Nat → Ω → Real} {Cfinite : Real} (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (fun ω => Y 0 ω) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (∀ (n : Nat),
            MeasureTheory.Integrable
              (fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                      (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
                        (fun n ω ωs =>
                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                            (instHSub.hSub
                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t)
                                ωs)
                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                        n ω)))
                  2)
              μ) →
          Filter.Eventually
              (fun n =>
                Real.instLE.le
                  (MeasureTheory.integral μ fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                          (HansenEconometrics.bootstrapVarianceRealIndexed
                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                            (fun n ω ωs =>
                              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                (instHSub.hSub
                                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                    (fun ωs t => ωs t) ωs)
                                  (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                            n ω)))
                      2)
                  (instHDiv.hDiv Cfinite n.cast))
              Filter.atTop →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim)
              Filter.atTop fun x => ProbabilityTheory.variance (fun ω => Y 0 ω) μ

Direct statement dependencies (4)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2641

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVariance_finSucc_l2_of_weak_distribution_cumulants

Theorem 10.9/10.11 ordinary nonparametric-bootstrap finite-replication scalar variance route from Hansen’s fourth-moment cumulant formula.

This combines the concrete Fin (n+1) cumulant route for conditional bootstrap variance consistency with an O(n⁻¹) mean-square simulation-error bound for the moment-form finite-replication variance estimator.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
  {Zsim : Nat → Nat → Ω → Real} {Z : Ωlim → Real} {σ2 Cfinite : Real} (Y : Nat → Ω → Real),
  MeasureTheory.MemLp Z 2 ν →
    HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs =>
          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
            (instHSub.hSub
              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        ν Z →
      (MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.empiricalCumulant2 fun i => Y i.val ω)
          Filter.atTop fun x => σ2) →
        (MeasureTheory.TendstoInMeasure μ
            (fun n ω =>
              instHDiv.hDiv (HansenEconometrics.empiricalCumulant4 fun i => Y i.val ω) (instHAdd.hAdd n.cast 1))
            Filter.atTop fun x => 0) →
          (∀ (n : Nat),
              MeasureTheory.Integrable
                (fun ω =>
                  instHPow.hPow
                    (Real.norm.norm
                      (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                        (HansenEconometrics.bootstrapVarianceRealIndexed
                          (fun n x => ProbabilityTheory.uniformOn Set.univ)
                          (fun n ω ωs =>
                            instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                              (instHSub.hSub
                                (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                  (fun ωs t => ωs t) ωs)
                                (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                          n ω)))
                    2)
                μ) →
            Filter.Eventually
                (fun n =>
                  Real.instLE.le
                    (MeasureTheory.integral μ fun ω =>
                      instHPow.hPow
                        (Real.norm.norm
                          (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                            (HansenEconometrics.bootstrapVarianceRealIndexed
                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                              (fun n ω ωs =>
                                instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                  (instHSub.hSub
                                    (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                      (fun ωs t => ωs t) ωs)
                                    (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                              n ω)))
                        2)
                    (instHDiv.hDiv Cfinite n.cast))
                Filter.atTop →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim)
                Filter.atTop fun x =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                  (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (7)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant4
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2694

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVarianceCenteredReal_finSucc_l2_of_weak_distribution_cumulants

Theorem 10.9/10.11 ordinary nonparametric-bootstrap centered finite-replication scalar variance route from Hansen’s fourth-moment cumulant formula.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
  {Zsim : Nat → Nat → Ω → Real} {Z : Ωlim → Real} {σ2 Cfinite : Real} (Y : Nat → Ω → Real),
  MeasureTheory.MemLp Z 2 ν →
    HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs =>
          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
            (instHSub.hSub
              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        ν Z →
      (MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.empiricalCumulant2 fun i => Y i.val ω)
          Filter.atTop fun x => σ2) →
        (MeasureTheory.TendstoInMeasure μ
            (fun n ω =>
              instHDiv.hDiv (HansenEconometrics.empiricalCumulant4 fun i => Y i.val ω) (instHAdd.hAdd n.cast 1))
            Filter.atTop fun x => 0) →
          (∀ (n : Nat),
              MeasureTheory.Integrable
                (fun ω =>
                  instHPow.hPow
                    (Real.norm.norm
                      (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                        (HansenEconometrics.bootstrapVarianceRealIndexed
                          (fun n x => ProbabilityTheory.uniformOn Set.univ)
                          (fun n ω ωs =>
                            instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                              (instHSub.hSub
                                (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                  (fun ωs t => ωs t) ωs)
                                (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                          n ω)))
                    2)
                μ) →
            Filter.Eventually
                (fun n =>
                  Real.instLE.le
                    (MeasureTheory.integral μ fun ω =>
                      instHPow.hPow
                        (Real.norm.norm
                          (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                            (HansenEconometrics.bootstrapVarianceRealIndexed
                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                              (fun n ω ωs =>
                                instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                  (instHSub.hSub
                                    (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                      (fun ωs t => ωs t) ωs)
                                    (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                              n ω)))
                        2)
                    (instHDiv.hDiv Cfinite n.cast))
                Filter.atTop →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim)
                Filter.atTop fun x =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                  (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (7)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant4
HansenEconometrics.empiricalMean
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2781

Theorem 10.106 of 24 linked endpoints

Bounded higher derivatives give uniform square integrability of Z_n^*=p_n(\hat\theta^*-\hat\theta), so the untrimmed bootstrap variance satisfies \hat V_{\theta}^{\mathrm{boot}}\xrightarrow{p}V_\theta.

theorem HansenEconometrics.chapter10_smooth_bootstrap_variance_consistency_of_gaussian_fourthMoment

Hansen Theorem 10.10, smooth-function Gaussian coordinate variance consistency from conditional fourth-moment convergence.

The fourth-moment premise is the formal endpoint of Hansen’s bounded-derivative Taylor/Rosenthal calculation before uniform square integrability is applied.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {r : Type u_7} [inst : Fintype r] [inst_1 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {S : Matrix r r Real}
  [MeasureTheory.IsFiniteMeasure (ProbabilityTheory.multivariateGaussian 0 S)] {B : Real} (a : r),
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
      MeasureTheory.MemLp (fun z => z.ofLp a) 2 (ProbabilityTheory.multivariateGaussian 0 S) →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar thetaStar
            (ProbabilityTheory.multivariateGaussian 0 S) fun z => z) →
          Real.instLE.le 0 B →
            (MeasureTheory.TendstoInMeasure μ
                (fun n ω => MeasureTheory.integral (Pstar n ω) fun ωs => instHPow.hPow ((thetaStar n ω ωs).ofLp a) 4)
                Filter.atTop fun x => B) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.Integrable (fun ωs => instHPow.hPow ((thetaStar n ω ωs).ofLp a) 4) (Pstar n ω)) →
                MeasureTheory.TendstoInMeasure μ
                  (HansenEconometrics.bootstrapVarianceReal Pstar fun n ω ωs => (thetaStar n ω ωs).ofLp a) Filter.atTop
                  fun x =>
                  instHSub.hSub
                    (MeasureTheory.integral (ProbabilityTheory.multivariateGaussian 0 S) fun z =>
                      instHPow.hPow (z.ofLp a) 2)
                    (instHPow.hPow
                      (MeasureTheory.integral (ProbabilityTheory.multivariateGaussian 0 S) fun z => z.ofLp a) 2)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal

HansenEconometrics/Chapter10Bootstrap/Variance.lean:5911

theorem HansenEconometrics.chapter10_smooth_bootstrap_variance_consistency_of_linearization_fourthMoment

Hansen Theorem 10.10, smooth-function variance consistency from exact derivative linearization and a linearized fourth-moment premise.

This is the exact-linearization face of the bounded-derivative calculation: the smooth statistic is first reduced to G T* using Hansen Theorem 10.7’s linearization route, then the scalar fourth-moment premise is rewritten through that equality and fed to the Gaussian fourth-moment variance wrapper above.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {B : Real} (a : r),
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          MeasureTheory.MemLp (fun z => z.ofLp a) 2
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              Real.instLE.le 0 B →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => B) →
                  (∀ (n : Nat) (ω : Ω),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    MeasureTheory.TendstoInMeasure μ
                      (HansenEconometrics.bootstrapVarianceReal Pstar fun n ω ωs => (thetaStar n ω ωs).ofLp a)
                      Filter.atTop fun x =>
                      instHSub.hSub
                        (MeasureTheory.integral
                          (ProbabilityTheory.multivariateGaussian 0
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                          fun z => instHPow.hPow (z.ofLp a) 2)
                        (instHPow.hPow
                          (MeasureTheory.integral
                            (ProbabilityTheory.multivariateGaussian 0
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                            fun z => z.ofLp a)
                          2)

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Variance.lean:7425

theorem HansenEconometrics.chapter10_smooth_bootstrap_covarianceMat_tendsto_of_linearization_fourthMoment

Hansen Theorem 10.10/10.12 smooth-function conditional covariance consistency from exact derivative linearization and linearized coordinate and coordinate-sum fourth-moment premises.

This is the coordinate-fourth-moment counterpart of the norm-fourth covariance route: exact linearization rewrites the smooth statistic’s coordinate and coordinate-sum fourth moments to the derivative-linearized statistic before the covariance fourth-moment constructor is applied.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.bootstrapCovarianceMat Pstar fun n ω ωs => (thetaStar n ω ωs).ofLp)
                            Filter.atTop fun x =>
                            Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:5909

theorem HansenEconometrics.chapter10_smooth_bootstrap_variance_consistency_of_linearization_normFourthMoment

Hansen Theorem 10.10, smooth-function variance consistency from exact derivative linearization and a norm fourth-moment premise on the underlying bootstrap statistic.

The norm fourth moment of T dominates each coordinate squared tail of G T; exact linearization transfers that tail control to the smooth statistic coordinate. This removes the coordinate-specific fourth-moment premise while leaving the genuinely nonlinear Taylor remainder step explicit.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {B : Real} (a : r),
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          MeasureTheory.MemLp (fun z => z.ofLp a) 2
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              Real.instLE.le 0 B →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                    Filter.atTop fun x => B) →
                  (∀ (n : Nat) (ω : Ω),
                      MeasureTheory.Integrable
                        (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4) (Pstar n ω)) →
                    MeasureTheory.TendstoInMeasure μ
                      (HansenEconometrics.bootstrapVarianceReal Pstar fun n ω ωs => (thetaStar n ω ωs).ofLp a)
                      Filter.atTop fun x =>
                      instHSub.hSub
                        (MeasureTheory.integral
                          (ProbabilityTheory.multivariateGaussian 0
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                          fun z => instHPow.hPow (z.ofLp a) 2)
                        (instHPow.hPow
                          (MeasureTheory.integral
                            (ProbabilityTheory.multivariateGaussian 0
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                            fun z => z.ofLp a)
                          2)

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Variance.lean:7618

theorem HansenEconometrics.chapter10_smooth_bootstrap_covarianceMat_tendsto_of_linearization_normFourthMoment

Hansen Theorem 10.10/10.12 smooth-function conditional covariance consistency from exact derivative linearization and a norm fourth-moment premise on the underlying bootstrap statistic.

The norm fourth-moment premise supplies the coordinate and coordinate-sum uniform-square-tail controls needed by the covariance-matrix version of Theorem 10.9; the exact linearization supplies the smooth Gaussian weak limit from Theorem 10.7.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {B : Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              Real.instLE.le 0 B →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                    Filter.atTop fun x => B) →
                  (∀ (n : Nat) (ω : Ω),
                      MeasureTheory.Integrable
                        (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4) (Pstar n ω)) →
                    MeasureTheory.TendstoInMeasure μ
                      (HansenEconometrics.bootstrapCovarianceMat Pstar fun n ω ωs => (thetaStar n ω ωs).ofLp)
                      Filter.atTop fun x =>
                      Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:5509

theorem HansenEconometrics.chapter10_smooth_bootstrap_variance_consistency_of_linearization_uniformSquareTail

Hansen Theorem 10.10, smooth-function variance consistency from exact derivative linearization and the Theorem 10.9 uniform-square-tail premise.

The exact linearization supplies the smooth Gaussian weak limit from Hansen Theorem 10.7. The scalar coordinate tail premise is still stated on the smooth statistic itself, so the model-specific Taylor/Rosenthal step remains explicit.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  (a : r),
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          MeasureTheory.MemLp (fun z => z.ofLp a) 2
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose)) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (HansenEconometrics.BootstrapUniformSquareTail μ Pstar (fun n ω ωs => (thetaStar n ω ωs).ofLp a)
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                  fun z => z.ofLp a) →
                MeasureTheory.TendstoInMeasure μ
                  (HansenEconometrics.bootstrapVarianceReal Pstar fun n ω ωs => (thetaStar n ω ωs).ofLp a) Filter.atTop
                  fun x =>
                  instHSub.hSub
                    (MeasureTheory.integral
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                      fun z => instHPow.hPow (z.ofLp a) 2)
                    (instHPow.hPow
                      (MeasureTheory.integral
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))
                        fun z => z.ofLp a)
                      2)

Direct statement dependencies (4)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Variance.lean:7207

Theorem 10.116 of 40 linked endpoints

For fixed n, the finite-replication trimmed covariance satisfies \hat V_{B,n}^{*,\tau}\xrightarrow{p^*}\operatorname{Var}^*(Z_n^{*,\tau}) as B\to\infty.

theorem HansenEconometrics.chapter10_finiteReplicationVariance_tendsto_of_l2_error_bounds

Hansen Theorem 10.11, finite-replication variance from bounded-trimmed L² WLLN bounds.

The displayed C / B mean-square bounds are the probability-theory premises supplied by the bounded trimmed bootstrap WLLN. This wrapper turns those bounds into the mean and second-moment convergence premises needed by chapter10_finiteReplicationVariance_tendsto_of_moments.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {Z : Nat → Nat → Ω → Real} {m m₂ Cmean Csecond : Real},
  (∀ (B : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal Z B ω) m)) 2)
        μ) →
    Filter.Eventually
        (fun B =>
          Real.instLE.le
            (MeasureTheory.integral μ fun ω =>
              instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal Z B ω) m)) 2)
            (instHDiv.hDiv Cmean B.cast))
        Filter.atTop →
      (∀ (B : Nat),
          MeasureTheory.Integrable
            (fun ω =>
              instHPow.hPow
                (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationSecondMomentReal Z B ω) m₂)) 2)
            μ) →
        Filter.Eventually
            (fun B =>
              Real.instLE.le
                (MeasureTheory.integral μ fun ω =>
                  instHPow.hPow
                    (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationSecondMomentReal Z B ω) m₂)) 2)
                (instHDiv.hDiv Csecond B.cast))
            Filter.atTop →
          MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Z) Filter.atTop
            fun x => instHSub.hSub m₂ (instHPow.hPow m 2)

Direct statement dependencies (3)

HansenEconometrics.finiteReplicationMeanReal
HansenEconometrics.finiteReplicationSecondMomentReal
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:480

theorem HansenEconometrics.chapter10_finiteReplicationVarianceCenteredReal_tendsto_of_l2_error_bounds

Hansen Theorem 10.11, centered finite-replication variance from bounded-trimmed L² WLLN bounds.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {Z : Nat → Nat → Ω → Real} {m m₂ Cmean Csecond : Real},
  (∀ (B : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal Z B ω) m)) 2)
        μ) →
    Filter.Eventually
        (fun B =>
          Real.instLE.le
            (MeasureTheory.integral μ fun ω =>
              instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal Z B ω) m)) 2)
            (instHDiv.hDiv Cmean B.cast))
        Filter.atTop →
      (∀ (B : Nat),
          MeasureTheory.Integrable
            (fun ω =>
              instHPow.hPow
                (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationSecondMomentReal Z B ω) m₂)) 2)
            μ) →
        Filter.Eventually
            (fun B =>
              Real.instLE.le
                (MeasureTheory.integral μ fun ω =>
                  instHPow.hPow
                    (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationSecondMomentReal Z B ω) m₂)) 2)
                (instHDiv.hDiv Csecond B.cast))
            Filter.atTop →
          MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Z) Filter.atTop
            fun x => instHSub.hSub m₂ (instHPow.hPow m 2)

Direct statement dependencies (3)

HansenEconometrics.finiteReplicationMeanReal
HansenEconometrics.finiteReplicationSecondMomentReal
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:537

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredMat_tendsto_of_l2_error_bounds

Hansen Theorem 10.11, centered finite-dimensional covariance from bounded-trimmed coordinatewise L² WLLN bounds.

This is the theorem-facing constructor for the finite-replication trimmed bootstrap covariance estimator: once bounded trimmed replications supply O(B⁻¹) mean-square errors for coordinate means and cross moments, the centered finite-replication covariance matrix converges to M₂ - m m’.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7}
  [inst : Fintype k] {Z : Nat → Nat → Ω → k → Real} {m : k → Real} {M₂ : Matrix k k Real} {Cmean : k → Real}
  {Ccross : k → k → Real},
  (∀ (a : k) (B : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanVec Z B ω a) (m a))) 2)
        μ) →
    (∀ (a : k),
        Filter.Eventually
          (fun B =>
            Real.instLE.le
              (MeasureTheory.integral μ fun ω =>
                instHPow.hPow
                  (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanVec Z B ω a) (m a))) 2)
              (instHDiv.hDiv (Cmean a) B.cast))
          Filter.atTop) →
      (∀ (a c : k) (B : Nat),
          MeasureTheory.Integrable
            (fun ω =>
              instHPow.hPow
                (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationCrossMomentMat Z B ω a c) (M₂ a c)))
                2)
            μ) →
        (∀ (a c : k),
            Filter.Eventually
              (fun B =>
                Real.instLE.le
                  (MeasureTheory.integral μ fun ω =>
                    instHPow.hPow
                      (Real.norm.norm
                        (instHSub.hSub (HansenEconometrics.finiteReplicationCrossMomentMat Z B ω a c) (M₂ a c)))
                      2)
                  (instHDiv.hDiv (Ccross a c) B.cast))
              Filter.atTop) →
          MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredMat Z) Filter.atTop
            fun x a c => instHSub.hSub (M₂ a c) (instHMul.hMul (m a) (m c))

Direct statement dependencies (3)

HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.finiteReplicationCrossMomentMat
HansenEconometrics.finiteReplicationMeanVec

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:6206

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredReal_tendsto_of_l2_error_bounds

Hansen Theorem 10.11, centered real finite-replication covariance from bounded-trimmed L² WLLN bounds.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {X Y : Nat → Nat → Ω → Real} {mX mY mXY CX CY CXY : Real},
  (∀ (B : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal X B ω) mX)) 2)
        μ) →
    Filter.Eventually
        (fun B =>
          Real.instLE.le
            (MeasureTheory.integral μ fun ω =>
              instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal X B ω) mX)) 2)
            (instHDiv.hDiv CX B.cast))
        Filter.atTop →
      (∀ (B : Nat),
          MeasureTheory.Integrable
            (fun ω =>
              instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal Y B ω) mY)) 2)
            μ) →
        Filter.Eventually
            (fun B =>
              Real.instLE.le
                (MeasureTheory.integral μ fun ω =>
                  instHPow.hPow (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationMeanReal Y B ω) mY))
                    2)
                (instHDiv.hDiv CY B.cast))
            Filter.atTop →
          (∀ (B : Nat),
              MeasureTheory.Integrable
                (fun ω =>
                  instHPow.hPow
                    (Real.norm.norm (instHSub.hSub (HansenEconometrics.finiteReplicationCrossMomentReal X Y B ω) mXY))
                    2)
                μ) →
            Filter.Eventually
                (fun B =>
                  Real.instLE.le
                    (MeasureTheory.integral μ fun ω =>
                      instHPow.hPow
                        (Real.norm.norm
                          (instHSub.hSub (HansenEconometrics.finiteReplicationCrossMomentReal X Y B ω) mXY))
                        2)
                    (instHDiv.hDiv CXY B.cast))
                Filter.atTop →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredReal X Y)
                Filter.atTop fun x => instHSub.hSub mXY (instHMul.hMul mX mY)

Direct statement dependencies (3)

HansenEconometrics.finiteReplicationCovarianceCenteredReal
HansenEconometrics.finiteReplicationCrossMomentReal
HansenEconometrics.finiteReplicationMeanReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:6133

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredMat_tendsto_of_trimmed_l2_simulation_error

Hansen Theorem 10.11/10.12 finite-replication trimmed covariance from coordinatewise L² simulation-error bounds.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7} [inst : Fintype k] {Zsim : Nat → Nat → Ω → k → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real} {τ : Nat → Real} {V : Matrix k k Real}
  {Cfinite : k → k → Real},
  (∀ (a c : k) (n : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow
            (Real.norm.norm
              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar Zstar τ n ω) a c))
            2)
        μ) →
    (∀ (a c : k),
        Filter.Eventually
          (fun n =>
            Real.instLE.le
              (MeasureTheory.integral μ fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                      (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar Zstar τ n ω) a c))
                  2)
              (instHDiv.hDiv (Cfinite a c) n.cast))
          Filter.atTop) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar Zstar τ) Filter.atTop
          fun x => V) →
        MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop
          fun x => V

Direct statement dependencies (2)

HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:12355

theorem HansenEconometrics.chapter10_finiteReplicationVariance_tendsto_of_moments

Hansen Theorem 10.11, finite-replication variance moment bridge.

If the finite-B replication mean and second moment converge in probability to their conditional limits, then the moment-form finite-replication variance converges in probability to m₂ - m². In applications, the moment premises are the bootstrap WLLN for bounded trimmed replications.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Z : Nat → Nat → Ω → Real} {m m₂ : Real},
  (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationMeanReal Z) Filter.atTop fun x => m) →
    (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationSecondMomentReal Z) Filter.atTop fun x =>
        m₂) →
      MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Z) Filter.atTop fun x =>
        instHSub.hSub m₂ (instHPow.hPow m 2)

Direct statement dependencies (3)

HansenEconometrics.finiteReplicationMeanReal
HansenEconometrics.finiteReplicationSecondMomentReal
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:409

Theorem 10.11 indexed scalar variance support6 of 10 linked endpoints

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVarianceCenteredReal_tendsto_of_bootstrap_zero_mean_moments

Indexed zero-mean finite-replication centered-variance wrapper for Hansen Theorem 10.11.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → Real} {σ2 : Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar) Filter.atTop fun x =>
          0) →
        (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar) Filter.atTop
            fun x => σ2) →
          (MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                  (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω))
              Filter.atTop fun x => 0) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim)
              Filter.atTop fun x => σ2

Direct statement dependencies (4)

HansenEconometrics.bootstrapMeanRealIndexed
HansenEconometrics.bootstrapSecondMomentRealIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2896

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVariance_tendsto_of_bootstrap_moments

Indexed Hansen Theorem 10.9/10.11 finite-replication variance from conditional bootstrap moment convergence.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → Real} {m m₂ : Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar) Filter.atTop fun x =>
          m) →
        (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar) Filter.atTop
            fun x => m₂) →
          (MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                  (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω))
              Filter.atTop fun x => 0) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim) Filter.atTop
              fun x => instHSub.hSub m₂ (instHPow.hPow m 2)

Direct statement dependencies (4)

HansenEconometrics.bootstrapMeanRealIndexed
HansenEconometrics.bootstrapSecondMomentRealIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2348

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVariance_tendsto_of_l2_simulation_error

Indexed Hansen Theorem 10.9/10.11 finite-replication variance from an L² simulation-error bound.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {Ωboot : Nat → Type u_7} [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → Real}
  {σ2 Cfinite : Real},
  (∀ (n : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow
            (Real.norm.norm
              (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω)))
            2)
        μ) →
    Filter.Eventually
        (fun n =>
          Real.instLE.le
            (MeasureTheory.integral μ fun ω =>
              instHPow.hPow
                (Real.norm.norm
                  (instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                    (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω)))
                2)
            (instHDiv.hDiv Cfinite n.cast))
        Filter.atTop →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar) Filter.atTop
          fun x => σ2) →
        MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim) Filter.atTop
          fun x => σ2

Direct statement dependencies (2)

HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2313

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVariance_tendsto_of_bootstrap_zero_mean_moments

Indexed zero-mean finite-replication variance wrapper for Hansen Theorem 10.11.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → Real} {σ2 : Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar) Filter.atTop fun x =>
          0) →
        (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar) Filter.atTop
            fun x => σ2) →
          (MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                  (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω))
              Filter.atTop fun x => 0) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim) Filter.atTop
              fun x => σ2

Direct statement dependencies (4)

HansenEconometrics.bootstrapMeanRealIndexed
HansenEconometrics.bootstrapSecondMomentRealIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2378

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVarianceCenteredReal_tendsto_of_bootstrap_moments

Indexed Hansen Theorem 10.9/10.11 centered finite-replication variance from conditional bootstrap moment convergence.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → Real} {m m₂ : Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanRealIndexed Pstar Zstar) Filter.atTop fun x =>
          m) →
        (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapSecondMomentRealIndexed Pstar Zstar) Filter.atTop
            fun x => m₂) →
          (MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                  (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω))
              Filter.atTop fun x => 0) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim)
              Filter.atTop fun x => instHSub.hSub m₂ (instHPow.hPow m 2)

Direct statement dependencies (4)

HansenEconometrics.bootstrapMeanRealIndexed
HansenEconometrics.bootstrapSecondMomentRealIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2866

theorem HansenEconometrics.chapter10_indexed_finiteReplicationVarianceCenteredReal_tendsto_of_uniformSquareTail

Indexed Hansen Theorem 10.9/10.11 centered finite-replication variance from bootstrap weak convergence and a named uniform-square-tail condition.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar Zstar ν Z →
            (MeasureTheory.TendstoInMeasure μ
                (fun n ω =>
                  instHSub.hSub (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim n ω)
                    (HansenEconometrics.bootstrapVarianceRealIndexed Pstar Zstar n ω))
                Filter.atTop fun x => 0) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceCenteredReal Zsim)
                Filter.atTop fun x =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                  (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (4)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.finiteReplicationVarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:2954

Theorem 10.11 indexed covariance support6 of 25 linked endpoints

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceCenteredMat_tendsto_of_trimmed_l2_simulation_error

Indexed Hansen Theorem 10.11/10.12 finite-replication trimmed covariance from coordinatewise L² simulation-error bounds.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7}
  [inst : Fintype k] {Ωboot : Nat → Type u_8} [inst_1 : (n : Nat) → MeasurableSpace (Ωboot n)]
  {Zsim : Nat → Nat → Ω → k → Real} {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → k → Real} {τ : Nat → Real} {V : Matrix k k Real} {Cfinite : k → k → Real},
  (∀ (a c : k) (n : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow
            (Real.norm.norm
              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar Zstar τ n ω) a c))
            2)
        μ) →
    (∀ (a c : k),
        Filter.Eventually
          (fun n =>
            Real.instLE.le
              (MeasureTheory.integral μ fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                      (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar Zstar τ n ω) a c))
                  2)
              (instHDiv.hDiv (Cfinite a c) n.cast))
          Filter.atTop) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar Zstar τ)
          Filter.atTop fun x => V) →
        MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop
          fun x => V

Direct statement dependencies (2)

HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:12415

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceCenteredReal_tendsto_of_bootstrap_zero_mean_moments

Indexed zero-mean scalar finite-replication covariance wrapper for Hansen Theorem 10.11.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] {Xsim Ysim : Nat → Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Xstar Ystar : (n : Nat) → Ω → Ωboot n → Real} {v : Real},
  (MeasureTheory.TendstoInMeasure μ (fun n ω => MeasureTheory.integral (Pstar n ω) fun x => Xstar n ω x) Filter.atTop
      fun x => 0) →
    (MeasureTheory.TendstoInMeasure μ (fun n ω => MeasureTheory.integral (Pstar n ω) fun x => Ystar n ω x) Filter.atTop
        fun x => 0) →
      (MeasureTheory.TendstoInMeasure μ
          (fun n ω =>
            MeasureTheory.integral (Pstar n ω) fun x => (fun ωs => instHMul.hMul (Xstar n ω ωs) (Ystar n ω ωs)) x)
          Filter.atTop fun x => v) →
        (MeasureTheory.TendstoInMeasure μ
            (fun n ω =>
              instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredReal Xsim Ysim n ω)
                (instHSub.hSub
                  (MeasureTheory.integral (Pstar n ω) fun x =>
                    (fun ωs => instHMul.hMul (Xstar n ω ωs) (Ystar n ω ωs)) x)
                  (instHMul.hMul (MeasureTheory.integral (Pstar n ω) fun x => Xstar n ω x)
                    (MeasureTheory.integral (Pstar n ω) fun x => Ystar n ω x))))
            Filter.atTop fun x => 0) →
          MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredReal Xsim Ysim)
            Filter.atTop fun x => v

Direct statement dependencies (1)

HansenEconometrics.finiteReplicationCovarianceCenteredReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:12176

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_tendsto_of_bootstrap_moments

Indexed Hansen Theorem 10.9/10.11 finite-replication covariance matrix from conditional bootstrap moment convergence.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {k : Type u_7} [inst : Fintype k]
  {Ωboot : Nat → Type u_8} [inst_1 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → k → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real} {m : k → Real}
  {M₂ : Matrix k k Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanVecIndexed Pstar Zstar) Filter.atTop fun x =>
          m) →
        (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCrossMomentMatIndexed Pstar Zstar) Filter.atTop
            fun x => M₂) →
          (MeasureTheory.TendstoInMeasure μ
              (fun n ω =>
                instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                  (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar n ω))
              Filter.atTop fun x => 0) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop
              fun x a c => instHSub.hSub (M₂ a c) (instHMul.hMul (m a) (m c))

Direct statement dependencies (4)

HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.bootstrapCrossMomentMatIndexed
HansenEconometrics.bootstrapMeanVecIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:6359

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceCenteredMat_tendsto_of_weak_tail

Indexed Hansen Theorem 10.9/10.11 centered finite-replication covariance from bootstrap weak convergence and indexed uniform-square-tail controls.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} {k : Type u_7} [inst : Fintype k] [MeasureTheory.IsFiniteMeasure ν]
  {Ωboot : Nat → Type u_8} [inst_2 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → k → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {Z : Ωlim → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (∀ (a : k), MeasureTheory.MemLp (fun ωlim => Z ωlim a) 2 ν) →
        HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Zstar ν Z →
          (∀ (a : k),
              HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar (fun n ω ωs => Zstar n ω ωs a) ν fun ωlim =>
                Z ωlim a) →
            (∀ (a c : k),
                HansenEconometrics.BootstrapUniformSquareTailIndexed μ Pstar
                  (fun n ω ωs => instHAdd.hAdd (Zstar n ω ωs a) (Zstar n ω ωs c)) ν fun ωlim =>
                  instHAdd.hAdd (Z ωlim a) (Z ωlim c)) →
              (MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                      (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar n ω))
                  Filter.atTop fun x => 0) →
                MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim)
                  Filter.atTop fun x a c =>
                  instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHMul.hMul (Z ωlim a) (Z ωlim c))
                    (instHMul.hMul (MeasureTheory.integral ν fun ωlim => Z ωlim a)
                      (MeasureTheory.integral ν fun ωlim => Z ωlim c))

Direct statement dependencies (4)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceCenteredMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:11387

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_tendsto_of_l2_simulation_error

Indexed Hansen Theorem 10.9/10.11 finite-replication covariance matrix from coordinatewise L² simulation-error bounds.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7}
  [inst : Fintype k] {Ωboot : Nat → Type u_8} [inst_1 : (n : Nat) → MeasurableSpace (Ωboot n)]
  {Zsim : Nat → Nat → Ω → k → Real} {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Zstar : (n : Nat) → Ω → Ωboot n → k → Real} {V : Matrix k k Real} {Cfinite : k → k → Real},
  (∀ (a c : k) (n : Nat),
      MeasureTheory.Integrable
        (fun ω =>
          instHPow.hPow
            (Real.norm.norm
              (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar n ω) a c))
            2)
        μ) →
    (∀ (a c : k),
        Filter.Eventually
          (fun n =>
            Real.instLE.le
              (MeasureTheory.integral μ fun ω =>
                instHPow.hPow
                  (Real.norm.norm
                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                      (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar n ω) a c))
                  2)
              (instHDiv.hDiv (Cfinite a c) n.cast))
          Filter.atTop) →
      (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar) Filter.atTop
          fun x => V) →
        MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop
          fun x => V

Direct statement dependencies (2)

HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:6322

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_tendsto_of_bootstrap_covariance

Indexed Hansen Theorem 10.9/10.11 finite-replication covariance matrix from conditional bootstrap covariance consistency.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {k : Type u_7} [inst : Fintype k]
  {Ωboot : Nat → Type u_8} [inst_1 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → k → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {V : Matrix k k Real},
  (MeasureTheory.TendstoInMeasure μ
      (fun n ω =>
        instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
          (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar n ω))
      Filter.atTop fun x => 0) →
    (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar Zstar) Filter.atTop
        fun x => V) →
      MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop
        fun x => V

Direct statement dependencies (2)

HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:6299

Theorem 10.126 of 20 linked endpoints

As n\to\infty, the trimmed conditional covariance satisfies \operatorname{Var}^*(Z_n^{*,\tau})\xrightarrow{p}V_\theta.

theorem HansenEconometrics.chapter10_trimmedBootstrapVariance_tendsto_of_moments

Hansen Theorem 10.12, trimmed conditional covariance moment bridge.

For the trimmed statistic Z** = Z* 1{‖Z*‖ ≤ τ}, convergence of its conditional mean vector and cross-moment matrix implies convergence of its conditional covariance matrix. The smooth-model proof of Theorem 10.12 supplies these moment premises by showing the trimming is asymptotically negligible and the trimmed sequence is uniformly square integrable.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} [inst : Fintype k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real}
  {τ : Nat → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k),
        MeasureTheory.MemLp (fun ωs => HansenEconometrics.trimmedBootstrapStatistic Zstar τ n ω ωs a) 2 (Pstar n ω)) →
      ∀ {m : k → Real} {M₂ : Matrix k k Real},
        (MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapMeanVec Pstar (HansenEconometrics.trimmedBootstrapStatistic Zstar τ))
            Filter.atTop fun x => m) →
          (MeasureTheory.TendstoInMeasure μ
              (HansenEconometrics.bootstrapCrossMomentMat Pstar (HansenEconometrics.trimmedBootstrapStatistic Zstar τ))
              Filter.atTop fun x => M₂) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar Zstar τ)
              Filter.atTop fun x a c => instHSub.hSub (M₂ a c) (instHMul.hMul (m a) (m c))

Direct statement dependencies (4)

HansenEconometrics.bootstrapCrossMomentMat
HansenEconometrics.bootstrapMeanVec
HansenEconometrics.trimmedBootstrapCovarianceMat
HansenEconometrics.trimmedBootstrapStatistic

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:8338

theorem HansenEconometrics.chapter10_trimmedBootstrapVariance_tendsto

Theorem 10.12 zero-mean covariance specialization.

In the asymptotically centered case, if the trimmed conditional mean converges to zero and the trimmed conditional cross moment converges to V, then the trimmed conditional covariance converges to V.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} [inst : Fintype k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real}
  {τ : Nat → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k),
        MeasureTheory.MemLp (fun ωs => HansenEconometrics.trimmedBootstrapStatistic Zstar τ n ω ωs a) 2 (Pstar n ω)) →
      ∀ {V : Matrix k k Real},
        (MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapMeanVec Pstar (HansenEconometrics.trimmedBootstrapStatistic Zstar τ))
            Filter.atTop fun x => 0) →
          (MeasureTheory.TendstoInMeasure μ
              (HansenEconometrics.bootstrapCrossMomentMat Pstar (HansenEconometrics.trimmedBootstrapStatistic Zstar τ))
              Filter.atTop fun x => V) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar Zstar τ)
              Filter.atTop fun x => V

Direct statement dependencies (4)

HansenEconometrics.bootstrapCrossMomentMat
HansenEconometrics.bootstrapMeanVec
HansenEconometrics.trimmedBootstrapCovarianceMat
HansenEconometrics.trimmedBootstrapStatistic

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:8369

theorem HansenEconometrics.chapter10_indexed_trimmedBootstrapVariance_tendsto

Indexed Theorem 10.12 zero-mean covariance specialization.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {k : Type u_7} [inst : Fintype k]
  {Ωboot : Nat → Type u_8} [inst_1 : (n : Nat) → MeasurableSpace (Ωboot n)]
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {τ : Nat → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k),
        MeasureTheory.MemLp (fun ωs => HansenEconometrics.trimmedBootstrapStatisticIndexed Zstar τ n ω ωs a) 2
          (Pstar n ω)) →
      ∀ {V : Matrix k k Real},
        (MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapMeanVecIndexed Pstar
              (HansenEconometrics.trimmedBootstrapStatisticIndexed Zstar τ))
            Filter.atTop fun x => 0) →
          (MeasureTheory.TendstoInMeasure μ
              (HansenEconometrics.bootstrapCrossMomentMatIndexed Pstar
                (HansenEconometrics.trimmedBootstrapStatisticIndexed Zstar τ))
              Filter.atTop fun x => V) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar Zstar τ)
              Filter.atTop fun x => V

Direct statement dependencies (4)

HansenEconometrics.bootstrapCrossMomentMatIndexed
HansenEconometrics.bootstrapMeanVecIndexed
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed
HansenEconometrics.trimmedBootstrapStatisticIndexed

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:8429

theorem HansenEconometrics.chapter10_indexed_trimmedBootstrapVariance_tendsto_of_moments

Indexed Hansen Theorem 10.12, trimmed conditional covariance moment bridge.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {k : Type u_7} [inst : Fintype k]
  {Ωboot : Nat → Type u_8} [inst_1 : (n : Nat) → MeasurableSpace (Ωboot n)]
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Zstar : (n : Nat) → Ω → Ωboot n → k → Real}
  {τ : Nat → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k),
        MeasureTheory.MemLp (fun ωs => HansenEconometrics.trimmedBootstrapStatisticIndexed Zstar τ n ω ωs a) 2
          (Pstar n ω)) →
      ∀ {m : k → Real} {M₂ : Matrix k k Real},
        (MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapMeanVecIndexed Pstar
              (HansenEconometrics.trimmedBootstrapStatisticIndexed Zstar τ))
            Filter.atTop fun x => m) →
          (MeasureTheory.TendstoInMeasure μ
              (HansenEconometrics.bootstrapCrossMomentMatIndexed Pstar
                (HansenEconometrics.trimmedBootstrapStatisticIndexed Zstar τ))
              Filter.atTop fun x => M₂) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar Zstar τ)
              Filter.atTop fun x a c => instHSub.hSub (M₂ a c) (instHMul.hMul (m a) (m c))

Direct statement dependencies (4)

HansenEconometrics.bootstrapCrossMomentMatIndexed
HansenEconometrics.bootstrapMeanVecIndexed
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed
HansenEconometrics.trimmedBootstrapStatisticIndexed

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:8397

theorem HansenEconometrics.chapter10_bootstrap_covarianceMat_tendsto_of_zero_mean_moments

Zero-mean conditional bootstrap covariance-matrix bridge, stated for cov.

This is the Theorem 10.12/10.19 covariance target in the asymptotically centered case: zero conditional means plus cross-moment convergence imply conditional bootstrap covariance convergence to V.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} [inst : Fintype k] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      ∀ {V : Matrix k k Real},
        (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanVec Pstar Zstar) Filter.atTop fun x => 0) →
          (MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCrossMomentMat Pstar Zstar) Filter.atTop
              fun x => V) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapCovarianceMat Pstar Zstar) Filter.atTop
              fun x => V

Direct statement dependencies (3)

HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.bootstrapCrossMomentMat
HansenEconometrics.bootstrapMeanVec

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:3409

theorem HansenEconometrics.chapter10_bootstrap_meanVec_tendsto_of_weak_distribution_of_uniformSquareTail

Hansen Theorem 10.9 finite-dimensional mean-vector wrapper.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim} [inst : Fintype k]
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → k → Real}
  {Z : Ωlim → k → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : k), MeasureTheory.MemLp (fun ωs => Zstar n ω ωs a) 2 (Pstar n ω)) →
      (∀ (a : k), MeasureTheory.MemLp (fun ωlim => Z ωlim a) 2 ν) →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          (∀ (a : k),
              HansenEconometrics.BootstrapUniformSquareTail μ Pstar (fun n ω ωs => Zstar n ω ωs a) ν fun ωlim =>
                Z ωlim a) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapMeanVec Pstar Zstar) Filter.atTop fun x a =>
              MeasureTheory.integral ν fun ωlim => Z ωlim a

Direct statement dependencies (3)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapMeanVec

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:2893

Theorem 10.11 smooth covariance coordinate-fourth-moment support4 endpoints

For fixed n, the finite-replication trimmed covariance satisfies \hat V_{B,n}^{*,\tau}\xrightarrow{p^*}\operatorname{Var}^*(Z_n^{*,\tau}) as B\to\infty. Smooth exact-linearization coordinate and coordinate-sum fourth-moment covariance consistency now feeds Hansen’s finite-replication covariance estimators with direct simulation-error or coordinatewise L² simulation-error premises

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceMat_tendsto_of_smooth_fourthMoment

Hansen Theorem 10.10/10.11 finite-replication covariance matrix for a smooth function under exact derivative linearization and coordinate plus coordinate-sum fourth-moment premises.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d} {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r}
  {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (MeasureTheory.TendstoInMeasure μ
                              (fun n ω =>
                                instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                  (HansenEconometrics.bootstrapCovarianceMat Pstar
                                    (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω))
                              Filter.atTop fun x => 0) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:7415

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredMat_tendsto_of_smooth_fourthMoment

Textbook-centered finite-replication covariance version of the smooth coordinate-fourth-moment finite-replication covariance bridge.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d} {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r}
  {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (MeasureTheory.TendstoInMeasure μ
                              (fun n ω =>
                                instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                  (HansenEconometrics.bootstrapCovarianceMat Pstar
                                    (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω))
                              Filter.atTop fun x => 0) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:7552

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceMat_tendsto_of_smooth_fourthMoment_l2

L² simulation-error version of the smooth coordinate-fourth-moment finite-replication covariance matrix bridge.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r]
  [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (∀ (a c : r) (n : Nat),
                              MeasureTheory.Integrable
                                (fun ω =>
                                  instHPow.hPow
                                    (Real.norm.norm
                                      (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                        (HansenEconometrics.bootstrapCovarianceMat Pstar
                                          (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                        a c))
                                    2)
                                μ) →
                            (∀ (a c : r),
                                Filter.Eventually
                                  (fun n =>
                                    Real.instLE.le
                                      (MeasureTheory.integral μ fun ω =>
                                        instHPow.hPow
                                          (Real.norm.norm
                                            (instHSub.hSub
                                              (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                              (HansenEconometrics.bootstrapCovarianceMat Pstar
                                                (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                              a c))
                                          2)
                                      (instHDiv.hDiv (Cfinite a c) n.cast))
                                  Filter.atTop) →
                              MeasureTheory.TendstoInMeasure μ
                                (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                                Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:8283

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredMat_tendsto_of_smooth_fourthMoment_l2

Textbook-centered L² simulation-error version of the smooth coordinate-fourth-moment finite-replication covariance bridge.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r]
  [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (∀ (a c : r) (n : Nat),
                              MeasureTheory.Integrable
                                (fun ω =>
                                  instHPow.hPow
                                    (Real.norm.norm
                                      (instHSub.hSub
                                        (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                        (HansenEconometrics.bootstrapCovarianceMat Pstar
                                          (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                        a c))
                                    2)
                                μ) →
                            (∀ (a c : r),
                                Filter.Eventually
                                  (fun n =>
                                    Real.instLE.le
                                      (MeasureTheory.integral μ fun ω =>
                                        instHPow.hPow
                                          (Real.norm.norm
                                            (instHSub.hSub
                                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                              (HansenEconometrics.bootstrapCovarianceMat Pstar
                                                (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                              a c))
                                          2)
                                      (instHDiv.hDiv (Cfinite a c) n.cast))
                                  Filter.atTop) →
                              MeasureTheory.TendstoInMeasure μ
                                (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                                Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:8442

Theorem 10.11 indexed smooth covariance coordinate-fourth-moment support4 endpoints

For fixed n, the finite-replication trimmed covariance satisfies \hat V_{B,n}^{*,\tau}\xrightarrow{p^*}\operatorname{Var}^*(Z_n^{*,\tau}) as B\to\infty. Sample-size-dependent smooth exact-linearization coordinate and coordinate-sum fourth-moment covariance consistency now feeds Hansen’s indexed finite-replication covariance estimators with direct simulation-error or coordinatewise L² simulation-error premises

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_tendsto_of_smooth_fourthMoment

Indexed finite-replication covariance matrix for a smooth function under exact derivative linearization and coordinate plus coordinate-sum fourth-moment premises.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} [inst : Fintype d]
  [inst_1 : Fintype r] [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
          (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (MeasureTheory.TendstoInMeasure μ
                              (fun n ω =>
                                instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                  (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                    (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω))
                              Filter.atTop fun x => 0) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:8827

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceCenteredMat_tendsto_of_smooth_fourthMoment

Indexed textbook-centered finite-replication covariance version of the smooth coordinate-fourth-moment finite-replication covariance bridge.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} [inst : Fintype d]
  [inst_1 : Fintype r] [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
          (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (MeasureTheory.TendstoInMeasure μ
                              (fun n ω =>
                                instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                  (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                    (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω))
                              Filter.atTop fun x => 0) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:8965

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_tendsto_of_smooth_fourthMoment_l2

Indexed L² simulation-error version of the smooth coordinate-fourth-moment finite-replication covariance matrix bridge.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7}
  {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d] [inst_3 : DecidableEq r]
  {Ωboot : Nat → Type u_9} [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
          (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (∀ (a c : r) (n : Nat),
                              MeasureTheory.Integrable
                                (fun ω =>
                                  instHPow.hPow
                                    (Real.norm.norm
                                      (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                        (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                          (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                        a c))
                                    2)
                                μ) →
                            (∀ (a c : r),
                                Filter.Eventually
                                  (fun n =>
                                    Real.instLE.le
                                      (MeasureTheory.integral μ fun ω =>
                                        instHPow.hPow
                                          (Real.norm.norm
                                            (instHSub.hSub
                                              (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                              (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                                (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                              a c))
                                          2)
                                      (instHDiv.hDiv (Cfinite a c) n.cast))
                                  Filter.atTop) →
                              MeasureTheory.TendstoInMeasure μ
                                (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                                Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:9705

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceCenteredMat_tendsto_of_smooth_fourthMoment_l2

Indexed textbook-centered L² simulation-error version of the smooth coordinate-fourth-moment finite-replication covariance bridge.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7}
  {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d] [inst_3 : DecidableEq r]
  {Ωboot : Nat → Type u_9} [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
          (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (a : r),
              MeasureTheory.MemLp (fun z => z.ofLp a) 2
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose))) →
            (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
                Eq (thetaStar n ω ωs)
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
              (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
                (∀ (a : r),
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                      Filter.atTop fun x => Bcoord a) →
                  (∀ (n : Nat) (ω : Ω) (a : r),
                      MeasureTheory.Integrable
                        (fun ωs =>
                          instHPow.hPow
                            ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                  (Tstar n ω ωs)).ofLp
                              a)
                            4)
                        (Pstar n ω)) →
                    (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                      (∀ (a c : r),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                            Filter.atTop fun x => Bsum a c) →
                        (∀ (n : Nat) (ω : Ω) (a c : r),
                            MeasureTheory.Integrable
                              (fun ωs =>
                                instHPow.hPow
                                  (instHAdd.hAdd
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      a)
                                    ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                          (Tstar n ω ωs)).ofLp
                                      c))
                                  4)
                              (Pstar n ω)) →
                          (∀ (a c : r) (n : Nat),
                              MeasureTheory.Integrable
                                (fun ω =>
                                  instHPow.hPow
                                    (Real.norm.norm
                                      (instHSub.hSub
                                        (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                        (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                          (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                        a c))
                                    2)
                                μ) →
                            (∀ (a c : r),
                                Filter.Eventually
                                  (fun n =>
                                    Real.instLE.le
                                      (MeasureTheory.integral μ fun ω =>
                                        instHPow.hPow
                                          (Real.norm.norm
                                            (instHSub.hSub
                                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                              (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                                (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                              a c))
                                          2)
                                      (instHDiv.hDiv (Cfinite a c) n.cast))
                                  Filter.atTop) →
                              MeasureTheory.TendstoInMeasure μ
                                (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                                Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:9866

Theorem 10.11 smooth covariance fourth-moment Gaussian-limit wrappers6 of 8 linked endpoints

For fixed n, the finite-replication trimmed covariance satisfies \hat V_{B,n}^{*,\tau}\xrightarrow{p^*}\operatorname{Var}^*(Z_n^{*,\tau}) as B\to\infty. Smooth coordinate-fourth-moment finite-replication covariance routes no longer require callers to pass automatic Gaussian-limit coordinate MemLp premises

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceMat_smooth_fourthMoment_gaussianLimit

Smooth coordinate-fourth-moment finite-replication covariance bridge with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d} {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r}
  {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
              Eq (thetaStar n ω ωs)
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
            (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
              (∀ (a : r),
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => Bcoord a) →
                (∀ (n : Nat) (ω : Ω) (a : r),
                    MeasureTheory.Integrable
                      (fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                      (Pstar n ω)) →
                  (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                    (∀ (a c : r),
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                          Filter.atTop fun x => Bsum a c) →
                      (∀ (n : Nat) (ω : Ω) (a c : r),
                          MeasureTheory.Integrable
                            (fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                            (Pstar n ω)) →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMat Pstar (fun n ω ωs => (thetaStar n ω ωs).ofLp)
                                  n ω))
                            Filter.atTop fun x => 0) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                            Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:10654

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceMat_smooth_fourthMoment_gaussianLimit_l2

L² simulation-error smooth coordinate-fourth-moment finite-replication covariance bridge with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r]
  [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
              Eq (thetaStar n ω ωs)
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
            (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
              (∀ (a : r),
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => Bcoord a) →
                (∀ (n : Nat) (ω : Ω) (a : r),
                    MeasureTheory.Integrable
                      (fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                      (Pstar n ω)) →
                  (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                    (∀ (a c : r),
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                          Filter.atTop fun x => Bsum a c) →
                      (∀ (n : Nat) (ω : Ω) (a c : r),
                          MeasureTheory.Integrable
                            (fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                            (Pstar n ω)) →
                        (∀ (a c : r) (n : Nat),
                            MeasureTheory.Integrable
                              (fun ω =>
                                instHPow.hPow
                                  (Real.norm.norm
                                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                      (HansenEconometrics.bootstrapCovarianceMat Pstar
                                        (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                      a c))
                                  2)
                              μ) →
                          (∀ (a c : r),
                              Filter.Eventually
                                (fun n =>
                                  Real.instLE.le
                                    (MeasureTheory.integral μ fun ω =>
                                      instHPow.hPow
                                        (Real.norm.norm
                                          (instHSub.hSub
                                            (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                            (HansenEconometrics.bootstrapCovarianceMat Pstar
                                              (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                            a c))
                                        2)
                                    (instHDiv.hDiv (Cfinite a c) n.cast))
                                Filter.atTop) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:10816

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredMat_smooth_fourthMoment_gaussianLimit

Textbook-centered smooth coordinate-fourth-moment finite-replication covariance bridge with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d} {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r}
  {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
              Eq (thetaStar n ω ωs)
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
            (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
              (∀ (a : r),
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => Bcoord a) →
                (∀ (n : Nat) (ω : Ω) (a : r),
                    MeasureTheory.Integrable
                      (fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                      (Pstar n ω)) →
                  (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                    (∀ (a c : r),
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                          Filter.atTop fun x => Bsum a c) →
                      (∀ (n : Nat) (ω : Ω) (a c : r),
                          MeasureTheory.Integrable
                            (fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                            (Pstar n ω)) →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMat Pstar (fun n ω ωs => (thetaStar n ω ωs).ofLp)
                                  n ω))
                            Filter.atTop fun x => 0) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                            Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:10735

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_smooth_fourthMoment_gaussianLimit

Indexed smooth coordinate-fourth-moment finite-replication covariance bridge with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} [inst : Fintype d]
  [inst_1 : Fintype r] [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
          (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
              Eq (thetaStar n ω ωs)
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
            (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
              (∀ (a : r),
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => Bcoord a) →
                (∀ (n : Nat) (ω : Ω) (a : r),
                    MeasureTheory.Integrable
                      (fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                      (Pstar n ω)) →
                  (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                    (∀ (a c : r),
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                          Filter.atTop fun x => Bsum a c) →
                      (∀ (n : Nat) (ω : Ω) (a c : r),
                          MeasureTheory.Integrable
                            (fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                            (Pstar n ω)) →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                  (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω))
                            Filter.atTop fun x => 0) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                            Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:10998

theorem HansenEconometrics.chapter10_finiteReplicationCovarianceCenteredMat_smooth_fourthMoment_gaussianLimit_l2

Textbook-centered L² simulation-error smooth coordinate-fourth-moment finite-replication covariance bridge with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r]
  [inst_2 : DecidableEq d] [inst_3 : DecidableEq r] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar (ProbabilityTheory.multivariateGaussian 0 V)
          fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
              Eq (thetaStar n ω ωs)
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
            (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
              (∀ (a : r),
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => Bcoord a) →
                (∀ (n : Nat) (ω : Ω) (a : r),
                    MeasureTheory.Integrable
                      (fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                      (Pstar n ω)) →
                  (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                    (∀ (a c : r),
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                          Filter.atTop fun x => Bsum a c) →
                      (∀ (n : Nat) (ω : Ω) (a c : r),
                          MeasureTheory.Integrable
                            (fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                            (Pstar n ω)) →
                        (∀ (a c : r) (n : Nat),
                            MeasureTheory.Integrable
                              (fun ω =>
                                instHPow.hPow
                                  (Real.norm.norm
                                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                      (HansenEconometrics.bootstrapCovarianceMat Pstar
                                        (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                      a c))
                                  2)
                              μ) →
                          (∀ (a c : r),
                              Filter.Eventually
                                (fun n =>
                                  Real.instLE.le
                                    (MeasureTheory.integral μ fun ω =>
                                      instHPow.hPow
                                        (Real.norm.norm
                                          (instHSub.hSub
                                            (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                            (HansenEconometrics.bootstrapCovarianceMat Pstar
                                              (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                            a c))
                                        2)
                                    (instHDiv.hDiv (Cfinite a c) n.cast))
                                Filter.atTop) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapCovarianceMat
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:10907

theorem HansenEconometrics.chapter10_indexed_finiteReplicationCovarianceMat_smooth_fourthMoment_gaussianLimit_l2

Indexed L² simulation-error smooth coordinate-fourth-moment finite-replication covariance bridge with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {d : Type u_7}
  {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d] [inst_3 : DecidableEq r]
  {Ωboot : Nat → Type u_9} [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → r → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V)
          G.transpose))]
  {Bcoord : r → Real} {Bsum Cfinite : r → r → Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
          (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
        (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
              Eq (thetaStar n ω ωs)
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
            (∀ (a : r), Real.instLE.le 0 (Bcoord a)) →
              (∀ (a : r),
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      MeasureTheory.integral (Pstar n ω) fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                    Filter.atTop fun x => Bcoord a) →
                (∀ (n : Nat) (ω : Ω) (a : r),
                    MeasureTheory.Integrable
                      (fun ωs =>
                        instHPow.hPow
                          ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                (Tstar n ω ωs)).ofLp
                            a)
                          4)
                      (Pstar n ω)) →
                  (∀ (a c : r), Real.instLE.le 0 (Bsum a c)) →
                    (∀ (a c : r),
                        MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                          Filter.atTop fun x => Bsum a c) →
                      (∀ (n : Nat) (ω : Ω) (a c : r),
                          MeasureTheory.Integrable
                            (fun ωs =>
                              instHPow.hPow
                                (instHAdd.hAdd
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    a)
                                  ((ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                                        (Tstar n ω ωs)).ofLp
                                    c))
                                4)
                            (Pstar n ω)) →
                        (∀ (a c : r) (n : Nat),
                            MeasureTheory.Integrable
                              (fun ω =>
                                instHPow.hPow
                                  (Real.norm.norm
                                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                      (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                        (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                      a c))
                                  2)
                              μ) →
                          (∀ (a c : r),
                              Filter.Eventually
                                (fun n =>
                                  Real.instLE.le
                                    (MeasureTheory.integral μ fun ω =>
                                      instHPow.hPow
                                        (Real.norm.norm
                                          (instHSub.hSub
                                            (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim n ω)
                                            (HansenEconometrics.bootstrapCovarianceMatIndexed Pstar
                                              (fun n ω ωs => (thetaStar n ω ωs).ofLp) n ω)
                                            a c))
                                        2)
                                    (instHDiv.hDiv (Cfinite a c) n.cast))
                                Filter.atTop) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceMomentMat Zsim) Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapCovarianceMatIndexed
HansenEconometrics.finiteReplicationCovarianceMomentMat
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:11162

Theorem 10.12 smooth Gaussian-limit wrappers4 endpoints

As n\to\infty, the trimmed conditional covariance satisfies \operatorname{Var}^*(Z_n^{*,\tau})\xrightarrow{p}V_\theta. Smooth exact-linearization trimmed covariance routes no longer require callers to pass automatic Gaussian-limit coordinate MemLp side conditions

theorem HansenEconometrics.chapter10_smooth_trimmedVariance_linearization_normFourth_gaussianLimit

Smooth exact-linearization trimmed covariance route with Gaussian-limit coordinate MemLp 2 premises discharged automatically.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real) {τ : Nat → Real}
  {B : Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar
            (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable fun ωs => (thetaStar n ω ωs).ofLp) →
            (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                  Eq (thetaStar n ω ωs)
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                          (setOf fun ωs => Real.instLT.lt (τ n) (Pi.normedRing.norm (thetaStar n ω ωs).ofLp))).toReal)
                    Filter.atTop fun x => 0) →
                  Real.instLE.le 0 B →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                        Filter.atTop fun x => B) →
                      (∀ (n : Nat) (ω : Ω),
                          MeasureTheory.Integrable
                            (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                            (Pstar n ω)) →
                        MeasureTheory.TendstoInMeasure μ
                          (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                            (fun n ω ωs => (thetaStar n ω ωs).ofLp) τ)
                          Filter.atTop fun x =>
                          Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:9587

theorem HansenEconometrics.chapter10_indexed_smooth_trimmedVariance_linearization_normFourth_gaussianLimit

Indexed smooth exact-linearization trimmed covariance route with automatic Gaussian-limit coordinate MemLp 2 premises.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} [inst : Fintype d]
  [inst_1 : Fintype r] [inst_2 : DecidableEq d] {Ωboot : Nat → Type u_9}
  [inst_3 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  {τ : Nat → Real} {B : Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
            (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable fun ωs => (thetaStar n ω ωs).ofLp) →
            (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
                  Eq (thetaStar n ω ωs)
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G) (Tstar n ω ωs))) →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                          (setOf fun ωs => Real.instLT.lt (τ n) (Pi.normedRing.norm (thetaStar n ω ωs).ofLp))).toReal)
                    Filter.atTop fun x => 0) →
                  Real.instLE.le 0 B →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                        Filter.atTop fun x => B) →
                      (∀ (n : Nat) (ω : Ω),
                          MeasureTheory.Integrable
                            (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                            (Pstar n ω)) →
                        MeasureTheory.TendstoInMeasure μ
                          (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                            (fun n ω ωs => (thetaStar n ω ωs).ofLp) τ)
                          Filter.atTop fun x =>
                          Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:9639

theorem HansenEconometrics.chapter10_smooth_trimmedVariance_normFourth_integral_norm_sq_gaussianLimit

Smooth trimmed covariance route with the trimming-tail probability and Gaussian-limit coordinate MemLp 2 premises discharged.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {d : Type u_7} {r : Type u_8} [inst : Fintype d] [inst_1 : Fintype r] [inst_2 : DecidableEq d]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → EuclideanSpace Real d}
  {thetaStar : Nat → Ω → Ωs → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real) {τ : Nat → Real}
  {Bsecond Bfourth : Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Tstar
              (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable fun ωs => (thetaStar n ω ωs).ofLp) →
              (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp) 2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
                  (∀ (n : Nat) (ω : Ω) (ωs : Ωs),
                      Eq (thetaStar n ω ωs)
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                          (Tstar n ω ωs))) →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow (Pi.normedRing.norm (thetaStar n ω ωs).ofLp) 2)
                        Filter.atTop fun x => Bsecond) →
                      Real.instLE.le 0 Bfourth →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                            Filter.atTop fun x => Bfourth) →
                          (∀ (n : Nat) (ω : Ω),
                              MeasureTheory.Integrable
                                (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                                (Pstar n ω)) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                (fun n ω ωs => (thetaStar n ω ωs).ofLp) τ)
                              Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:9692

theorem HansenEconometrics.chapter10_indexed_smooth_trimmedVariance_normFourth_integral_norm_sq_gaussianLimit

Indexed smooth trimmed covariance route with the trimming-tail probability and Gaussian-limit coordinate MemLp 2 premises discharged.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {d : Type u_7} {r : Type u_8} [inst : Fintype d]
  [inst_1 : Fintype r] [inst_2 : DecidableEq d] {Ωboot : Nat → Type u_9}
  [inst_3 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Tstar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real d}
  {thetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real r} {V : Matrix d d Real} (G : Matrix r d Real)
  {τ : Nat → Real} {Bsecond Bfourth : Real},
  V.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar Tstar
              (ProbabilityTheory.multivariateGaussian 0 V) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable fun ωs => (thetaStar n ω ωs).ofLp) →
              (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp) 2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : r), MeasureTheory.MemLp (fun ωs => (thetaStar n ω ωs).ofLp a) 2 (Pstar n ω)) →
                  (∀ (n : Nat) (ω : Ω) (ωs : Ωboot n),
                      Eq (thetaStar n ω ωs)
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap G)
                          (Tstar n ω ωs))) →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow (Pi.normedRing.norm (thetaStar n ω ωs).ofLp) 2)
                        Filter.atTop fun x => Bsecond) →
                      Real.instLE.le 0 Bfourth →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                            Filter.atTop fun x => Bfourth) →
                          (∀ (n : Nat) (ω : Ω),
                              MeasureTheory.Integrable
                                (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (Tstar n ω ωs)) 4)
                                (Pstar n ω)) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                                (fun n ω ωs => (thetaStar n ω ωs).ofLp) τ)
                              Filter.atTop fun x =>
                              Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul G V) G.transpose

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/Covariance.lean:9748

Equation 10.146 of 33 linked endpoints

The third through sixth conditional moments of \sqrt n(\bar Y^*-\bar Y) are exact polynomials in the empirical cumulants.

theorem HansenEconometrics.integral_cube_normalized_empiricalBootstrapResampleMean_uniformOn_fun_sub_eq

Hansen equation (10.14), third conditional moment of the normalized ordinary-bootstrap sample mean.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (Fintype.card κ).cast.sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        3)
    (instHDiv.hDiv (HansenEconometrics.empiricalCumulant3 Y) (Fintype.card κ).cast.sqrt)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCumulant3
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:3982

theorem HansenEconometrics.integral_fifth_normalized_empiricalBootstrapResampleMean_uniformOn_fun_sub_eq

Hansen equation (10.14), fifth conditional moment of the normalized ordinary-bootstrap sample mean, before rewriting the fifth central moment as a sample cumulant.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (Fintype.card κ).cast.sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        5)
    (instHAdd.hAdd
      (instHDiv.hDiv (HansenEconometrics.empiricalCentralMoment Y 5)
        (instHMul.hMul (Fintype.card κ).cast (Fintype.card κ).cast.sqrt))
      (instHMul.hMul
        (instHMul.hMul
          (instHDiv.hDiv (instHMul.hMul 10 (instHSub.hSub (Fintype.card κ).cast 1))
            (instHMul.hMul (Fintype.card κ).cast (Fintype.card κ).cast.sqrt))
          (HansenEconometrics.empiricalCumulant3 Y))
        (HansenEconometrics.empiricalCumulant2 Y)))

Direct statement dependencies (5)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCentralMoment
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant3
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4141

theorem HansenEconometrics.integral_sixth_normalized_empiricalBootstrapResampleMean_uniformOn_fun_sub_eq

Hansen equation (10.14), sixth conditional moment of the normalized ordinary-bootstrap sample mean, before rewriting central moments as sample cumulants.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (Fintype.card κ).cast.sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        6)
    (instHAdd.hAdd
      (instHAdd.hAdd
        (instHAdd.hAdd
          (instHDiv.hDiv (HansenEconometrics.empiricalCentralMoment Y 6) (instHPow.hPow (Fintype.card κ).cast 2))
          (instHMul.hMul
            (instHMul.hMul
              (instHDiv.hDiv (instHMul.hMul 15 (instHSub.hSub (Fintype.card κ).cast 1))
                (instHPow.hPow (Fintype.card κ).cast 2))
              (HansenEconometrics.empiricalCentralMoment Y 4))
            (HansenEconometrics.empiricalCumulant2 Y)))
        (instHMul.hMul
          (instHDiv.hDiv (instHMul.hMul 10 (instHSub.hSub (Fintype.card κ).cast 1))
            (instHPow.hPow (Fintype.card κ).cast 2))
          (instHPow.hPow (HansenEconometrics.empiricalCumulant3 Y) 2)))
      (instHMul.hMul
        (instHDiv.hDiv
          (instHMul.hMul (instHMul.hMul 15 (instHSub.hSub (Fintype.card κ).cast 1))
            (instHSub.hSub (Fintype.card κ).cast 2))
          (instHPow.hPow (Fintype.card κ).cast 2))
        (instHPow.hPow (HansenEconometrics.empiricalCumulant2 Y) 3)))

Direct statement dependencies (5)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCentralMoment
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant3
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4240

theorem HansenEconometrics.integral_fourth_normalized_empiricalBootstrapResampleMean_uniformOn_fun_sub_eq

Hansen equation (10.14), fourth conditional moment of the normalized ordinary-bootstrap sample mean, before rewriting the fourth central moment as a sample cumulant.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (Fintype.card κ).cast.sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        4)
    (instHAdd.hAdd (instHDiv.hDiv (HansenEconometrics.empiricalCentralMoment Y 4) (Fintype.card κ).cast)
      (instHMul.hMul (instHDiv.hDiv (instHMul.hMul 3 (instHSub.hSub (Fintype.card κ).cast 1)) (Fintype.card κ).cast)
        (instHPow.hPow (HansenEconometrics.empiricalCumulant2 Y) 2)))

Direct statement dependencies (4)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCentralMoment
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4052

theorem HansenEconometrics.integral_cube_normalized_empiricalBootstrapResampleMean_uniformOn_fun_sub_eq_formula

Hansen equation (10.14), third conditional moment in the named formula surface.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (Fintype.card κ).cast.sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        3)
    (HansenEconometrics.normalizedBootstrapMeanMoment3Formula (Fintype.card κ).cast Y)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.normalizedBootstrapMeanMoment3Formula

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4401

theorem HansenEconometrics.integral_fifth_normalized_empiricalBootstrapResampleMean_uniformOn_fun_sub_eq_formula

Hansen equation (10.14), fifth conditional moment in the named formula surface.

Formal statement

∀ {ι : Type u_7} [inst : MeasurableSpace ι] [inst_1 : Fintype ι] [MeasurableSingletonClass ι] {κ : Type u_8}
  [inst_3 : Fintype κ] [Nonempty κ] [Nonempty ι] [MeasurableSingletonClass (κ → ι)] (Y : ι → Real),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (Fintype.card κ).cast.sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean Y (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean Y)))
        5)
    (HansenEconometrics.normalizedBootstrapMeanMoment5Formula (Fintype.card κ).cast Y)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.normalizedBootstrapMeanMoment5Formula

HansenEconometrics/Chapter10Bootstrap/Empirical.lean:4438

Equation 10.14 shifted ordinary-bootstrap wrappers4 endpoints

The third through sixth conditional moments of \sqrt n(\bar Y^*-\bar Y) are exact polynomials in the empirical cumulants. The same exact third-through-sixth conditional moment formulas specialized to the sample-size-indexed ordinary resampling space Fin (n+1) -> Fin (n+1)

theorem HansenEconometrics.integral_cube_normalized_finSucc_resampleMean_sub_empiricalMean_eq_formula

Scalar Fin (n+1) CLT-scale third conditional moment formula for the ordinary nonparametric bootstrap.

This is the shifted sample-size-indexed face of Hansen equation (10.14) for the ordinary empirical resampling space.

Formal statement

∀ {Ω : Type u_1} (Y : Nat → Ω → Real) (n : Nat) (ω : Ω),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        3)
    (HansenEconometrics.normalizedBootstrapMeanMoment3Formula (instHAdd.hAdd n.cast 1) fun i => Y i.val ω)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.normalizedBootstrapMeanMoment3Formula

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1788

theorem HansenEconometrics.integral_fourth_normalized_finSucc_resampleMean_sub_empiricalMean_eq_formula

Scalar Fin (n+1) CLT-scale fourth conditional moment formula for the ordinary nonparametric bootstrap.

This is the shifted sample-size-indexed face of Hansen equation (10.14) used by the fourth-moment route to uniform square integrability in (10.17).

Formal statement

∀ {Ω : Type u_1} (Y : Nat → Ω → Real) (n : Nat) (ω : Ω),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        4)
    (HansenEconometrics.normalizedBootstrapMeanMoment4Formula (instHAdd.hAdd n.cast 1) fun i => Y i.val ω)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.normalizedBootstrapMeanMoment4Formula

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1810

theorem HansenEconometrics.integral_fifth_normalized_finSucc_resampleMean_sub_empiricalMean_eq_formula

Scalar Fin (n+1) CLT-scale fifth conditional moment formula for the ordinary nonparametric bootstrap.

This is the shifted sample-size-indexed face of Hansen equation (10.14) for the ordinary empirical resampling space.

Formal statement

∀ {Ω : Type u_1} (Y : Nat → Ω → Real) (n : Nat) (ω : Ω),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        5)
    (HansenEconometrics.normalizedBootstrapMeanMoment5Formula (instHAdd.hAdd n.cast 1) fun i => Y i.val ω)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.normalizedBootstrapMeanMoment5Formula

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1924

theorem HansenEconometrics.integral_sixth_normalized_finSucc_resampleMean_sub_empiricalMean_eq_formula

Scalar Fin (n+1) CLT-scale sixth conditional moment formula for the ordinary nonparametric bootstrap.

This is the shifted sample-size-indexed face of Hansen equation (10.14) for the ordinary empirical resampling space.

Formal statement

∀ {Ω : Type u_1} (Y : Nat → Ω → Real) (n : Nat) (ω : Ω),
  Eq
    (MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
      instHPow.hPow
        (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
          (instHSub.hSub (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
            (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        6)
    (HansenEconometrics.normalizedBootstrapMeanMoment6Formula (instHAdd.hAdd n.cast 1) fun i => Y i.val ω)

Direct statement dependencies (3)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.normalizedBootstrapMeanMoment6Formula

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1946

Equations 10.15/10.164 endpoints

The sample and bootstrap statistics share the limit Z_n\xrightarrow{d}Z and Z_n^*\xrightarrow{d^*}Z.

def HansenEconometrics.TendstoInBootstrapWeakDistribution

Bootstrap convergence in distribution in bounded-continuous-test-function form.

For every bounded continuous real test function, the conditional bootstrap expectation converges in ordinary probability to the corresponding expectation under the limiting law.

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {Ωlim : Type u_3} →
      {E : Type u_4} →
        {mΩ : MeasurableSpace Ω} →
          {mΩs : MeasurableSpace Ωs} →
            {mΩlim : MeasurableSpace Ωlim} →
              [TopologicalSpace E] →
                MeasureTheory.Measure Ω →
                  (Nat → Ω → MeasureTheory.Measure Ωs) →
                    (Nat → Ω → Ωs → E) → MeasureTheory.Measure Ωlim → (Ωlim → E) → Prop

HansenEconometrics/Chapter10Bootstrap/WeakDistribution.lean:54

def HansenEconometrics.TendstoInBootstrapDistribution

Hansen Definition 10.2: convergence in bootstrap distribution.

The conditional CDF of Zstar n converges in ordinary probability, under the original-sample law μ, to the limit CDF at every continuity point of the limit CDF.

Formal statement

{Ω : Type u_1} →
  {Ωs : Type u_2} →
    {Ωlim : Type u_3} →
      {k : Type u_6} →
        {mΩ : MeasurableSpace Ω} →
          {mΩs : MeasurableSpace Ωs} →
            {mΩlim : MeasurableSpace Ωlim} →
              MeasureTheory.Measure Ω →
                (Nat → Ω → MeasureTheory.Measure Ωs) →
                  (Nat → Ω → Ωs → k → Real) → MeasureTheory.Measure Ωlim → (Ωlim → k → Real) → Prop

HansenEconometrics/Chapter10Bootstrap/Distribution.lean:37

theorem HansenEconometrics.chapter10_bootstrap_variance_consistency_of_weak_distribution_uniform_square_tail

Hansen Theorem 10.9, weak-distribution plus uniform-square-tail variance bridge.

This is the theorem-facing uniform-integrability assembly: for every tolerance one chooses a large threshold whose squared tail is small for the limit law and small in probability for the conditional bootstrap law.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real}
  {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          (∀ (ε : Real),
              Real.instLT.lt 0 ε →
                Exists fun R =>
                  And (Real.instLE.le 1 R)
                    (And
                      (Real.instLE.le
                        (MeasureTheory.integral ν fun ωlim =>
                          (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                            (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                        ε)
                      (Filter.Tendsto
                        (fun n =>
                          MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω =>
                              Real.instLE.le ε
                                (Real.pseudoMetricSpace.dist
                                  (MeasureTheory.integral (Pstar n ω) fun ωs =>
                                    (setOf fun ωs => Real.instLE.le R (abs (Zstar n ω ωs))).indicator
                                      (fun ωs => instHPow.hPow (Zstar n ω ωs) 2) ωs)
                                  0)))
                        Filter.atTop (nhds 0)))) →
            MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapVarianceReal Pstar Zstar) Filter.atTop
              fun x =>
              instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4300

theorem HansenEconometrics.chapter10_finiteReplicationVariance_tendsto_of_weak_distribution_uniformSquareTail

Hansen Theorem 10.9/10.11 finite-replication variance from bootstrap weak convergence and a uniform-square-tail condition.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Zsim : Nat → Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → Real} {Z : Ωlim → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          HansenEconometrics.BootstrapUniformSquareTail μ Pstar Zstar ν Z →
            (MeasureTheory.TendstoInMeasure μ
                (fun n ω =>
                  instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                    (HansenEconometrics.bootstrapVarianceReal Pstar Zstar n ω))
                Filter.atTop fun x => 0) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim)
                Filter.atTop fun x =>
                instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                  (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (4)

HansenEconometrics.BootstrapUniformSquareTail
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:802

Equation 10.176 endpoints

The normalized bootstrap mean has bounded fourth conditional moment: \mathbb E^*[(Z_n^*)^4]=O_p(1).

theorem HansenEconometrics.bootstrapUniformSquareTail_of_fourthMoment_tendstoInMeasure

Uniform square-tail constructor from a fourth-moment convergence premise.

The conditional fourth moment controls the conditional squared tail by R⁻² E*[Z*⁴]. If that fourth moment converges in probability to B, and the chosen threshold also makes the limit squared tail small, then Hansen’s named uniform square-tail condition follows.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → Real} {ν : MeasureTheory.Measure Ωlim} {Z : Ωlim → Real} {B : Real},
  (MeasureTheory.TendstoInMeasure μ
      (fun n ω => MeasureTheory.integral (Pstar n ω) fun ωs => instHPow.hPow (Zstar n ω ωs) 4) Filter.atTop fun x =>
      B) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.Integrable (fun ωs => instHPow.hPow (Zstar n ω ωs) 4) (Pstar n ω)) →
      (∀ (ε : Real),
          Real.instLT.lt 0 ε →
            Exists fun R =>
              And (Real.instLE.le 1 R)
                (And (Real.instLT.lt (instHMul.hMul (instHPow.hPow (Real.instInv.inv R) 2) (instHAdd.hAdd B 1)) ε)
                  (Real.instLE.le
                    (MeasureTheory.integral ν fun ωlim =>
                      (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                        (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                    ε))) →
        HansenEconometrics.BootstrapUniformSquareTail μ Pstar Zstar ν Z

Direct statement dependencies (1)

HansenEconometrics.BootstrapUniformSquareTail

HansenEconometrics/Chapter10Bootstrap/Variance.lean:3850

theorem HansenEconometrics.chapter10_bootstrap_variance_consistency_of_weak_distribution_fourthMoment_tail

Hansen Theorem 10.9 from fourth-moment tail controls.

Bootstrap weak convergence plus conditional fourth-moment convergence supplies conditional bootstrap variance consistency once the weak-limit squared tails are eventually small at large thresholds.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real}
  {Z : Ωlim → Real} {B : Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          Real.instLE.le 0 B →
            (MeasureTheory.TendstoInMeasure μ
                (fun n ω => MeasureTheory.integral (Pstar n ω) fun ωs => instHPow.hPow (Zstar n ω ωs) 4) Filter.atTop
                fun x => B) →
              (∀ (n : Nat) (ω : Ω), MeasureTheory.Integrable (fun ωs => instHPow.hPow (Zstar n ω ωs) 4) (Pstar n ω)) →
                (∀ (ε : Real),
                    Real.instLT.lt 0 ε →
                      Exists fun R₀ =>
                        And (Real.instLE.le 1 R₀)
                          (∀ (R : Real),
                            Real.instLE.le R₀ R →
                              Real.instLE.le
                                (MeasureTheory.integral ν fun ωlim =>
                                  (setOf fun ωlim => Real.instLE.le R (abs (Z ωlim))).indicator
                                    (fun ωlim => instHPow.hPow (Z ωlim) 2) ωlim)
                                ε)) →
                  MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapVarianceReal Pstar Zstar) Filter.atTop
                    fun x =>
                    instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                      (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal

HansenEconometrics/Chapter10Bootstrap/Variance.lean:4358

theorem HansenEconometrics.chapter10_finiteReplicationVariance_tendsto_of_fourthMoment_memLp_limit

Hansen Theorem 10.9/10.11 finite-replication variance from bootstrap weak convergence and fourth-moment convergence, with the weak-limit tail premise discharged by MemLp Z 2 ν.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [MeasureTheory.IsFiniteMeasure ν] {Zsim : Nat → Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {Zstar : Nat → Ω → Ωs → Real} {Z : Ωlim → Real} {B : Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.MemLp (Zstar n ω) 2 (Pstar n ω)) →
      MeasureTheory.MemLp Z 2 ν →
        HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar Zstar ν Z →
          Real.instLE.le 0 B →
            (MeasureTheory.TendstoInMeasure μ
                (fun n ω => MeasureTheory.integral (Pstar n ω) fun ωs => instHPow.hPow (Zstar n ω ωs) 4) Filter.atTop
                fun x => B) →
              (∀ (n : Nat) (ω : Ω), MeasureTheory.Integrable (fun ωs => instHPow.hPow (Zstar n ω ωs) 4) (Pstar n ω)) →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      instHSub.hSub (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim n ω)
                        (HansenEconometrics.bootstrapVarianceReal Pstar Zstar n ω))
                    Filter.atTop fun x => 0) →
                  MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationVarianceMomentReal Zsim)
                    Filter.atTop fun x =>
                    instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
                      (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapVarianceReal
HansenEconometrics.finiteReplicationVarianceMomentReal

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:1124

theorem HansenEconometrics.integral_fourth_normalized_finSucc_resampleMean_sub_empiricalMean_tendstoInMeasure_of_cumulants

Fourth conditional moment convergence for the ordinary Fin (n+1) bootstrap sample mean from Hansen’s cumulant formula.

If the empirical variance converges to σ2 and the scaled fourth cumulant is negligible, the exact equation (10.14) formula gives E*[(sqrt (n+1) (Ybar* - Ybar))^4] ->p 3 σ2^2. This is the sample-mean fourth-moment route behind Hansen equation (10.17).

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (Y : Nat → Ω → Real) {σ2 : Real},
  (MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.empiricalCumulant2 fun i => Y i.val ω) Filter.atTop
      fun x => σ2) →
    (MeasureTheory.TendstoInMeasure μ
        (fun n ω => instHDiv.hDiv (HansenEconometrics.empiricalCumulant4 fun i => Y i.val ω) (instHAdd.hAdd n.cast 1))
        Filter.atTop fun x => 0) →
      MeasureTheory.TendstoInMeasure μ
        (fun n ω =>
          MeasureTheory.integral (ProbabilityTheory.uniformOn Set.univ) fun ωs =>
            instHPow.hPow
              (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                (instHSub.hSub
                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                  (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
              4)
        Filter.atTop fun x => instHMul.hMul 3 (instHPow.hPow σ2 2)

Direct statement dependencies (4)

HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant4
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:1835

theorem HansenEconometrics.bootstrapUniformSquareTailIndexed_normalized_finSucc_resampleMean_sub_empiricalMean_of_cumulants

Indexed ordinary-bootstrap uniform-square-tail route from Hansen’s fourth-moment cumulant formula.

For the concrete Fin (n+1) ordinary resampling space, convergence of the empirical variance and negligibility of the scaled fourth cumulant supply the conditional fourth-moment premise in the indexed uniform-square-tail constructor.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] (Y : Nat → Ω → Real) {Z : Ωlim → Real} {σ2 : Real},
  MeasureTheory.MemLp Z 2 ν →
    (MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.empiricalCumulant2 fun i => Y i.val ω) Filter.atTop
        fun x => σ2) →
      (MeasureTheory.TendstoInMeasure μ
          (fun n ω => instHDiv.hDiv (HansenEconometrics.empiricalCumulant4 fun i => Y i.val ω) (instHAdd.hAdd n.cast 1))
          Filter.atTop fun x => 0) →
        HansenEconometrics.BootstrapUniformSquareTailIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
          (fun n ω ωs =>
            instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
              (instHSub.hSub
                (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
          ν Z

Direct statement dependencies (5)

HansenEconometrics.BootstrapUniformSquareTailIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant4
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Variance.lean:5147

theorem HansenEconometrics.chapter10_indexed_bootstrap_variance_finSucc_resampleMean_of_weak_distribution_cumulants

Indexed Hansen Theorem 10.9 for the concrete ordinary bootstrap sample mean, using Hansen’s fourth-moment cumulant formula to discharge uniform square integrability.

This is the sample-mean fourth-moment route behind equation (10.17): once the ordinary normalized bootstrap mean has the indexed weak limit and the empirical cumulants satisfy the exact-formula convergence premises, the conditional bootstrap variance is consistent.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [MeasureTheory.IsFiniteMeasure ν] (Y : Nat → Ω → Real) {Z : Ωlim → Real} {σ2 : Real},
  MeasureTheory.MemLp Z 2 ν →
    HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs =>
          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
            (instHSub.hSub
              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
        ν Z →
      (MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.empiricalCumulant2 fun i => Y i.val ω)
          Filter.atTop fun x => σ2) →
        (MeasureTheory.TendstoInMeasure μ
            (fun n ω =>
              instHDiv.hDiv (HansenEconometrics.empiricalCumulant4 fun i => Y i.val ω) (instHAdd.hAdd n.cast 1))
            Filter.atTop fun x => 0) →
          MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapVarianceRealIndexed (fun n x => ProbabilityTheory.uniformOn Set.univ)
              fun n ω ωs =>
              instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                (instHSub.hSub
                  (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                  (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
            Filter.atTop fun x =>
            instHSub.hSub (MeasureTheory.integral ν fun ωlim => instHPow.hPow (Z ωlim) 2)
              (instHPow.hPow (MeasureTheory.integral ν fun ωlim => Z ωlim) 2)

Direct statement dependencies (6)

HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapVarianceRealIndexed
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalCumulant2
HansenEconometrics.empiricalCumulant4
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Variance.lean:5205

Equations 10.18/10.193 endpoints

The percentile method assumes a_n(\hat\theta-\theta)\xrightarrow{d}\xi and a_n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}\xi.

theorem HansenEconometrics.percentileCoverageVector_tendstoInDistribution_of_components

Componentwise Slutsky constructor for the percentile-coverage joint vector.

This assembles the joint convergence premise in chapter10_percentileCI_coverage_tendsto_of_joint_quantile_limit from the scaled estimator-error limit and the two bootstrap endpoint limits.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {a : Nat → Real} {θ : Real} {θhat qLower qUpper : Nat → Ω → Real}
  {ξ : Ωlim → Real} {qLowerLim qUpperLim : Real},
  MeasureTheory.TendstoInDistribution (fun n ω => instHMul.hMul (a n) (instHSub.hSub (θhat n ω) θ)) Filter.atTop ξ
      (fun x => μ) ν →
    (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω)))
        Filter.atTop fun x => qLowerLim) →
      (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω)))
          Filter.atTop fun x => qUpperLim) →
        (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω))) μ) →
          (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω))) μ) →
            MeasureTheory.TendstoInDistribution (HansenEconometrics.percentileCoverageVector a θ θhat qLower qUpper)
              Filter.atTop (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) (fun x => μ) ν

Direct statement dependencies (2)

HansenEconometrics.percentileCoverageLimitVector
HansenEconometrics.percentileCoverageVector

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:58

theorem HansenEconometrics.chapter10_percentileCI_coverage_tendsto_of_joint_quantile_limit

Hansen Theorem 10.13, percentile-interval coverage bridge.

If the scaled estimator error and the scaled bootstrap percentile endpoints jointly converge to (ξ, qL, qU), and the limiting coverage boundary has zero probability, then the percentile interval coverage converges to P[qL <= -ξ <= qU]. Hansen’s symmetric continuous-limit conclusion 1 - α is obtained by instantiating this bridge with the appropriate bootstrap quantile limits and symmetry identity for the limit law.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θ : Real} {θhat qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real} {qLowerLim qUpperLim : Real},
      MeasureTheory.TendstoInDistribution (HansenEconometrics.percentileCoverageVector a θ θhat qLower qUpper)
          Filter.atTop (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) (fun x => μ) ν →
        Eq
            (MeasureTheory.Measure.instFunLike.coe
              (MeasureTheory.Measure.map (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) ν)
              (frontier HansenEconometrics.percentileCoverageSet))
            0 →
          Filter.Tendsto
            (fun n =>
              MeasureTheory.Measure.instFunLike.coe μ
                (setOf fun ω => HansenEconometrics.percentileCIEvent θ (qLower n ω) (qUpper n ω)))
            Filter.atTop
            (nhds
              (MeasureTheory.Measure.instFunLike.coe
                (MeasureTheory.Measure.map (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) ν)
                HansenEconometrics.percentileCoverageSet))

Direct statement dependencies (4)

HansenEconometrics.percentileCIEvent
HansenEconometrics.percentileCoverageLimitVector
HansenEconometrics.percentileCoverageSet
HansenEconometrics.percentileCoverageVector

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:299

theorem HansenEconometrics.chapter10_percentileCI_coverage_tendsto_one_sub_alpha_of_components_symmetric_cdf

Symmetric endpoint-CDF percentile-interval calibration.

This is the Hansen Theorem 10.13 specialization where the limiting bootstrap percentile endpoints are -q and q, and the scalar limit law has endpoint CDF masses α / 2 and 1 - α / 2.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θ : Real} {θhat qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
      MeasureTheory.TendstoInDistribution (fun n ω => instHMul.hMul (a n) (instHSub.hSub (θhat n ω) θ)) Filter.atTop ξ
          (fun x => μ) ν →
        (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω)))
            Filter.atTop fun x => Real.instNeg.neg q) →
          (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω)))
              Filter.atTop fun x => q) →
            (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω))) μ) →
              (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω))) μ) →
                ProbabilityTheory.HasLaw ξ η ν →
                  Real.instLE.le 0 q →
                    Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                      Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                        Filter.Tendsto
                          (fun n =>
                            MeasureTheory.Measure.instFunLike.coe μ
                              (setOf fun ω => HansenEconometrics.percentileCIEvent θ (qLower n ω) (qUpper n ω)))
                          Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (1)

HansenEconometrics.percentileCIEvent

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:592

Equation 10.203 endpoints

The bootstrap quantile endpoint satisfies a_n(q_\alpha^*-\hat\theta)\xrightarrow{p}q_\alpha.

theorem HansenEconometrics.chapter10_percentileEndpoint_scaled_sub_tendstoInMeasure

Hansen equation (10.20), algebraic endpoint form.

If a bootstrap offset Qstar converges to the limiting quantile q, then the original-scale endpoint θhat + Qstar / a has the displayed scaled endpoint convergence aₙ(q*ₙ - θhatₙ) →p q.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θhat Qstar : Nat → Ω → Real} {q : Real},
      (MeasureTheory.TendstoInMeasure μ Qstar Filter.atTop fun x => q) →
        MeasureTheory.TendstoInMeasure μ
          (fun n ω =>
            instHMul.hMul (a n) (instHSub.hSub (instHAdd.hAdd (θhat n ω) (instHDiv.hDiv (Qstar n ω) (a n))) (θhat n ω)))
          Filter.atTop fun x => q

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:639

theorem HansenEconometrics.chapter10_percentileEndpoint_scaled_sub_tendstoInMeasure_of_lowerQuantile

Hansen equation (10.20), lower-generalized-inverse route.

Pointwise conditional-CDF convergence identifies the bootstrap lower quantile; the original-scale endpoint then satisfies aₙ(q*ₙ - θhatₙ) →p q.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → Real} {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θhat : Nat → Ω → Real} {G : Real → Real} {p q : Real},
      (∀ (n : Nat) (ω : Ω), Monotone fun x => HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) →
        (∀ (n : Nat) (ω : Ω),
            (setOf fun x => Real.instLE.le p (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
          (∀ (n : Nat) (ω : Ω),
              BddBelow (setOf fun x => Real.instLE.le p (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
            (∀ (n : Nat) (ω : Ω) (x : Real),
                Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) p →
                  Exists fun δ =>
                    And (Real.instLT.lt 0 δ)
                      (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω) p)) →
              StrictMono G →
                Eq (G q) p →
                  (∀ (x : Real),
                      MeasureTheory.TendstoInMeasure μ
                        (fun n ω => HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) Filter.atTop fun x_1 =>
                        G x) →
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        instHMul.hMul (a n)
                          (instHSub.hSub
                            (instHAdd.hAdd (θhat n ω)
                              (instHDiv.hDiv (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar p n ω) (a n)))
                            (θhat n ω)))
                      Filter.atTop fun x => q

Direct statement dependencies (2)

HansenEconometrics.bootstrapScalarCDF
HansenEconometrics.bootstrapScalarLowerQuantile

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:663

theorem HansenEconometrics.bootstrapScalarLowerQuantile_tendsto_of_strictMono_cdf

Bootstrap scalar lower-quantile convergence with a strictly increasing limiting CDF.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Zstar : Nat → Ω → Ωs → Real} {G : Real → Real} {p q : Real},
  (∀ (n : Nat) (ω : Ω), Monotone fun x => HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω) →
    (∀ (n : Nat) (ω : Ω),
        (setOf fun x => Real.instLE.le p (HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω)).Nonempty) →
      (∀ (n : Nat) (ω : Ω),
          BddBelow (setOf fun x => Real.instLE.le p (HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω))) →
        (∀ (n : Nat) (ω : Ω) (x : Real),
            Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω) p →
              Exists fun δ =>
                And (Real.instLT.lt 0 δ)
                  (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Zstar (instHAdd.hAdd x δ) n ω) p)) →
          StrictMono G →
            Eq (G q) p →
              (∀ (x : Real),
                  MeasureTheory.TendstoInMeasure μ (fun n ω => HansenEconometrics.bootstrapScalarCDF Pstar Zstar x n ω)
                    Filter.atTop fun x_1 => G x) →
                MeasureTheory.TendstoInMeasure μ (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Zstar p)
                  Filter.atTop fun x => q

Direct statement dependencies (2)

HansenEconometrics.bootstrapScalarCDF
HansenEconometrics.bootstrapScalarLowerQuantile

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:932

Theorem 10.136 of 24 linked endpoints

If the sample and bootstrap centered statistics share a continuous symmetric limit, then \Pr(\theta\in C_{\mathrm{pct}})\to1-\alpha.

theorem HansenEconometrics.chapter10_percentileCI_coverage_tendsto_of_joint_quantile_limit

Hansen Theorem 10.13, percentile-interval coverage bridge.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θ : Real} {θhat qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real} {qLowerLim qUpperLim : Real},
      MeasureTheory.TendstoInDistribution (HansenEconometrics.percentileCoverageVector a θ θhat qLower qUpper)
          Filter.atTop (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) (fun x => μ) ν →
        Eq
            (MeasureTheory.Measure.instFunLike.coe
              (MeasureTheory.Measure.map (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) ν)
              (frontier HansenEconometrics.percentileCoverageSet))
            0 →
          Filter.Tendsto
            (fun n =>
              MeasureTheory.Measure.instFunLike.coe μ
                (setOf fun ω => HansenEconometrics.percentileCIEvent θ (qLower n ω) (qUpper n ω)))
            Filter.atTop
            (nhds
              (MeasureTheory.Measure.instFunLike.coe
                (MeasureTheory.Measure.map (HansenEconometrics.percentileCoverageLimitVector ξ qLowerLim qUpperLim) ν)
                HansenEconometrics.percentileCoverageSet))

Direct statement dependencies (4)

HansenEconometrics.percentileCIEvent
HansenEconometrics.percentileCoverageLimitVector
HansenEconometrics.percentileCoverageSet
HansenEconometrics.percentileCoverageVector

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:299

theorem HansenEconometrics.chapter10_percentileCI_coverage_tendsto_one_sub_alpha_of_components_symmetric_cdf

Symmetric endpoint-CDF percentile-interval calibration.

This is the Hansen Theorem 10.13 specialization where the limiting bootstrap percentile endpoints are -q and q, and the scalar limit law has endpoint CDF masses α / 2 and 1 - α / 2.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θ : Real} {θhat qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
      MeasureTheory.TendstoInDistribution (fun n ω => instHMul.hMul (a n) (instHSub.hSub (θhat n ω) θ)) Filter.atTop ξ
          (fun x => μ) ν →
        (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω)))
            Filter.atTop fun x => Real.instNeg.neg q) →
          (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω)))
              Filter.atTop fun x => q) →
            (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω))) μ) →
              (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω))) μ) →
                ProbabilityTheory.HasLaw ξ η ν →
                  Real.instLE.le 0 q →
                    Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                      Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                        Filter.Tendsto
                          (fun n =>
                            MeasureTheory.Measure.instFunLike.coe μ
                              (setOf fun ω => HansenEconometrics.percentileCIEvent θ (qLower n ω) (qUpper n ω)))
                          Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (1)

HansenEconometrics.percentileCIEvent

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:592

theorem HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_resampleMean_of_iIndep_tail_posDef_brackets

Indexed ordinary nonparametric-bootstrap percentile-interval coverage from the concrete normalized scalar Fin (n+1) resample-mean CLT.

The bootstrap endpoint statistic is no longer abstract: it is the lower generalized inverse of the conditional CDF of sqrt(n+1) (Ybar*_n - Ybar_n) under the finite ordinary resampling law. The sample-side limit and percentile calibration remain explicit, as in Hansen Theorem 10.13.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (Y 0) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.covMat μ fun ω x => Y 0 ω).PosDef →
          ∀ {a : Nat → Real},
            (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
              ∀ {θ : Real} {θhat : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
                MeasureTheory.TendstoInDistribution (fun n ω => instHMul.hMul (a n) (instHSub.hSub (θhat n ω) θ))
                    Filter.atTop ξ (fun x => μ) ν →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub (Real.instNeg.neg q) ε))
                              (instHDiv.hDiv α 2)) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt (instHDiv.hDiv α 2)
                                ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub q ε))
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                    ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd q ε))) →
                              (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                        (fun n ω ωs =>
                                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                            (instHSub.hSub
                                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                                (fun ωs t => ωs t) ωs)
                                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                          (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                          (fun n ω ωs =>
                                            instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                              (instHSub.hSub
                                                (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                                  (fun ωs t => ωs t) ωs)
                                                (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    ProbabilityTheory.HasLaw ξ η ν →
                                      ProbabilityTheory.HasLaw (fun z => z.ofLp Unit.unit) η
                                          (ProbabilityTheory.multivariateGaussian 0
                                            (HansenEconometrics.covMat μ fun ω x => Y 0 ω)) →
                                        Real.instLE.le 0 q →
                                          Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q))
                                              (instHDiv.hDiv α 2) →
                                            Eq ((ProbabilityTheory.cdf η).toFun q)
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                              Filter.Tendsto
                                                (fun n =>
                                                  MeasureTheory.Measure.instFunLike.coe μ
                                                    (setOf fun ω =>
                                                      HansenEconometrics.percentileCIEvent θ
                                                        (instHAdd.hAdd (θhat n ω)
                                                          (instHDiv.hDiv
                                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                              (fun n ω ωs =>
                                                                instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                                                  (instHSub.hSub
                                                                    (HansenEconometrics.empiricalBootstrapResampleMean
                                                                      (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                                                                    (HansenEconometrics.empiricalMean fun i =>
                                                                      Y i.val ω)))
                                                              (instHDiv.hDiv α 2) n ω)
                                                            (a n)))
                                                        (instHAdd.hAdd (θhat n ω)
                                                          (instHDiv.hDiv
                                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                              (fun n ω ωs =>
                                                                instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                                                  (instHSub.hSub
                                                                    (HansenEconometrics.empiricalBootstrapResampleMean
                                                                      (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                                                                    (HansenEconometrics.empiricalMean fun i =>
                                                                      Y i.val ω)))
                                                              (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                                            (a n)))))
                                                Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (5)

HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.percentileCIEvent

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:1954

theorem HansenEconometrics.chapter10_percentileCI_coverage_tendsto_one_sub_alpha_of_components_law_cdf

Componentwise endpoint-CDF percentile-interval calibration with limiting coverage 1 - α.

This is the Theorem 10.13 coverage bridge stated directly from scalar estimator-error convergence and bootstrap endpoint convergence in probability.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {θ : Real} {θhat qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real} {qLowerLim qUpperLim α : Real},
      MeasureTheory.TendstoInDistribution (fun n ω => instHMul.hMul (a n) (instHSub.hSub (θhat n ω) θ)) Filter.atTop ξ
          (fun x => μ) ν →
        (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω)))
            Filter.atTop fun x => qLowerLim) →
          (MeasureTheory.TendstoInMeasure μ (fun n ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω)))
              Filter.atTop fun x => qUpperLim) →
            (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qLower n ω) (θhat n ω))) μ) →
              (∀ (n : Nat), AEMeasurable (fun ω => instHMul.hMul (a n) (instHSub.hSub (qUpper n ω) (θhat n ω))) μ) →
                ProbabilityTheory.HasLaw ξ η ν →
                  Real.instLE.le qLowerLim qUpperLim →
                    Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg qUpperLim)) (instHDiv.hDiv α 2) →
                      Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg qLowerLim))
                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                        Filter.Tendsto
                          (fun n =>
                            MeasureTheory.Measure.instFunLike.coe μ
                              (setOf fun ω => HansenEconometrics.percentileCIEvent θ (qLower n ω) (qUpper n ω)))
                          Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (1)

HansenEconometrics.percentileCIEvent

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:546

theorem HansenEconometrics.percentileCoverageLimit_measure_set_eq

The vector-law probability of the percentile-coverage limit set is the scalar event probability P[qL <= -ξ <= qU].

Formal statement

∀ {Ωlim : Type u_3} {mΩlim : MeasurableSpace Ωlim} {ν : MeasureTheory.Measure Ωlim} {ξ : Ωlim → Real}
  {qLower qUpper : Real},
  AEMeasurable ξ ν →
    Eq
      (MeasureTheory.Measure.instFunLike.coe
        (MeasureTheory.Measure.map (HansenEconometrics.percentileCoverageLimitVector ξ qLower qUpper) ν)
        HansenEconometrics.percentileCoverageSet)
      (MeasureTheory.Measure.instFunLike.coe ν
        (setOf fun ω =>
          And (Real.instLE.le qLower (Real.instNeg.neg (ξ ω))) (Real.instLE.le (Real.instNeg.neg (ξ ω)) qUpper)))

Direct statement dependencies (2)

HansenEconometrics.percentileCoverageLimitVector
HansenEconometrics.percentileCoverageSet

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:183

theorem HansenEconometrics.percentileCoverage_scalar_event_eq_law

The scalar percentile-coverage event can be read from the law of the limit statistic as the interval [-qU, -qL].

Formal statement

∀ {Ωlim : Type u_3} {mΩlim : MeasurableSpace Ωlim} {ν : MeasureTheory.Measure Ωlim} {ξ : Ωlim → Real}
  {η : MeasureTheory.Measure Real},
  ProbabilityTheory.HasLaw ξ η ν →
    ∀ (qLower qUpper : Real),
      Eq
        (MeasureTheory.Measure.instFunLike.coe ν
          (setOf fun ω =>
            And (Real.instLE.le qLower (Real.instNeg.neg (ξ ω))) (Real.instLE.le (Real.instNeg.neg (ξ ω)) qUpper)))
        (MeasureTheory.Measure.instFunLike.coe η (Set.Icc (Real.instNeg.neg qUpper) (Real.instNeg.neg qLower)))

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:249

Theorem 10.13 finite OLS percentile wrappers6 of 12 linked endpoints

If the sample and bootstrap centered statistics share a continuous symmetric limit, then \Pr(\theta\in C_{\mathrm{pct}})\to1-\alpha. Concrete finite ordinary-bootstrap OLS linear-restriction numerators can feed the percentile endpoint route once the sample-side OLS limit and scalar law calibration are supplied

theorem HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_olsBetaOrZero_gapEnvelope_bounds_brackets

Local-CDF bracketing finite OLS percentile-interval wrapper.

The bootstrap endpoint statistic is the concrete one-row ordinary-bootstrap OLS numerator sqrt(n+1)(R βhat* - R βhat). The sample-side OLS distribution limit and scalar limit-law calibration remain explicit, matching Hansen Theorem 10.13.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {k : Type u_8} [inst_2 : Fintype k] [inst_3 : DecidableEq k]
  {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η]
  {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {ξ : Ωlim → Real} {Clin Cbeta q α : Real} (β : k → Real) (R : Matrix Unit k Real),
      MeasureTheory.TendstoInDistribution
          (fun n ω =>
            instHMul.hMul (a n)
              (instHSub.hSub
                (HansenEconometrics.linearRestrictionEstimate R
                  (HansenEconometrics.olsBetaOrZero (HansenEconometrics.stackRegressors X n ω)
                    (HansenEconometrics.stackOutcomes y n ω)))
                (HansenEconometrics.linearRestrictionEstimate R β)))
          Filter.atTop ξ (fun x => μ) ν →
        (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
          HansenEconometrics.ScoreCLTConditions μ X e →
            (HansenEconometrics.scoreCovMat μ X e).PosDef →
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                    R.transpose).PosDef →
                Filter.Eventually
                    (fun n =>
                      ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                        Real.instLE.le
                          ((PiLp.instNorm 2 fun x => Real).norm
                            (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                          Clin)
                    Filter.atTop →
                  Filter.Eventually
                      (fun n =>
                        ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                          Real.instLE.le
                            ((PiLp.instNorm 2 fun x => Real).norm
                              (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                            Cbeta)
                      Filter.atTop →
                    (∀ (δ : Real),
                        Real.instLT.lt 0 δ →
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              (ProbabilityTheory.uniformOn Set.univ).real
                                (setOf fun ωs =>
                                  Real.instLE.le δ
                                    (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω
                                      ωs)))
                            Filter.atTop fun x => 0) →
                      Real.instLT.lt 0 α →
                        Real.instLT.lt α 1 →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub (Real.instNeg.neg q) ε))
                                  (instHDiv.hDiv α 2)) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt (instHDiv.hDiv α 2)
                                    ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub q ε))
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                                (∀ (ε : Real),
                                    Real.instLT.lt 0 ε →
                                      Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                        ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd q ε))) →
                                  (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                                    (∀ (n : Nat),
                                        AEMeasurable
                                          (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                            (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                            (fun n ω ωs =>
                                              HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R
                                                X y n ω ωs)
                                            (instHDiv.hDiv α 2) n)
                                          μ) →
                                      (∀ (n : Nat),
                                          AEMeasurable
                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                              (fun n ω ωs =>
                                                HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                  R X y n ω ωs)
                                              (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                            μ) →
                                        ProbabilityTheory.HasLaw ξ η ν →
                                          ProbabilityTheory.HasLaw (fun z => z.ofLp Unit.unit) η
                                              (ProbabilityTheory.multivariateGaussian 0
                                                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                                                    (HansenEconometrics.heteroAsymCov μ X e))
                                                  R.transpose)) →
                                            Real.instLE.le 0 q →
                                              Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q))
                                                  (instHDiv.hDiv α 2) →
                                                Eq ((ProbabilityTheory.cdf η).toFun q)
                                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                                  Filter.Tendsto
                                                    (fun n =>
                                                      MeasureTheory.Measure.instFunLike.coe μ
                                                        (setOf fun ω =>
                                                          HansenEconometrics.percentileCIEvent
                                                            (HansenEconometrics.linearRestrictionEstimate R β)
                                                            (instHAdd.hAdd
                                                              (HansenEconometrics.linearRestrictionEstimate R
                                                                (HansenEconometrics.olsBetaOrZero
                                                                  (HansenEconometrics.stackRegressors X n ω)
                                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                                              ⋯)
                                                            (instHAdd.hAdd
                                                              (HansenEconometrics.linearRestrictionEstimate R
                                                                (HansenEconometrics.olsBetaOrZero
                                                                  (HansenEconometrics.stackRegressors X n ω)
                                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                                              ⋯)))
                                                    Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (13)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.olsBetaOrZero
HansenEconometrics.percentileCIEvent
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:2218

theorem HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_olsBetaOrZero_gapEnvelope_bounds

Strict-CDF finite OLS percentile-interval wrapper.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {k : Type u_8} [inst_2 : Fintype k] [inst_3 : DecidableEq k]
  {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η]
  {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {a : Nat → Real},
  (∀ (n : Nat), Real.instLT.lt 0 (a n)) →
    ∀ {ξ : Ωlim → Real} {Clin Cbeta q α : Real} (β : k → Real) (R : Matrix Unit k Real),
      MeasureTheory.TendstoInDistribution
          (fun n ω =>
            instHMul.hMul (a n)
              (instHSub.hSub
                (HansenEconometrics.linearRestrictionEstimate R
                  (HansenEconometrics.olsBetaOrZero (HansenEconometrics.stackRegressors X n ω)
                    (HansenEconometrics.stackOutcomes y n ω)))
                (HansenEconometrics.linearRestrictionEstimate R β)))
          Filter.atTop ξ (fun x => μ) ν →
        (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
          HansenEconometrics.ScoreCLTConditions μ X e →
            (HansenEconometrics.scoreCovMat μ X e).PosDef →
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                    R.transpose).PosDef →
                Filter.Eventually
                    (fun n =>
                      ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                        Real.instLE.le
                          ((PiLp.instNorm 2 fun x => Real).norm
                            (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                          Clin)
                    Filter.atTop →
                  Filter.Eventually
                      (fun n =>
                        ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                          Real.instLE.le
                            ((PiLp.instNorm 2 fun x => Real).norm
                              (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                            Cbeta)
                      Filter.atTop →
                    (∀ (δ : Real),
                        Real.instLT.lt 0 δ →
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              (ProbabilityTheory.uniformOn Set.univ).real
                                (setOf fun ωs =>
                                  Real.instLE.le δ
                                    (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω
                                      ωs)))
                            Filter.atTop fun x => 0) →
                      Real.instLT.lt 0 α →
                        Real.instLT.lt α 1 →
                          (StrictMono fun x => (ProbabilityTheory.cdf η).toFun x) →
                            (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                              (∀ (n : Nat),
                                  AEMeasurable
                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                      (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                      (fun n ω ωs =>
                                        HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n
                                          ω ωs)
                                      (instHDiv.hDiv α 2) n)
                                    μ) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                        (fun n ω ωs =>
                                          HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y
                                            n ω ωs)
                                        (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                      μ) →
                                  ProbabilityTheory.HasLaw ξ η ν →
                                    ProbabilityTheory.HasLaw (fun z => z.ofLp Unit.unit) η
                                        (ProbabilityTheory.multivariateGaussian 0
                                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                                              (HansenEconometrics.heteroAsymCov μ X e))
                                            R.transpose)) →
                                      Real.instLE.le 0 q →
                                        Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                          Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                            Filter.Tendsto
                                              (fun n =>
                                                MeasureTheory.Measure.instFunLike.coe μ
                                                  (setOf fun ω =>
                                                    HansenEconometrics.percentileCIEvent
                                                      (HansenEconometrics.linearRestrictionEstimate R β)
                                                      (instHAdd.hAdd
                                                        (HansenEconometrics.linearRestrictionEstimate R
                                                          (HansenEconometrics.olsBetaOrZero
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        (instHDiv.hDiv
                                                          (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                            (fun n x => ⋯)
                                                            (fun n ω ωs =>
                                                              HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                                R X y n ω ωs)
                                                            (instHDiv.hDiv α 2) n ω)
                                                          (a n)))
                                                      (instHAdd.hAdd
                                                        (HansenEconometrics.linearRestrictionEstimate R
                                                          (HansenEconometrics.olsBetaOrZero
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        (instHDiv.hDiv
                                                          (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                            (fun n x => ⋯)
                                                            (fun n ω ωs =>
                                                              HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                                R X y n ω ωs)
                                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                                          (a n)))))
                                              Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (13)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.olsBetaOrZero
HansenEconometrics.percentileCIEvent
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:2393

theorem HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_olsBetaOrZero_gapEnvelope_bounds_sampleCLT

Strict-CDF version of the finite OLS percentile-interval wrapper whose sample-side OLS linear-restriction CLT is supplied by Chapter 7.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν]
  {k : Type u_8} [inst_2 : Fintype k] [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {ξ : Ωlim → Real} {Clin Cbeta q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  ProbabilityTheory.HasLaw ξ
      (ProbabilityTheory.gaussianReal 0
        (HansenEconometrics.olsProjectionAsymVar μ X e (R.transpose.mulVec fun x => 1)).toNNReal)
      ν →
    (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
      HansenEconometrics.ScoreCLTConditions μ X e →
        (HansenEconometrics.scoreCovMat μ X e).PosDef →
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                R.transpose).PosDef →
            Filter.Eventually
                (fun n =>
                  ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                    Real.instLE.le
                      ((PiLp.instNorm 2 fun x => Real).norm
                        (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                      Clin)
                Filter.atTop →
              Filter.Eventually
                  (fun n =>
                    ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                      Real.instLE.le
                        ((PiLp.instNorm 2 fun x => Real).norm
                          (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                        Cbeta)
                  Filter.atTop →
                (∀ (δ : Real),
                    Real.instLT.lt 0 δ →
                      MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          (ProbabilityTheory.uniformOn Set.univ).real
                            (setOf fun ωs =>
                              Real.instLE.le δ
                                (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω
                                  ωs)))
                        Filter.atTop fun x => 0) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (StrictMono fun x =>
                          (ProbabilityTheory.cdf
                                (ProbabilityTheory.gaussianReal 0
                                  (HansenEconometrics.olsProjectionAsymVar μ X e
                                      (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                            x) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                (fun n ω ωs =>
                                  HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                                (instHDiv.hDiv α 2) n)
                              μ) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                  (fun n ω ωs =>
                                    HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω
                                      ωs)
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                μ) →
                            Real.instLE.le 0 q →
                              Eq
                                  ((ProbabilityTheory.cdf
                                        (ProbabilityTheory.gaussianReal 0
                                          (HansenEconometrics.olsProjectionAsymVar μ X e
                                              (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                    (Real.instNeg.neg q))
                                  (instHDiv.hDiv α 2) →
                                Eq
                                    ((ProbabilityTheory.cdf
                                          (ProbabilityTheory.gaussianReal 0
                                            (HansenEconometrics.olsProjectionAsymVar μ X e
                                                (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                      q)
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.percentileCIEvent
                                            (HansenEconometrics.linearRestrictionEstimate R β)
                                            (instHAdd.hAdd
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (instHDiv.hDiv
                                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                  (fun n ω ωs =>
                                                    HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                      R X y n ω ωs)
                                                  (instHDiv.hDiv α 2) n ω)
                                                (ite (Eq n 0) 1 n.cast.sqrt)))
                                            (instHAdd.hAdd
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (instHDiv.hDiv
                                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                  (fun n ω ωs =>
                                                    HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                      R X y n ω ωs)
                                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                                (ite (Eq n 0) 1 n.cast.sqrt)))))
                                    Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (14)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsProjectionAsymVar
HansenEconometrics.percentileCIEvent
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3211

theorem HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_olsBetaOrZero_gapEnvelope_bounds_sampleCLT_brackets

Finite OLS percentile-interval wrapper with the sample-side OLS linear-restriction CLT discharged by Chapter 7.

The scale is if n = 0 then 1 else sqrt n: it is positive for every n, as required by the percentile endpoint API, and agrees eventually with Hansen’s usual sqrt n scaling.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν]
  {k : Type u_8} [inst_2 : Fintype k] [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {ξ : Ωlim → Real} {Clin Cbeta q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  ProbabilityTheory.HasLaw ξ
      (ProbabilityTheory.gaussianReal 0
        (HansenEconometrics.olsProjectionAsymVar μ X e (R.transpose.mulVec fun x => 1)).toNNReal)
      ν →
    (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
      HansenEconometrics.ScoreCLTConditions μ X e →
        (HansenEconometrics.scoreCovMat μ X e).PosDef →
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                R.transpose).PosDef →
            Filter.Eventually
                (fun n =>
                  ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                    Real.instLE.le
                      ((PiLp.instNorm 2 fun x => Real).norm
                        (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                      Clin)
                Filter.atTop →
              Filter.Eventually
                  (fun n =>
                    ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                      Real.instLE.le
                        ((PiLp.instNorm 2 fun x => Real).norm
                          (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                        Cbeta)
                  Filter.atTop →
                (∀ (δ : Real),
                    Real.instLT.lt 0 δ →
                      MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          (ProbabilityTheory.uniformOn Set.univ).real
                            (setOf fun ωs =>
                              Real.instLE.le δ
                                (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω
                                  ωs)))
                        Filter.atTop fun x => 0) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt
                              ((ProbabilityTheory.cdf
                                    (ProbabilityTheory.gaussianReal 0
                                      (HansenEconometrics.olsProjectionAsymVar μ X e
                                          (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                (instHSub.hSub (Real.instNeg.neg q) ε))
                              (instHDiv.hDiv α 2)) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt (instHDiv.hDiv α 2)
                                ((ProbabilityTheory.cdf
                                      (ProbabilityTheory.gaussianReal 0
                                        (HansenEconometrics.olsProjectionAsymVar μ X e
                                            (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                  (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt
                                  ((ProbabilityTheory.cdf
                                        (ProbabilityTheory.gaussianReal 0
                                          (HansenEconometrics.olsProjectionAsymVar μ X e
                                              (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                    (instHSub.hSub q ε))
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                    ((ProbabilityTheory.cdf
                                          (ProbabilityTheory.gaussianReal 0
                                            (HansenEconometrics.olsProjectionAsymVar μ X e
                                                (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                      (instHAdd.hAdd q ε))) →
                              (∀ (n : Nat),
                                  AEMeasurable
                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                      (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                      (fun n ω ωs =>
                                        HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n
                                          ω ωs)
                                      (instHDiv.hDiv α 2) n)
                                    μ) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                        (fun n ω ωs =>
                                          HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y
                                            n ω ωs)
                                        (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                      μ) →
                                  Real.instLE.le 0 q →
                                    Eq
                                        ((ProbabilityTheory.cdf
                                              (ProbabilityTheory.gaussianReal 0
                                                (HansenEconometrics.olsProjectionAsymVar μ X e
                                                    (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                          (Real.instNeg.neg q))
                                        (instHDiv.hDiv α 2) →
                                      Eq
                                          ((ProbabilityTheory.cdf
                                                (ProbabilityTheory.gaussianReal 0
                                                  (HansenEconometrics.olsProjectionAsymVar μ X e
                                                      (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                            q)
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                        Filter.Tendsto
                                          (fun n =>
                                            MeasureTheory.Measure.instFunLike.coe μ
                                              (setOf fun ω =>
                                                HansenEconometrics.percentileCIEvent
                                                  (HansenEconometrics.linearRestrictionEstimate R β)
                                                  (instHAdd.hAdd
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (instHDiv.hDiv
                                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                        (fun n ω ωs =>
                                                          HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                            R X y n ω ωs)
                                                        (instHDiv.hDiv α 2) n ω)
                                                      (ite (Eq n 0) 1 n.cast.sqrt)))
                                                  (instHAdd.hAdd
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (instHDiv.hDiv
                                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                        (fun n ω ωs =>
                                                          HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                            R X y n ω ωs)
                                                        (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                                      (ite (Eq n 0) 1 n.cast.sqrt)))))
                                          Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (14)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsProjectionAsymVar
HansenEconometrics.percentileCIEvent
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:2861

theorem HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_olsBetaOrZero_gapEnvelope_bounds_sampleCLT_gaussian

Direct Gaussian-law version of the strict-CDF finite OLS percentile-interval wrapper whose sample-side OLS linear-restriction CLT is supplied by Chapter 7.

This fixes the auxiliary limit space to the Gaussian law itself and the limit random variable to the identity map.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_8} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Clin Cbeta q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
    HansenEconometrics.ScoreCLTConditions μ X e →
      (HansenEconometrics.scoreCovMat μ X e).PosDef →
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
              R.transpose).PosDef →
          Filter.Eventually
              (fun n =>
                ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                  Real.instLE.le
                    ((PiLp.instNorm 2 fun x => Real).norm
                      (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                    Clin)
              Filter.atTop →
            Filter.Eventually
                (fun n =>
                  ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                    Real.instLE.le
                      ((PiLp.instNorm 2 fun x => Real).norm
                        (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                      Cbeta)
                Filter.atTop →
              (∀ (δ : Real),
                  Real.instLT.lt 0 δ →
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Real.instLE.le δ
                              (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                      Filter.atTop fun x => 0) →
                Real.instLT.lt 0 α →
                  Real.instLT.lt α 1 →
                    (StrictMono fun x =>
                        (ProbabilityTheory.cdf
                              (ProbabilityTheory.gaussianReal 0
                                (HansenEconometrics.olsProjectionAsymVar μ X e
                                    (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                          x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                              (fun n x => ProbabilityTheory.uniformOn Set.univ)
                              (fun n ω ωs =>
                                HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                              (instHDiv.hDiv α 2) n)
                            μ) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                (fun n ω ωs =>
                                  HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                              μ) →
                          Real.instLE.le 0 q →
                            Eq
                                ((ProbabilityTheory.cdf
                                      (ProbabilityTheory.gaussianReal 0
                                        (HansenEconometrics.olsProjectionAsymVar μ X e
                                            (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                  (Real.instNeg.neg q))
                                (instHDiv.hDiv α 2) →
                              Eq
                                  ((ProbabilityTheory.cdf
                                        (ProbabilityTheory.gaussianReal 0
                                          (HansenEconometrics.olsProjectionAsymVar μ X e
                                              (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                    q)
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                Filter.Tendsto
                                  (fun n =>
                                    MeasureTheory.Measure.instFunLike.coe μ
                                      (setOf fun ω =>
                                        HansenEconometrics.percentileCIEvent
                                          (HansenEconometrics.linearRestrictionEstimate R β)
                                          (instHAdd.hAdd
                                            (HansenEconometrics.linearRestrictionEstimate R
                                              (HansenEconometrics.olsBetaOrZero
                                                (HansenEconometrics.stackRegressors X n ω)
                                                (HansenEconometrics.stackOutcomes y n ω)))
                                            (instHDiv.hDiv
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                (fun n ω ωs =>
                                                  HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                    R X y n ω ωs)
                                                (instHDiv.hDiv α 2) n ω)
                                              (ite (Eq n 0) 1 n.cast.sqrt)))
                                          (instHAdd.hAdd
                                            (HansenEconometrics.linearRestrictionEstimate R
                                              (HansenEconometrics.olsBetaOrZero
                                                (HansenEconometrics.stackRegressors X n ω)
                                                (HansenEconometrics.stackOutcomes y n ω)))
                                            (instHDiv.hDiv
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                (fun n ω ωs =>
                                                  HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                    R X y n ω ωs)
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                              (ite (Eq n 0) 1 n.cast.sqrt)))))
                                  Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (14)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsProjectionAsymVar
HansenEconometrics.percentileCIEvent
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3709

theorem

HansenEconometrics.chapter10_percentileCI_coverage_indexed_finSucc_olsBetaOrZero_gapEnvelope_bounds_sampleCLT_brackets_gaussian

Direct Gaussian-law version of the local-CDF finite OLS percentile-interval wrapper whose sample-side OLS linear-restriction CLT is supplied by Chapter 7.

This fixes the auxiliary limit space to the Gaussian law itself and the limit random variable to the identity map, while retaining local CDF bracketing at the percentile endpoints.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_8} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Clin Cbeta q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
    HansenEconometrics.ScoreCLTConditions μ X e →
      (HansenEconometrics.scoreCovMat μ X e).PosDef →
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
              R.transpose).PosDef →
          Filter.Eventually
              (fun n =>
                ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                  Real.instLE.le
                    ((PiLp.instNorm 2 fun x => Real).norm
                      (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                    Clin)
              Filter.atTop →
            Filter.Eventually
                (fun n =>
                  ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                    Real.instLE.le
                      ((PiLp.instNorm 2 fun x => Real).norm
                        (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                      Cbeta)
                Filter.atTop →
              (∀ (δ : Real),
                  Real.instLT.lt 0 δ →
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Real.instLE.le δ
                              (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                      Filter.atTop fun x => 0) →
                Real.instLT.lt 0 α →
                  Real.instLT.lt α 1 →
                    (∀ (ε : Real),
                        Real.instLT.lt 0 ε →
                          Real.instLT.lt
                            ((ProbabilityTheory.cdf
                                  (ProbabilityTheory.gaussianReal 0
                                    (HansenEconometrics.olsProjectionAsymVar μ X e
                                        (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                              (instHSub.hSub (Real.instNeg.neg q) ε))
                            (instHDiv.hDiv α 2)) →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt (instHDiv.hDiv α 2)
                              ((ProbabilityTheory.cdf
                                    (ProbabilityTheory.gaussianReal 0
                                      (HansenEconometrics.olsProjectionAsymVar μ X e
                                          (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf
                                      (ProbabilityTheory.gaussianReal 0
                                        (HansenEconometrics.olsProjectionAsymVar μ X e
                                            (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                  (instHSub.hSub q ε))
                                (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                  ((ProbabilityTheory.cdf
                                        (ProbabilityTheory.gaussianReal 0
                                          (HansenEconometrics.olsProjectionAsymVar μ X e
                                              (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                    (instHAdd.hAdd q ε))) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                    (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                    (fun n ω ωs =>
                                      HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω
                                        ωs)
                                    (instHDiv.hDiv α 2) n)
                                  μ) →
                              (∀ (n : Nat),
                                  AEMeasurable
                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                      (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                      (fun n ω ωs =>
                                        HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n
                                          ω ωs)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                    μ) →
                                Real.instLE.le 0 q →
                                  Eq
                                      ((ProbabilityTheory.cdf
                                            (ProbabilityTheory.gaussianReal 0
                                              (HansenEconometrics.olsProjectionAsymVar μ X e
                                                  (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                        (Real.instNeg.neg q))
                                      (instHDiv.hDiv α 2) →
                                    Eq
                                        ((ProbabilityTheory.cdf
                                              (ProbabilityTheory.gaussianReal 0
                                                (HansenEconometrics.olsProjectionAsymVar μ X e
                                                    (R.transpose.mulVec fun x => 1)).toNNReal)).toFun
                                          q)
                                        (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                      Filter.Tendsto
                                        (fun n =>
                                          MeasureTheory.Measure.instFunLike.coe μ
                                            (setOf fun ω =>
                                              HansenEconometrics.percentileCIEvent
                                                (HansenEconometrics.linearRestrictionEstimate R β)
                                                (instHAdd.hAdd
                                                  (HansenEconometrics.linearRestrictionEstimate R
                                                    (HansenEconometrics.olsBetaOrZero
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  (instHDiv.hDiv
                                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                      (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                      (fun n ω ωs =>
                                                        HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                          R X y n ω ωs)
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (ite (Eq n 0) 1 n.cast.sqrt)))
                                                (instHAdd.hAdd
                                                  (HansenEconometrics.linearRestrictionEstimate R
                                                    (HansenEconometrics.olsBetaOrZero
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  (instHDiv.hDiv
                                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                      (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                      (fun n ω ωs =>
                                                        HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
                                                          R X y n ω ωs)
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)
                                                    (ite (Eq n 0) 1 n.cast.sqrt)))))
                                        Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (14)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsProjectionAsymVar
HansenEconometrics.percentileCIEvent
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3063

Equations 10.21-10.276 of 10 linked endpoints

The bias-corrected interval uses z_0=\Phi^{-1}(P^*(\hat\theta^*\le\hat\theta)) and adjusted levels \Phi(2z_0+z_\alpha).

theorem HansenEconometrics.biasCorrectedIdealCIEvent_probability_eq_one_sub_alpha

Hansen BC exact coverage: if the pivot critical values have endpoint CDF masses alpha / 2 and 1 - alpha / 2, then the ideal BC interval has coverage 1 - alpha.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {eta : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure eta] [MeasureTheory.NoAtoms eta]
  {psi : Real → Real} {theta z0 zLower zUpper alpha : Real} {thetaHat : Ω → Real},
  ProbabilityTheory.HasLaw (fun ω => HansenEconometrics.biasCorrectedPivot psi theta z0 (thetaHat ω)) eta μ →
    Real.instLE.le zLower zUpper →
      Eq ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zUpper)) (instHDiv.hDiv alpha 2) →
        Eq ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zLower)) (instHSub.hSub 1 (instHDiv.hDiv alpha 2)) →
          Eq
            (MeasureTheory.Measure.instFunLike.coe μ
              (setOf fun ω => HansenEconometrics.biasCorrectedIdealCIEvent psi theta (thetaHat ω) z0 zLower zUpper))
            (ENNReal.ofReal (instHSub.hSub 1 alpha))

Direct statement dependencies (2)

HansenEconometrics.biasCorrectedIdealCIEvent
HansenEconometrics.biasCorrectedPivot

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4026

theorem HansenEconometrics.biasCorrectedIdealCIEvent_probability_eq_cdf_sub

Hansen BC exact-coverage bridge in CDF-increment form.

Under the transformed pivotal model (10.21), the ideal BC interval coverage is the probability that the pivot lies in [-zUpper, -zLower]. A non-atomic limit law reads this probability as a CDF increment.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {eta : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure eta] [MeasureTheory.NoAtoms eta]
  {psi : Real → Real} {theta z0 zLower zUpper : Real} {thetaHat : Ω → Real},
  ProbabilityTheory.HasLaw (fun ω => HansenEconometrics.biasCorrectedPivot psi theta z0 (thetaHat ω)) eta μ →
    Real.instLE.le zLower zUpper →
      Eq
        (MeasureTheory.Measure.instFunLike.coe μ
          (setOf fun ω => HansenEconometrics.biasCorrectedIdealCIEvent psi theta (thetaHat ω) z0 zLower zUpper))
        (ENNReal.ofReal
          (instHSub.hSub ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zLower))
            ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zUpper))))

Direct statement dependencies (2)

HansenEconometrics.biasCorrectedIdealCIEvent
HansenEconometrics.biasCorrectedPivot

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4003

def HansenEconometrics.biasCorrectedPivot

Hansen equation (10.21): transformed BC pivot psi(thetaHat) - psi(theta) + z0.

Formal statement

(Real → Real) → Real → Real → Real → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3969

def HansenEconometrics.bootstrapBiasCorrection

Hansen equation (10.23): normal-scale bias correction from a quantile function. For the BC interval this quantile function is Phi^{-1}.

Formal statement

(Real → Real) → Real → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3946

def HansenEconometrics.bootstrapMedianBiasShare

Hansen equation (10.22): finite-replication bootstrap median-bias share p* = B^{-1} sum_b 1{theta*_b <= thetaHat}.

Formal statement

{B : Type u_8} → [Fintype B] → Real → (B → Real) → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3939

def HansenEconometrics.biasCorrectedAdjustedLevel

Hansen equation (10.24): BC adjusted percentile level x(alpha) = Phi(z_alpha + 2 z0).

Formal statement

(Real → Real) → (Real → Real) → Real → Real → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:3952

Equations 10.28/10.296 of 13 linked endpoints

The BCa interval adds acceleration a and uses adjusted level \Phi(z_0+(z_0+z_\alpha)/(1-a(z_0+z_\alpha))).

theorem HansenEconometrics.bcaIdealCIEvent_probability_eq_one_sub_alpha

Hansen BCa exact coverage: endpoint CDF masses alpha / 2 and 1 - alpha / 2 imply coverage 1 - alpha.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {eta : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure eta] [MeasureTheory.NoAtoms eta]
  {psi : Real → Real} {theta accel z0 zLower zUpper alpha : Real} {thetaHat : Ω → Real},
  ProbabilityTheory.HasLaw (fun ω => HansenEconometrics.bcaPivot psi theta accel z0 (thetaHat ω)) eta μ →
    Real.instLT.lt 0 (instHAdd.hAdd 1 (instHMul.hMul accel (psi theta))) →
      Real.instLT.lt 0 (instHSub.hSub 1 (instHMul.hMul accel (instHAdd.hAdd zLower z0))) →
        Real.instLT.lt 0 (instHSub.hSub 1 (instHMul.hMul accel (instHAdd.hAdd zUpper z0))) →
          Real.instLE.le zLower zUpper →
            Eq ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zUpper)) (instHDiv.hDiv alpha 2) →
              Eq ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zLower))
                  (instHSub.hSub 1 (instHDiv.hDiv alpha 2)) →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe μ
                    (setOf fun ω => HansenEconometrics.bcaIdealCIEvent psi theta (thetaHat ω) accel z0 zLower zUpper))
                  (ENNReal.ofReal (instHSub.hSub 1 alpha))

Direct statement dependencies (2)

HansenEconometrics.bcaIdealCIEvent
HansenEconometrics.bcaPivot

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4285

theorem HansenEconometrics.bcaIdealCIEvent_probability_eq_cdf_sub

Hansen BCa exact-coverage bridge in CDF-increment form.

Under the transformed pivotal model (10.28), the ideal BCa interval coverage is the probability that the BCa pivot lies in [-zUpper, -zLower].

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
  {eta : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure eta] [MeasureTheory.NoAtoms eta]
  {psi : Real → Real} {theta accel z0 zLower zUpper : Real} {thetaHat : Ω → Real},
  ProbabilityTheory.HasLaw (fun ω => HansenEconometrics.bcaPivot psi theta accel z0 (thetaHat ω)) eta μ →
    Real.instLT.lt 0 (instHAdd.hAdd 1 (instHMul.hMul accel (psi theta))) →
      Real.instLT.lt 0 (instHSub.hSub 1 (instHMul.hMul accel (instHAdd.hAdd zLower z0))) →
        Real.instLT.lt 0 (instHSub.hSub 1 (instHMul.hMul accel (instHAdd.hAdd zUpper z0))) →
          Real.instLE.le zLower zUpper →
            Eq
              (MeasureTheory.Measure.instFunLike.coe μ
                (setOf fun ω => HansenEconometrics.bcaIdealCIEvent psi theta (thetaHat ω) accel z0 zLower zUpper))
              (ENNReal.ofReal
                (instHSub.hSub ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zLower))
                  ((ProbabilityTheory.cdf eta).toFun (Real.instNeg.neg zUpper))))

Direct statement dependencies (2)

HansenEconometrics.bcaIdealCIEvent
HansenEconometrics.bcaPivot

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4259

def HansenEconometrics.bcaPivot

Hansen equation (10.28): BCa transformed pivot (psi(thetaHat) - psi(theta)) / (1 + a * psi(theta)) + z0.

Formal statement

(Real → Real) → Real → Real → Real → Real → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4107

def HansenEconometrics.bcaAdjustedLevel

Hansen’s BCa adjusted percentile level: Phi(z0 + (z_alpha + z0) / (1 - a (z_alpha + z0))).

Formal statement

(Real → Real) → (Real → Real) → Real → Real → Real → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4074

def HansenEconometrics.bcaBootstrapPivot

Hansen equation (10.29): the bootstrap analogue of the BCa transformed pivot, centered at the sample estimate.

Formal statement

(Real → Real) → Real → Real → Real → Real → Real

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4113

def HansenEconometrics.bcaPercentileCIEvent

Hansen’s BCa percentile interval event formed from bootstrap quantiles at the accelerated adjusted endpoint levels.

Formal statement

Real → (Real → Real) → (Real → Real) → (Real → Real) → Real → Real → Real → Prop

HansenEconometrics/Chapter10Bootstrap/Percentile.lean:4096

Equations 10.30/10.314 endpoints

The sample and bootstrap t ratios share the limit T_n\xrightarrow{d}\xi and T_n^*\xrightarrow{d^*}\xi.

theorem HansenEconometrics.percentileTCoverageVector_tendstoInDistribution_of_components

Componentwise Slutsky constructor for the percentile-t coverage joint vector.

This assembles the joint convergence premise in chapter10_percentileTCI_coverage_tendsto_of_joint_quantile_limit from the sample t-ratio limit and the two bootstrap percentile-t endpoint limits.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {θ : Real} {θhat se qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real}
  {qLowerLim qUpperLim : Real},
  MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
      Filter.atTop ξ (fun x => μ) ν →
    (MeasureTheory.TendstoInMeasure μ qLower Filter.atTop fun x => qLowerLim) →
      (MeasureTheory.TendstoInMeasure μ qUpper Filter.atTop fun x => qUpperLim) →
        (∀ (n : Nat), AEMeasurable (qLower n) μ) →
          (∀ (n : Nat), AEMeasurable (qUpper n) μ) →
            MeasureTheory.TendstoInDistribution (HansenEconometrics.percentileTCoverageVector θ θhat se qLower qUpper)
              Filter.atTop (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) (fun x => μ) ν

Direct statement dependencies (3)

HansenEconometrics.percentileTCoverageLimitVector
HansenEconometrics.percentileTCoverageVector
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:60

theorem HansenEconometrics.chapter10_percentileTCI_coverage_tendsto_of_joint_quantile_limit

Hansen Theorem 10.14, percentile-t interval coverage bridge.

If the sample t-ratio and bootstrap percentile-t critical values jointly converge to (ξ, qL, qU), and the limiting coverage boundary has zero probability, then percentile-t interval coverage converges to P[qL <= ξ <= qU]. Hansen’s first-order validity conclusion 1 - α is obtained by instantiating this bridge with the bootstrap quantile limits from (10.31).

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {θ : Real} {θhat se qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real}
  {qLowerLim qUpperLim : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (HansenEconometrics.percentileTCoverageVector θ θhat se qLower qUpper)
        Filter.atTop (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) (fun x => μ) ν →
      Eq
          (MeasureTheory.Measure.instFunLike.coe
            (MeasureTheory.Measure.map (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) ν)
            (frontier HansenEconometrics.percentileTCoverageSet))
          0 →
        Filter.Tendsto
          (fun n =>
            MeasureTheory.Measure.instFunLike.coe μ
              (setOf fun ω => HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω) (qUpper n ω)))
          Filter.atTop
          (nhds
            (MeasureTheory.Measure.instFunLike.coe
              (MeasureTheory.Measure.map (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) ν)
              HansenEconometrics.percentileTCoverageSet))

Direct statement dependencies (4)

HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTCoverageLimitVector
HansenEconometrics.percentileTCoverageSet
HansenEconometrics.percentileTCoverageVector

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:303

theorem HansenEconometrics.chapter10_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_lowerQuantiles

Symmetric percentile-t coverage from bootstrap lower quantiles.

Pointwise convergence in probability of the conditional bootstrap CDF, plus the concrete lower-generalized-inverse bracketing assumptions, identifies the bootstrap percentile-t endpoints at levels α / 2 and 1 - α / 2. The result then feeds those endpoint limits into the symmetric [-q, q] coverage wrapper.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν]
  {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → Real} {θ : Real} {θhat se : Nat → Ω → Real}
  {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (∀ (n : Nat) (ω : Ω), Monotone fun x => HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) →
        (∀ (n : Nat) (ω : Ω),
            (setOf fun x =>
                Real.instLE.le (instHDiv.hDiv α 2)
                  (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
          (∀ (n : Nat) (ω : Ω),
              BddBelow
                (setOf fun x =>
                  Real.instLE.le (instHDiv.hDiv α 2) (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
            (∀ (n : Nat) (ω : Ω) (x : Real),
                Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) (instHDiv.hDiv α 2) →
                  Exists fun δ =>
                    And (Real.instLT.lt 0 δ)
                      (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω)
                        (instHDiv.hDiv α 2))) →
              (∀ (n : Nat) (ω : Ω),
                  (setOf fun x =>
                      Real.instLE.le (instHSub.hSub 1 (instHDiv.hDiv α 2))
                        (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
                (∀ (n : Nat) (ω : Ω),
                    BddBelow
                      (setOf fun x =>
                        Real.instLE.le (instHSub.hSub 1 (instHDiv.hDiv α 2))
                          (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
                  (∀ (n : Nat) (ω : Ω) (x : Real),
                      Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)
                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                        Exists fun δ =>
                          And (Real.instLT.lt 0 δ)
                            (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω)
                              (instHSub.hSub 1 (instHDiv.hDiv α 2)))) →
                    (StrictMono fun x => (ProbabilityTheory.cdf η).toFun x) →
                      (∀ (x : Real),
                          MeasureTheory.TendstoInMeasure μ
                            (fun n ω => HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) Filter.atTop fun x_1 =>
                            (ProbabilityTheory.cdf η).toFun x) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar (instHDiv.hDiv α 2) n) μ) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                μ) →
                            ProbabilityTheory.HasLaw ξ η ν →
                              Real.instLE.le 0 q →
                                Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (4)

HansenEconometrics.bootstrapScalarCDF
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:734

theorem HansenEconometrics.chapter10_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrapDistribution_lowerQuantiles

Symmetric percentile-t coverage from bootstrap-distribution convergence of the bootstrap t-ratio statistic.

This is the Definition 10.2-facing version of chapter10_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_lowerQuantiles.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν]
  {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η]
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → Real} {θ : Real} {θhat se : Nat → Ω → Real}
  {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω),
            (setOf fun x =>
                Real.instLE.le (instHDiv.hDiv α 2)
                  (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
          (∀ (n : Nat) (ω : Ω),
              BddBelow
                (setOf fun x =>
                  Real.instLE.le (instHDiv.hDiv α 2) (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
            (∀ (n : Nat) (ω : Ω) (x : Real),
                Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω) (instHDiv.hDiv α 2) →
                  Exists fun δ =>
                    And (Real.instLT.lt 0 δ)
                      (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω)
                        (instHDiv.hDiv α 2))) →
              (∀ (n : Nat) (ω : Ω),
                  (setOf fun x =>
                      Real.instLE.le (instHSub.hSub 1 (instHDiv.hDiv α 2))
                        (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)).Nonempty) →
                (∀ (n : Nat) (ω : Ω),
                    BddBelow
                      (setOf fun x =>
                        Real.instLE.le (instHSub.hSub 1 (instHDiv.hDiv α 2))
                          (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω))) →
                  (∀ (n : Nat) (ω : Ω) (x : Real),
                      Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar x n ω)
                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                        Exists fun δ =>
                          And (Real.instLT.lt 0 δ)
                            (Real.instLT.lt (HansenEconometrics.bootstrapScalarCDF Pstar Tstar (instHAdd.hAdd x δ) n ω)
                              (instHSub.hSub 1 (instHDiv.hDiv α 2)))) →
                    (StrictMono fun x => (ProbabilityTheory.cdf η).toFun x) →
                      (HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs x => Tstar n ω ωs) η
                          fun x x_1 => x) →
                        (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar (instHDiv.hDiv α 2) n) μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              ProbabilityTheory.HasLaw ξ η ν →
                                Real.instLE.le 0 q →
                                  Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                    Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                      Filter.Tendsto
                                        (fun n =>
                                          MeasureTheory.Measure.instFunLike.coe μ
                                            (setOf fun ω =>
                                              HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                  (instHDiv.hDiv α 2) n ω)
                                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                        Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapScalarCDF
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:824

Theorem 10.146 of 32 linked endpoints

If the sample and bootstrap t ratios share a continuous limit, then \Pr(\theta\in C_{\mathrm{pt}})\to1-\alpha.

theorem HansenEconometrics.chapter10_percentileTCI_coverage_tendsto_of_joint_quantile_limit

Hansen Theorem 10.14, percentile-t interval coverage bridge.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {θ : Real} {θhat se qLower qUpper : Nat → Ω → Real} {ξ : Ωlim → Real}
  {qLowerLim qUpperLim : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (HansenEconometrics.percentileTCoverageVector θ θhat se qLower qUpper)
        Filter.atTop (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) (fun x => μ) ν →
      Eq
          (MeasureTheory.Measure.instFunLike.coe
            (MeasureTheory.Measure.map (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) ν)
            (frontier HansenEconometrics.percentileTCoverageSet))
          0 →
        Filter.Tendsto
          (fun n =>
            MeasureTheory.Measure.instFunLike.coe μ
              (setOf fun ω => HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω) (qUpper n ω)))
          Filter.atTop
          (nhds
            (MeasureTheory.Measure.instFunLike.coe
              (MeasureTheory.Measure.map (HansenEconometrics.percentileTCoverageLimitVector ξ qLowerLim qUpperLim) ν)
              HansenEconometrics.percentileTCoverageSet))

Direct statement dependencies (4)

HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTCoverageLimitVector
HansenEconometrics.percentileTCoverageSet
HansenEconometrics.percentileTCoverageVector

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:303

theorem HansenEconometrics.chapter10_percentileTCI_coverage_indexed_finSucc_resampleMean_brackets

Indexed ordinary nonparametric-bootstrap percentile-t coverage from the concrete normalized scalar Fin (n+1) resample-mean CLT.

The bootstrap percentile-t critical values are the lower generalized inverses of the conditional CDF of sqrt(n+1) (Ybar*_n - Ybar_n) under the finite ordinary resampling law. The sample-side t-ratio convergence, positive-standard-error premise, and endpoint calibration remain explicit, as in Hansen Theorem 10.14.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (Y 0) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.covMat μ fun ω x => Y 0 ω).PosDef →
          ∀ {θ : Real} {θhat se : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
            (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
              MeasureTheory.TendstoInDistribution
                  (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω)) Filter.atTop ξ (fun x => μ)
                  ν →
                Real.instLT.lt 0 α →
                  Real.instLT.lt α 1 →
                    (∀ (ε : Real),
                        Real.instLT.lt 0 ε →
                          Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub (Real.instNeg.neg q) ε))
                            (instHDiv.hDiv α 2)) →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt (instHDiv.hDiv α 2)
                              ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub q ε))
                                (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                  ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd q ε))) →
                            (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                              (∀ (n : Nat),
                                  AEMeasurable
                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                      (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                      (fun n ω ωs =>
                                        instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                          (instHSub.hSub
                                            (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                              (fun ωs t => ωs t) ωs)
                                            (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                                      (instHDiv.hDiv α 2) n)
                                    μ) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                        (fun n ω ωs =>
                                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                            (instHSub.hSub
                                              (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                                (fun ωs t => ωs t) ωs)
                                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                                        (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                      μ) →
                                  ProbabilityTheory.HasLaw ξ η ν →
                                    ProbabilityTheory.HasLaw (fun z => z.ofLp Unit.unit) η
                                        (ProbabilityTheory.multivariateGaussian 0
                                          (HansenEconometrics.covMat μ fun ω x => Y 0 ω)) →
                                      Real.instLE.le 0 q →
                                        Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                          Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                            Filter.Tendsto
                                              (fun n =>
                                                MeasureTheory.Measure.instFunLike.coe μ
                                                  (setOf fun ω =>
                                                    HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                        (fun n ω ωs =>
                                                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                                            (instHSub.hSub
                                                              (HansenEconometrics.empiricalBootstrapResampleMean
                                                                (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                                                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                                                        (instHDiv.hDiv α 2) n ω)
                                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                        (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                        (fun n ω ωs =>
                                                          instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                                            (instHSub.hSub
                                                              (HansenEconometrics.empiricalBootstrapResampleMean
                                                                (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                                                              (HansenEconometrics.empiricalMean fun i => Y i.val ω)))
                                                        (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                              Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (6)

HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:1648

theorem HansenEconometrics.chapter10_percentileTCI_coverage_tendsto_one_sub_alpha_of_components_symmetric_cdf

Symmetric endpoint-CDF percentile-t calibration.

This is the Hansen Theorem 10.14 specialization where the limiting bootstrap percentile-t endpoints are -q and q, and the scalar t-ratio limit law has endpoint CDF masses α / 2 and 1 - α / 2.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] {θ : Real} {θhat se qLower qUpper : Nat → Ω → Real}
  {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (MeasureTheory.TendstoInMeasure μ qLower Filter.atTop fun x => Real.instNeg.neg q) →
        (MeasureTheory.TendstoInMeasure μ qUpper Filter.atTop fun x => q) →
          (∀ (n : Nat), AEMeasurable (qLower n) μ) →
            (∀ (n : Nat), AEMeasurable (qUpper n) μ) →
              ProbabilityTheory.HasLaw ξ η ν →
                Real.instLE.le 0 q →
                  Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                    Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                      Filter.Tendsto
                        (fun n =>
                          MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω =>
                              HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω) (qUpper n ω)))
                        Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (2)

HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:603

theorem HansenEconometrics.chapter10_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Regression-facing percentile-t coverage from the Theorem 10.18 bootstrap t-statistic route.

The bootstrap lower quantiles are computed from the studentized transformed statistic TthetaStar / seThetaStar. The joint numerator/standard-error weak limit and scale consistency feed Theorem 10.18’s standard-normal bootstrap CDF wrapper; the existing Theorem 10.14 quantile route then gives 1 - α coverage.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {θ : Real} {θhat se : Nat → Ω → Real} {seθ q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop (fun x => x) (fun x => μ) (ProbabilityTheory.gaussianReal 0 1) →
      Real.instLT.lt 0 seθ →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
            (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
            fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
          (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                (instHDiv.hDiv α 2) n)
                              μ) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                μ) →
                            Real.instLE.le 0 q →
                              Eq
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (Real.instNeg.neg q))
                                  (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                              (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                              (instHDiv.hDiv α 2) n ω)
                                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                              (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                              (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:1910

theorem HansenEconometrics.chapter10_olsHC0_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC0 OLS scalar restriction.

The displayed interval uses the ordinary OLS estimate and HC0 standard error scaled by sqrt n; the harmless n = 0 branch keeps the totalized standard error positive while the proof identifies the statistic with the Chapter 7 HC0 t-statistic eventually.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovStar (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHDiv.hDiv α 2) n)
                                μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                    (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              Real.instLE.le 0 q →
                                Eq
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (Real.instNeg.neg q))
                                    (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent
                                              (HansenEconometrics.linearRestrictionEstimate R β)
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (ite (Eq n 0) 1
                                                (instHDiv.hDiv
                                                  (HansenEconometrics.linearRestrictionStdError R
                                                    (HansenEconometrics.olsHetCovStar
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  n.cast.sqrt))
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovStar
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5283

theorem HansenEconometrics.chapter10_olsHC1_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC1 OLS scalar restriction.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC1Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHDiv.hDiv α 2) n)
                                μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                    (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              Real.instLE.le 0 q →
                                Eq
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (Real.instNeg.neg q))
                                    (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent
                                              (HansenEconometrics.linearRestrictionEstimate R β)
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (ite (Eq n 0) 1
                                                (instHDiv.hDiv
                                                  (HansenEconometrics.linearRestrictionStdError R
                                                    (HansenEconometrics.olsHetCovHC1Star
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  n.cast.sqrt))
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC1Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5451

Theorem 10.14 auxiliary-limit quantile routes4 endpoints

If the sample and bootstrap t ratios share a continuous limit, then \Pr(\theta\in C_{\mathrm{pt}})\to1-\alpha. Auxiliary bootstrap t-ratio limits can feed the percentile-t lower-quantile route by HasLaw, with strict-CDF or local-limit-CDF endpoint bracketing

theorem HansenEconometrics.chapter10_percentileTCI_coverage_bootstrapDistribution_law_quantile_prob

Symmetric percentile-t coverage from a one-dimensional bootstrap distribution whose scalar limit has law η.

This law-facing variant lets the bootstrap t-ratio limit live on an auxiliary probability space while HasLaw supplies the scalar CDF used to identify the lower generalized-inverse endpoints.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_7}
  [inst_2 : MeasurableSpace Ωstar] {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] {νstar : MeasureTheory.Measure Ωstar} {Zlim : Ωstar → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → Real} {θ : Real} {θhat se : Nat → Ω → Real}
  {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), AEMeasurable (Tstar n ω) (Pstar n ω)) →
          Real.instLT.lt 0 α →
            Real.instLT.lt α 1 →
              (StrictMono fun x => (ProbabilityTheory.cdf η).toFun x) →
                (HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs x => Tstar n ω ωs) νstar
                    fun ωstar x => Zlim ωstar) →
                  ProbabilityTheory.HasLaw Zlim η νstar →
                    (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar (instHDiv.hDiv α 2) n) μ) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                              μ) →
                          ProbabilityTheory.HasLaw ξ η ν →
                            Real.instLE.le 0 q →
                              Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                              (instHDiv.hDiv α 2) n ω)
                                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                              (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:1070

theorem HansenEconometrics.chapter10_percentileTCI_coverage_bootstrapDistribution_law_quantile_prob_brackets

Symmetric percentile-t coverage from an auxiliary one-dimensional bootstrap limit, retaining local CDF bracketing at the lower generalized inverse endpoints.

This law-facing variant is the local-bracketing counterpart of chapter10_percentileTCI_coverage_bootstrapDistribution_law_quantile_prob: the bootstrap t-ratio limit may live on an auxiliary probability space, and HasLaw identifies its scalar CDF with cdf η.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_7}
  [inst_2 : MeasurableSpace Ωstar] {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] {νstar : MeasureTheory.Measure Ωstar} {Zlim : Ωstar → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Tstar : Nat → Ω → Ωs → Real} {θ : Real} {θhat se : Nat → Ω → Real}
  {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), AEMeasurable (Tstar n ω) (Pstar n ω)) →
          Real.instLT.lt 0 α →
            Real.instLT.lt α 1 →
              (∀ (ε : Real),
                  Real.instLT.lt 0 ε →
                    Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub (Real.instNeg.neg q) ε))
                      (instHDiv.hDiv α 2)) →
                (∀ (ε : Real),
                    Real.instLT.lt 0 ε →
                      Real.instLT.lt (instHDiv.hDiv α 2)
                        ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                  (∀ (ε : Real),
                      Real.instLT.lt 0 ε →
                        Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub q ε))
                          (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                    (∀ (ε : Real),
                        Real.instLT.lt 0 ε →
                          Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                            ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd q ε))) →
                      (HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs x => Tstar n ω ωs) νstar
                          fun ωstar x => Zlim ωstar) →
                        ProbabilityTheory.HasLaw Zlim η νstar →
                          (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar (instHDiv.hDiv α 2) n)
                                  μ) →
                              (∀ (n : Nat),
                                  AEMeasurable
                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                    μ) →
                                ProbabilityTheory.HasLaw ξ η ν →
                                  Real.instLE.le 0 q →
                                    Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                      Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                        Filter.Tendsto
                                          (fun n =>
                                            MeasureTheory.Measure.instFunLike.coe μ
                                              (setOf fun ω =>
                                                HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                    (instHDiv.hDiv α 2) n ω)
                                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Tstar
                                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                          Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:1162

theorem HansenEconometrics.chapter10_indexed_percentileTCI_coverage_bootstrapDistribution_law_quantile_prob

Indexed symmetric percentile-t coverage from a one-dimensional bootstrap distribution whose scalar limit has law η.

This sample-size-dependent law-facing wrapper lets the bootstrap t-ratio limit live on an auxiliary probability space while HasLaw supplies the scalar CDF used to identify the lower generalized-inverse endpoints.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} {Ωboot : Nat → Type u_7} [inst : (n : Nat) → MeasurableSpace (Ωboot n)]
  [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_8}
  [inst_3 : MeasurableSpace Ωstar] {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] {νstar : MeasureTheory.Measure Ωstar} {Zlim : Ωstar → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → Real} {θ : Real}
  {θhat se : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), AEMeasurable (Tstar n ω) (Pstar n ω)) →
          Real.instLT.lt 0 α →
            Real.instLT.lt α 1 →
              (StrictMono fun x => (ProbabilityTheory.cdf η).toFun x) →
                (HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar (fun n ω ωs x => Tstar n ω ωs) νstar
                    fun ωstar x => Zlim ωstar) →
                  ProbabilityTheory.HasLaw Zlim η νstar →
                    (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar (instHDiv.hDiv α 2) n)
                            μ) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                              μ) →
                          ProbabilityTheory.HasLaw ξ η ν →
                            Real.instLE.le 0 q →
                              Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                              (instHDiv.hDiv α 2) n ω)
                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                              (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:1465

theorem HansenEconometrics.chapter10_indexed_percentileTCI_coverage_bootstrapDistribution_law_quantile_prob_brackets

Indexed symmetric percentile-t coverage from an auxiliary one-dimensional bootstrap limit, retaining local CDF bracketing at the lower generalized-inverse endpoints.

This sample-size-dependent law-facing wrapper is the indexed counterpart of chapter10_percentileTCI_coverage_bootstrapDistribution_law_quantile_prob_brackets.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} {Ωboot : Nat → Type u_7} [inst : (n : Nat) → MeasurableSpace (Ωboot n)]
  [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_8}
  [inst_3 : MeasurableSpace Ωstar] {η : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] {νstar : MeasureTheory.Measure Ωstar} {Zlim : Ωstar → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {Tstar : (n : Nat) → Ω → Ωboot n → Real} {θ : Real}
  {θhat se : Nat → Ω → Real} {ξ : Ωlim → Real} {q α : Real},
  (∀ (n : Nat) (ω : Ω), Real.instLT.lt 0 (se n ω)) →
    MeasureTheory.TendstoInDistribution (fun n ω => HansenEconometrics.percentileTStatistic θ (θhat n ω) (se n ω))
        Filter.atTop ξ (fun x => μ) ν →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), AEMeasurable (Tstar n ω) (Pstar n ω)) →
          Real.instLT.lt 0 α →
            Real.instLT.lt α 1 →
              (∀ (ε : Real),
                  Real.instLT.lt 0 ε →
                    Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub (Real.instNeg.neg q) ε))
                      (instHDiv.hDiv α 2)) →
                (∀ (ε : Real),
                    Real.instLT.lt 0 ε →
                      Real.instLT.lt (instHDiv.hDiv α 2)
                        ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                  (∀ (ε : Real),
                      Real.instLT.lt 0 ε →
                        Real.instLT.lt ((ProbabilityTheory.cdf η).toFun (instHSub.hSub q ε))
                          (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                    (∀ (ε : Real),
                        Real.instLT.lt 0 ε →
                          Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                            ((ProbabilityTheory.cdf η).toFun (instHAdd.hAdd q ε))) →
                      (HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar (fun n ω ωs x => Tstar n ω ωs)
                          νstar fun ωstar x => Zlim ωstar) →
                        ProbabilityTheory.HasLaw Zlim η νstar →
                          (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf η).toFun y) x) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                    (instHDiv.hDiv α 2) n)
                                  μ) →
                              (∀ (n : Nat),
                                  AEMeasurable
                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                    μ) →
                                ProbabilityTheory.HasLaw ξ η ν →
                                  Real.instLE.le 0 q →
                                    Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg q)) (instHDiv.hDiv α 2) →
                                      Eq ((ProbabilityTheory.cdf η).toFun q) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                        Filter.Tendsto
                                          (fun n =>
                                            MeasureTheory.Measure.instFunLike.coe μ
                                              (setOf fun ω =>
                                                HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω)
                                                  (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                                    (instHDiv.hDiv α 2) n ω)
                                                  (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Tstar
                                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                          Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.percentileTCIEvent
HansenEconometrics.percentileTStatistic

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:1556

Theorem 10.14 HC2/HC3 OLS wrappers4 endpoints

If the sample and bootstrap t ratios share a continuous limit, then \Pr(\theta\in C_{\mathrm{pt}})\to1-\alpha. Ordinary HC2/HC3 scalar OLS restrictions can use the same percentile-t route once the bootstrap t-statistic premise is supplied

theorem HansenEconometrics.chapter10_olsHC2_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC2 OLS scalar restriction.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC2Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHDiv.hDiv α 2) n)
                                μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                    (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              Real.instLE.le 0 q →
                                Eq
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (Real.instNeg.neg q))
                                    (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent
                                              (HansenEconometrics.linearRestrictionEstimate R β)
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (ite (Eq n 0) 1
                                                (instHDiv.hDiv
                                                  (HansenEconometrics.linearRestrictionStdError R
                                                    (HansenEconometrics.olsHetCovHC2Star
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  n.cast.sqrt))
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC2Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5619

theorem HansenEconometrics.chapter10_indexed_olsHC2_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Indexed Theorem 10.14 with the actual sample statistic specialized to the ordinary HC2 OLS scalar restriction.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {TthetaStar seThetaStar : (n : Nat) → Ω → Ωboot n → Real}
  {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC2Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHDiv.hDiv α 2) n)
                                μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                    (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              Real.instLE.le 0 q →
                                Eq
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (Real.instNeg.neg q))
                                    (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent
                                              (HansenEconometrics.linearRestrictionEstimate R β)
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (ite (Eq n 0) 1
                                                (instHDiv.hDiv
                                                  (HansenEconometrics.linearRestrictionStdError R
                                                    (HansenEconometrics.olsHetCovHC2Star
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  n.cast.sqrt))
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC2Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5703

theorem HansenEconometrics.chapter10_olsHC3_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC3 OLS scalar restriction.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC3Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHDiv.hDiv α 2) n)
                                μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                    (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              Real.instLE.le 0 q →
                                Eq
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (Real.instNeg.neg q))
                                    (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent
                                              (HansenEconometrics.linearRestrictionEstimate R β)
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (ite (Eq n 0) 1
                                                (instHDiv.hDiv
                                                  (HansenEconometrics.linearRestrictionStdError R
                                                    (HansenEconometrics.olsHetCovHC3Star
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  n.cast.sqrt))
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC3Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5787

theorem HansenEconometrics.chapter10_indexed_olsHC3_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat

Indexed Theorem 10.14 with the actual sample statistic specialized to the ordinary HC3 OLS scalar restriction.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {TthetaStar seThetaStar : (n : Nat) → Ω → Ωboot n → Real}
  {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC3Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (StrictMono fun x => (ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun x) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                  (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                  (instHDiv.hDiv α 2) n)
                                μ) →
                            (∀ (n : Nat),
                                AEMeasurable
                                  (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                    (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                  μ) →
                              Real.instLE.le 0 q →
                                Eq
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (Real.instNeg.neg q))
                                    (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.percentileTCIEvent
                                              (HansenEconometrics.linearRestrictionEstimate R β)
                                              (HansenEconometrics.linearRestrictionEstimate R
                                                (HansenEconometrics.olsBetaOrZero
                                                  (HansenEconometrics.stackRegressors X n ω)
                                                  (HansenEconometrics.stackOutcomes y n ω)))
                                              (ite (Eq n 0) 1
                                                (instHDiv.hDiv
                                                  (HansenEconometrics.linearRestrictionStdError R
                                                    (HansenEconometrics.olsHetCovHC3Star
                                                      (HansenEconometrics.stackRegressors X n ω)
                                                      (HansenEconometrics.stackOutcomes y n ω)))
                                                  n.cast.sqrt))
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHDiv.hDiv α 2) n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC3Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5871

Theorem 10.14 OLS local-CDF wrappers6 of 8 linked endpoints

If the sample and bootstrap t ratios share a continuous limit, then \Pr(\theta\in C_{\mathrm{pt}})\to1-\alpha. Ordinary HC0-HC3 scalar OLS restrictions can use the regression-facing percentile-t route with local standard-normal CDF endpoint bracketing instead of global strict CDF monotonicity

theorem HansenEconometrics.chapter10_olsHC0_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat_brackets

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC0 OLS scalar restriction, using local standard-normal CDF bracketing.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovStar (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (instHSub.hSub (Real.instNeg.neg q) ε))
                                (instHDiv.hDiv α 2)) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHDiv.hDiv α 2)
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (instHSub.hSub q ε))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                      ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                        (instHAdd.hAdd q ε))) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                        (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                          (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    Real.instLE.le 0 q →
                                      Eq
                                          ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                            (Real.instNeg.neg q))
                                          (instHDiv.hDiv α 2) →
                                        Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                          Filter.Tendsto
                                            (fun n =>
                                              MeasureTheory.Measure.instFunLike.coe μ
                                                (setOf fun ω =>
                                                  HansenEconometrics.percentileTCIEvent
                                                    (HansenEconometrics.linearRestrictionEstimate R β)
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (ite (Eq n 0) 1
                                                      (instHDiv.hDiv
                                                        (HansenEconometrics.linearRestrictionStdError R
                                                          (HansenEconometrics.olsHetCovStar
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        n.cast.sqrt))
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                            Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovStar
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:5955

theorem HansenEconometrics.chapter10_olsHC1_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat_brackets

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC1 OLS scalar restriction, using local standard-normal CDF bracketing.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC1Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (instHSub.hSub (Real.instNeg.neg q) ε))
                                (instHDiv.hDiv α 2)) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHDiv.hDiv α 2)
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (instHSub.hSub q ε))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                      ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                        (instHAdd.hAdd q ε))) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                        (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                          (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    Real.instLE.le 0 q →
                                      Eq
                                          ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                            (Real.instNeg.neg q))
                                          (instHDiv.hDiv α 2) →
                                        Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                          Filter.Tendsto
                                            (fun n =>
                                              MeasureTheory.Measure.instFunLike.coe μ
                                                (setOf fun ω =>
                                                  HansenEconometrics.percentileTCIEvent
                                                    (HansenEconometrics.linearRestrictionEstimate R β)
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (ite (Eq n 0) 1
                                                      (instHDiv.hDiv
                                                        (HansenEconometrics.linearRestrictionStdError R
                                                          (HansenEconometrics.olsHetCovHC1Star
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        n.cast.sqrt))
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                            Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC1Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:6138

theorem HansenEconometrics.chapter10_olsHC2_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat_brackets

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC2 OLS scalar restriction, using local standard-normal CDF bracketing.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC2Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (instHSub.hSub (Real.instNeg.neg q) ε))
                                (instHDiv.hDiv α 2)) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHDiv.hDiv α 2)
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (instHSub.hSub q ε))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                      ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                        (instHAdd.hAdd q ε))) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                        (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                          (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    Real.instLE.le 0 q →
                                      Eq
                                          ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                            (Real.instNeg.neg q))
                                          (instHDiv.hDiv α 2) →
                                        Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                          Filter.Tendsto
                                            (fun n =>
                                              MeasureTheory.Measure.instFunLike.coe μ
                                                (setOf fun ω =>
                                                  HansenEconometrics.percentileTCIEvent
                                                    (HansenEconometrics.linearRestrictionEstimate R β)
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (ite (Eq n 0) 1
                                                      (instHDiv.hDiv
                                                        (HansenEconometrics.linearRestrictionStdError R
                                                          (HansenEconometrics.olsHetCovHC2Star
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        n.cast.sqrt))
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                            Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC2Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:6321

theorem HansenEconometrics.chapter10_olsHC3_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat_brackets

Theorem 10.14 with the actual sample statistic specialized to the ordinary HC3 OLS scalar restriction, using local standard-normal CDF bracketing.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC3Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (instHSub.hSub (Real.instNeg.neg q) ε))
                                (instHDiv.hDiv α 2)) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHDiv.hDiv α 2)
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (instHSub.hSub q ε))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                      ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                        (instHAdd.hAdd q ε))) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                        (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                          (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    Real.instLE.le 0 q →
                                      Eq
                                          ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                            (Real.instNeg.neg q))
                                          (instHDiv.hDiv α 2) →
                                        Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                          Filter.Tendsto
                                            (fun n =>
                                              MeasureTheory.Measure.instFunLike.coe μ
                                                (setOf fun ω =>
                                                  HansenEconometrics.percentileTCIEvent
                                                    (HansenEconometrics.linearRestrictionEstimate R β)
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (ite (Eq n 0) 1
                                                      (instHDiv.hDiv
                                                        (HansenEconometrics.linearRestrictionStdError R
                                                          (HansenEconometrics.olsHetCovHC3Star
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        n.cast.sqrt))
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                            Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC3Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:6504

theorem

HansenEconometrics.chapter10_indexed_olsHC0_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat_brackets

Indexed Theorem 10.14 with the actual sample statistic specialized to the ordinary HC0 OLS scalar restriction, using local standard-normal CDF bracketing.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {TthetaStar seThetaStar : (n : Nat) → Ω → Ωboot n → Real}
  {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovStar (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (instHSub.hSub (Real.instNeg.neg q) ε))
                                (instHDiv.hDiv α 2)) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHDiv.hDiv α 2)
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (instHSub.hSub q ε))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                      ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                        (instHAdd.hAdd q ε))) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                        (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                          (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    Real.instLE.le 0 q →
                                      Eq
                                          ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                            (Real.instNeg.neg q))
                                          (instHDiv.hDiv α 2) →
                                        Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                          Filter.Tendsto
                                            (fun n =>
                                              MeasureTheory.Measure.instFunLike.coe μ
                                                (setOf fun ω =>
                                                  HansenEconometrics.percentileTCIEvent
                                                    (HansenEconometrics.linearRestrictionEstimate R β)
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (ite (Eq n 0) 1
                                                      (instHDiv.hDiv
                                                        (HansenEconometrics.linearRestrictionStdError R
                                                          (HansenEconometrics.olsHetCovStar
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        n.cast.sqrt))
                                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                            Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovStar
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:6047

theorem

HansenEconometrics.chapter10_indexed_olsHC1_percentileTCI_coverage_tendsto_one_sub_alpha_of_bootstrap_regression_tstat_brackets

Indexed Theorem 10.14 with the actual sample statistic specialized to the ordinary HC1 OLS scalar restriction, using local standard-normal CDF bracketing.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {TthetaStar seThetaStar : (n : Nat) → Ω → Ωboot n → Real}
  {seθ q α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      (∀ (n : Nat) (ω : Ω),
          Ne n 0 →
            Real.instLT.lt 0
              (HansenEconometrics.linearRestrictionStdError R
                (HansenEconometrics.olsHetCovHC1Star (HansenEconometrics.stackRegressors X n ω)
                  (HansenEconometrics.stackOutcomes y n ω)))) →
        Real.instLT.lt 0 seθ →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
              (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs })
              (ProbabilityTheory.gaussianReal 0 1) fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
            (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
                (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                  (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar seThetaStar fun x => seθ) →
                    Real.instLT.lt 0 α →
                      Real.instLT.lt α 1 →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (instHSub.hSub (Real.instNeg.neg q) ε))
                                (instHDiv.hDiv α 2)) →
                          (∀ (ε : Real),
                              Real.instLT.lt 0 ε →
                                Real.instLT.lt (instHDiv.hDiv α 2)
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (instHAdd.hAdd (Real.instNeg.neg q) ε))) →
                            (∀ (ε : Real),
                                Real.instLT.lt 0 ε →
                                  Real.instLT.lt
                                    ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                      (instHSub.hSub q ε))
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2))) →
                              (∀ (ε : Real),
                                  Real.instLT.lt 0 ε →
                                    Real.instLT.lt (instHSub.hSub 1 (instHDiv.hDiv α 2))
                                      ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                        (instHAdd.hAdd q ε))) →
                                (∀ (n : Nat),
                                    AEMeasurable
                                      (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                        (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                        (instHDiv.hDiv α 2) n)
                                      μ) →
                                  (∀ (n : Nat),
                                      AEMeasurable
                                        (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                          (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                          (instHSub.hSub 1 (instHDiv.hDiv α 2)) n)
                                        μ) →
                                    Real.instLE.le 0 q →
                                      Eq
                                          ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                            (Real.instNeg.neg q))
                                          (instHDiv.hDiv α 2) →
                                        Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun q)
                                            (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                          Filter.Tendsto
                                            (fun n =>
                                              MeasureTheory.Measure.instFunLike.coe μ
                                                (setOf fun ω =>
                                                  HansenEconometrics.percentileTCIEvent
                                                    (HansenEconometrics.linearRestrictionEstimate R β)
                                                    (HansenEconometrics.linearRestrictionEstimate R
                                                      (HansenEconometrics.olsBetaOrZero
                                                        (HansenEconometrics.stackRegressors X n ω)
                                                        (HansenEconometrics.stackOutcomes y n ω)))
                                                    (ite (Eq n 0) 1
                                                      (instHDiv.hDiv
                                                        (HansenEconometrics.linearRestrictionStdError R
                                                          (HansenEconometrics.olsHetCovHC1Star
                                                            (HansenEconometrics.stackRegressors X n ω)
                                                            (HansenEconometrics.stackOutcomes y n ω)))
                                                        n.cast.sqrt))
                                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHDiv.hDiv α 2) n ω)
                                                    (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                                      (fun n ω ωs =>
                                                        instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                                                      (instHSub.hSub 1 (instHDiv.hDiv α 2)) n ω)))
                                            Filter.atTop (nhds (ENNReal.ofReal (instHSub.hSub 1 α)))

Direct statement dependencies (12)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionEstimate
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsBetaOrZero
HansenEconometrics.olsHetCovHC1Star
HansenEconometrics.percentileTCIEvent
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/PercentileT.lean:6230

Theorem 10.156 of 12 linked endpoints

Under the stated Edgeworth conditions, \Pr(\theta\in C_{\mathrm{pt}})=1-\alpha+O(n^{-1}).

theorem HansenEconometrics.chapter10_percentileT_secondOrder_interval_expansion

Hansen Theorem 10.15, Edgeworth component of the percentile-t refinement.

A second-order Edgeworth expansion for a scalar t-ratio gives the symmetric interval probability expansion used by the percentile-t bootstrap interval. The even p₁ and odd p₂ hypotheses encode the cancellation of the n^{-1/2} Edgeworth term in two-sided intervals.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c coverage : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) coverage →
            Filter.Tendsto
              (fun n =>
                instHMul.hMul n.cast
                  (instHSub.hSub
                    (instHSub.hSub
                      (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                        (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c)))
                      coverage)
                    (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
              Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:98

theorem HansenEconometrics.chapter10_percentileT_secondOrder_interval_expansion_of_transfer

Hansen Theorem 10.15 transfer form.

Once the bootstrap percentile-t quantile argument supplies an o(n⁻¹) difference between the random interval coverage and the fixed symmetric interval coverage, the fixed-critical Edgeworth expansion transfers to the random/bootstrap interval.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c coverage : Real} {randomCoverage : Nat → Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) coverage →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub (randomCoverage n)
                      (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                        (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c)))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub (instHSub.hSub (randomCoverage n) coverage)
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:122

theorem HansenEconometrics.chapter10_percentileT_secondOrder_interval_expansion_one_sub_alpha

Hansen Theorem 10.15, percentile-t second-order interval expansion in the textbook 1 - α coverage form.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c alpha : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) (instHSub.hSub 1 alpha) →
            Filter.Tendsto
              (fun n =>
                instHMul.hMul n.cast
                  (instHSub.hSub
                    (instHSub.hSub
                      (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                        (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c)))
                      (instHSub.hSub 1 alpha))
                    (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
              Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:196

theorem HansenEconometrics.chapter10_percentileT_secondOrder_interval_event_expansion_of_transfer

Hansen Theorem 10.15 transfer form for the actual percentile-t interval event.

The only remaining premise is the theorem’s higher-order bootstrap-quantile replacement step: the event probability of the random percentile-t interval differs from the fixed symmetric interval probability by o(n⁻¹).

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {θ : Real} {θhat se qLower qUpper : Nat → Ω → Real}
  {c coverage : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) coverage →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub
                      (MeasureTheory.Measure.instFunLike.coe μ
                          (setOf fun ω =>
                            HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω)
                              (qUpper n ω))).toReal
                      (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                        (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c)))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub
                      (instHSub.hSub
                        (MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω =>
                              HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω)
                                (qUpper n ω))).toReal
                        coverage)
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (3)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.percentileTCIEvent
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:161

theorem HansenEconometrics.chapter10_percentileT_secondOrder_interval_expansion_one_sub_alpha_of_transfer

Hansen Theorem 10.15 transfer form in the textbook 1 - α coverage normalization.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c alpha : Real} {randomCoverage : Nat → Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) (instHSub.hSub 1 alpha) →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub (randomCoverage n)
                      (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                        (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c)))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub (instHSub.hSub (randomCoverage n) (instHSub.hSub 1 alpha))
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:218

theorem HansenEconometrics.chapter10_percentileT_secondOrder_interval_event_expansion_one_sub_alpha_of_transfer

Event-probability form of Hansen Theorem 10.15 with limiting coverage 1 - α.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {θ : Real} {θhat se qLower qUpper : Nat → Ω → Real}
  {c alpha : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) (instHSub.hSub 1 alpha) →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub
                      (MeasureTheory.Measure.instFunLike.coe μ
                          (setOf fun ω =>
                            HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω)
                              (qUpper n ω))).toReal
                      (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                        (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c)))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub
                      (instHSub.hSub
                        (MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω =>
                              HansenEconometrics.percentileTCIEvent θ (θhat n ω) (se n ω) (qLower n ω)
                                (qUpper n ω))).toReal
                        (instHSub.hSub 1 alpha))
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (3)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.percentileTCIEvent
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:249

Theorem 10.166 of 23 linked endpoints

The bootstrap critical value satisfies q_{1-\alpha}^*\xrightarrow{p}q_{1-\alpha} and \Pr(\lvert T\rvert\gt q_{1-\alpha}^*\mid H_0)\to\alpha.

theorem HansenEconometrics.chapter10_bootstrap_abs_test_rejectionProb_tendsto_of_joint_critical_value_limit

Hansen Theorem 10.16, bootstrap critical-value rejection-probability bridge.

If the test statistic and bootstrap critical value jointly converge to (ξ, q), and the rejection boundary has zero limit mass, then the rejection probability converges to P[q < |ξ|].

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {T crit : Nat → Ω → Real} {ξ : Ωlim → Real} {critLim : Real},
  MeasureTheory.TendstoInDistribution (HansenEconometrics.bootstrapAbsTestVector T crit) Filter.atTop
      (HansenEconometrics.bootstrapAbsTestLimitVector ξ critLim) (fun x => μ) ν →
    Eq
        (MeasureTheory.Measure.instFunLike.coe
          (MeasureTheory.Measure.map (HansenEconometrics.bootstrapAbsTestLimitVector ξ critLim) ν)
          (frontier HansenEconometrics.bootstrapAbsRejectionSet))
        0 →
      Filter.Tendsto
        (fun n =>
          MeasureTheory.Measure.instFunLike.coe μ
            (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (T n ω) (crit n ω)))
        Filter.atTop
        (nhds
          (MeasureTheory.Measure.instFunLike.coe
            (MeasureTheory.Measure.map (HansenEconometrics.bootstrapAbsTestLimitVector ξ critLim) ν)
            HansenEconometrics.bootstrapAbsRejectionSet))

Direct statement dependencies (4)

HansenEconometrics.bootstrapAbsRejectionSet
HansenEconometrics.bootstrapAbsTestLimitVector
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapAbsTestVector

HansenEconometrics/Chapter10Bootstrap/Tests.lean:239

theorem HansenEconometrics.chapter10_indexed_abs_test_resampleMean_of_iIndep_tail_posDef_brackets

Indexed ordinary nonparametric-bootstrap two-sided critical-value test from the concrete normalized scalar Fin (n+1) resample-mean CLT.

The bootstrap critical value is the lower generalized inverse of the conditional CDF of |sqrt(n+1) (Ybar*_n - Ybar_n)| under the finite ordinary resampling law. The sample-side statistic convergence and endpoint calibration are kept explicit, matching Hansen Theorem 10.16.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η ηAbs : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] [MeasureTheory.IsProbabilityMeasure ηAbs]
  [MeasureTheory.NoAtoms ηAbs] (Y : Nat → Ω → Real),
  MeasureTheory.MemLp (Y 0) 2 μ →
    ProbabilityTheory.iIndepFun Y μ →
      (∀ (i : Nat), ProbabilityTheory.IdentDistrib (Y i) (Y 0) μ μ) →
        (HansenEconometrics.covMat μ fun ω x => Y 0 ω).PosDef →
          ∀ {T : Nat → Ω → Real} {ξ : Ωlim → Real} {critLim α : Real},
            MeasureTheory.TendstoInDistribution T Filter.atTop ξ (fun x => μ) ν →
              Real.instLT.lt 0 α →
                Real.instLT.lt α 1 →
                  (∀ (ε : Real),
                      Real.instLT.lt 0 ε →
                        Real.instLT.lt ((ProbabilityTheory.cdf ηAbs).toFun (instHSub.hSub critLim ε))
                          (instHSub.hSub 1 α)) →
                    (∀ (ε : Real),
                        Real.instLT.lt 0 ε →
                          Real.instLT.lt (instHSub.hSub 1 α)
                            ((ProbabilityTheory.cdf ηAbs).toFun (instHAdd.hAdd critLim ε))) →
                      (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf ηAbs).toFun y) x) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                (fun n ω ωs =>
                                  abs
                                    (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                      (instHSub.hSub
                                        (HansenEconometrics.empiricalBootstrapResampleMean (fun i => Y i.val ω)
                                          (fun ωs t => ωs t) ωs)
                                        (HansenEconometrics.empiricalMean fun i => Y i.val ω))))
                                (instHSub.hSub 1 α) n)
                              μ) →
                          ProbabilityTheory.HasLaw ξ η ν →
                            ProbabilityTheory.HasLaw (fun z => abs (z.ofLp Unit.unit)) ηAbs
                                (ProbabilityTheory.multivariateGaussian 0
                                  (HansenEconometrics.covMat μ fun ω x => Y 0 ω)) →
                              Real.instLE.le 0 critLim →
                                Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg critLim)) (instHDiv.hDiv α 2) →
                                  Eq ((ProbabilityTheory.cdf η).toFun critLim) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                    Filter.Tendsto
                                      (fun n =>
                                        MeasureTheory.Measure.instFunLike.coe μ
                                          (setOf fun ω =>
                                            HansenEconometrics.bootstrapAbsTestReject (T n ω)
                                              (HansenEconometrics.bootstrapScalarLowerQuantileIndexed
                                                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                                                (fun n ω ωs =>
                                                  abs
                                                    (instHMul.hMul (instHAdd.hAdd n.cast 1).sqrt
                                                      (instHSub.hSub
                                                        (HansenEconometrics.empiricalBootstrapResampleMean
                                                          (fun i => Y i.val ω) (fun ωs t => ωs t) ωs)
                                                        (HansenEconometrics.empiricalMean fun i => Y i.val ω))))
                                                (instHSub.hSub 1 α) n ω)))
                                      Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (5)

HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.covMat
HansenEconometrics.empiricalBootstrapResampleMean
HansenEconometrics.empiricalMean

HansenEconometrics/Chapter10Bootstrap/Tests.lean:1505

theorem HansenEconometrics.chapter10_bootstrap_abs_test_rejectionProb_tendsto_alpha_of_bootstrap_regression_tstat

Regression-facing two-sided bootstrap-test calibration from the Theorem 10.18 t-statistic route.

The bootstrap critical value is the lower generalized inverse of the conditional law of |TthetaStar / seThetaStar|. The joint numerator/standard-error bootstrap weak limit and scale consistency give the absolute-standard-normal bootstrap CDF limit, and the existing Theorem 10.16 quantile route then gives rejection probability α.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {T : Nat → Ω → Real} {seθ critLim α : Real},
  MeasureTheory.TendstoInDistribution T Filter.atTop (fun x => x) (fun x => μ) (ProbabilityTheory.gaussianReal 0 1) →
    Real.instLT.lt 0 seθ →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
          (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
          fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
              (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                Real.instLT.lt 0 α →
                  Real.instLT.lt α 1 →
                    (∀ (ε : Real),
                        Real.instLT.lt 0 ε →
                          Real.instLT.lt
                            ((ProbabilityTheory.cdf
                                  (MeasureTheory.Measure.map (fun z => abs z)
                                    (ProbabilityTheory.gaussianReal 0 1))).toFun
                              (instHSub.hSub critLim ε))
                            (instHSub.hSub 1 α)) →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt (instHSub.hSub 1 α)
                              ((ProbabilityTheory.cdf
                                    (MeasureTheory.Measure.map (fun z => abs z)
                                      (ProbabilityTheory.gaussianReal 0 1))).toFun
                                (instHAdd.hAdd critLim ε))) →
                        (∀ (n : Nat),
                            AEMeasurable
                              (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                (instHSub.hSub 1 α) n)
                              μ) →
                          Real.instLE.le 0 critLim →
                            Eq
                                ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                  (Real.instNeg.neg critLim))
                                (instHDiv.hDiv α 2) →
                              Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun critLim)
                                  (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                Filter.Tendsto
                                  (fun n =>
                                    MeasureTheory.Measure.instFunLike.coe μ
                                      (setOf fun ω =>
                                        HansenEconometrics.bootstrapAbsTestReject (T n ω)
                                          (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                            (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                            (instHSub.hSub 1 α) n ω)))
                                  Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantile

HansenEconometrics/Chapter10Bootstrap/Tests.lean:1771

theorem HansenEconometrics.chapter10_bootstrap_abs_test_rejectionProb_tendsto_alpha_of_components_law_cdf_endpoints

Componentwise endpoint-CDF two-sided bootstrap-test calibration with limiting size α.

This is the Theorem 10.16 rejection bridge stated directly from statistic convergence and bootstrap critical-value convergence in probability.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} [inst : MeasureTheory.IsProbabilityMeasure μ]
  [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {η : MeasureTheory.Measure Real}
  [MeasureTheory.IsProbabilityMeasure η] [MeasureTheory.NoAtoms η] {T crit : Nat → Ω → Real} {ξ : Ωlim → Real}
  {critLim α : Real},
  MeasureTheory.TendstoInDistribution T Filter.atTop ξ (fun x => μ) ν →
    (MeasureTheory.TendstoInMeasure μ crit Filter.atTop fun x => critLim) →
      (∀ (n : Nat), AEMeasurable (crit n) μ) →
        ProbabilityTheory.HasLaw ξ η ν →
          Real.instLE.le 0 critLim →
            Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg critLim)) (instHDiv.hDiv α 2) →
              Eq ((ProbabilityTheory.cdf η).toFun critLim) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                Filter.Tendsto
                  (fun n =>
                    MeasureTheory.Measure.instFunLike.coe μ
                      (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (T n ω) (crit n ω)))
                  Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (1)

HansenEconometrics.bootstrapAbsTestReject

HansenEconometrics/Chapter10Bootstrap/Tests.lean:446

theorem HansenEconometrics.bootstrapAbsTest_scalar_rejection_eq_law

The scalar two-sided rejection event can be read from the law of the limit statistic.

Formal statement

∀ {Ωlim : Type u_3} {mΩlim : MeasurableSpace Ωlim} {ν : MeasureTheory.Measure Ωlim} {ξ : Ωlim → Real}
  {η : MeasureTheory.Measure Real},
  ProbabilityTheory.HasLaw ξ η ν →
    ∀ (critLim : Real),
      Eq
        (MeasureTheory.Measure.instFunLike.coe ν
          (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (ξ ω) critLim))
        (MeasureTheory.Measure.instFunLike.coe η (setOf fun x => HansenEconometrics.bootstrapAbsTestReject x critLim))

Direct statement dependencies (1)

HansenEconometrics.bootstrapAbsTestReject

HansenEconometrics/Chapter10Bootstrap/Tests.lean:169

theorem HansenEconometrics.tendstoInMeasure_quantile_of_cdf_brackets

Quantile convergence from pointwise CDF convergence at strict bracketing points.

If the random CDFs Gseq n converge in probability to G at every fixed point, the target q is strictly bracketed by the limiting CDF around level p, and qseq is a lower-quantile selection for each random CDF, then qseq ->p q. This is the reusable quantile-convergence constructor behind the percentile, percentile-t, and bootstrap critical-value endpoints in Hansen Theorems 10.13, 10.14, and 10.16.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Gseq : Nat → Ω → Real → Real} {G : Real → Real}
  {p q : Real} {qseq : Nat → Ω → Real},
  HansenEconometrics.CDFQuantileBracket Gseq p qseq →
    (∀ (ε : Real), Real.instLT.lt 0 ε → Real.instLT.lt (G (instHSub.hSub q ε)) p) →
      (∀ (ε : Real), Real.instLT.lt 0 ε → Real.instLT.lt p (G (instHAdd.hAdd q ε))) →
        (∀ (x : Real), MeasureTheory.TendstoInMeasure μ (fun n ω => Gseq n ω x) Filter.atTop fun x_1 => G x) →
          MeasureTheory.TendstoInMeasure μ qseq Filter.atTop fun x => q

Direct statement dependencies (1)

HansenEconometrics.CDFQuantileBracket

HansenEconometrics/Chapter10Bootstrap/Quantiles.lean:138

Theorem 10.16 auxiliary-limit critical-value routes4 endpoints

The bootstrap critical value satisfies q_{1-\alpha}^*\xrightarrow{p}q_{1-\alpha} and \Pr(\lvert T\rvert\gt q_{1-\alpha}^*\mid H_0)\to\alpha. Auxiliary absolute-statistic limits can feed the bootstrap critical-value lower-quantile route by HasLaw, with strict-CDF or local-limit-CDF endpoint bracketing

theorem HansenEconometrics.chapter10_bootstrap_abs_test_rejectionProb_law_quantile_prob

Two-sided bootstrap-test calibration from a bootstrap distribution whose absolute-statistic scalar limit has law ηAbs.

The absolute bootstrap statistic may converge on an auxiliary probability space; HasLaw identifies its scalar CDF for the lower critical-value quantile route.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_7}
  [inst_2 : MeasurableSpace Ωstar] {η ηAbs : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] [MeasureTheory.IsProbabilityMeasure ηAbs] {νstar : MeasureTheory.Measure Ωstar}
  {Alim : Ωstar → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Astar : Nat → Ω → Ωs → Real} {T : Nat → Ω → Real}
  {ξ : Ωlim → Real} {critLim α : Real},
  MeasureTheory.TendstoInDistribution T Filter.atTop ξ (fun x => μ) ν →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat) (ω : Ω), AEMeasurable (Astar n ω) (Pstar n ω)) →
        Real.instLT.lt 0 α →
          Real.instLT.lt α 1 →
            (StrictMono fun x => (ProbabilityTheory.cdf ηAbs).toFun x) →
              Eq ((ProbabilityTheory.cdf ηAbs).toFun critLim) (instHSub.hSub 1 α) →
                (HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs x => Astar n ω ωs) νstar
                    fun ωstar x => Alim ωstar) →
                  ProbabilityTheory.HasLaw Alim ηAbs νstar →
                    (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf ηAbs).toFun y) x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Astar (instHSub.hSub 1 α) n) μ) →
                        ProbabilityTheory.HasLaw ξ η ν →
                          Real.instLE.le 0 critLim →
                            Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg critLim)) (instHDiv.hDiv α 2) →
                              Eq ((ProbabilityTheory.cdf η).toFun critLim) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                Filter.Tendsto
                                  (fun n =>
                                    MeasureTheory.Measure.instFunLike.coe μ
                                      (setOf fun ω =>
                                        HansenEconometrics.bootstrapAbsTestReject (T n ω)
                                          (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Astar
                                            (instHSub.hSub 1 α) n ω)))
                                  Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantile

HansenEconometrics/Chapter10Bootstrap/Tests.lean:862

theorem HansenEconometrics.chapter10_bootstrap_abs_test_rejectionProb_law_quantile_prob_brackets

Two-sided bootstrap-test calibration from an auxiliary absolute-statistic limit, retaining local CDF bracketing at the lower critical-value endpoint.

This law-facing variant is the local-bracketing counterpart of chapter10_bootstrap_abs_test_rejectionProb_law_quantile_prob: HasLaw identifies the auxiliary absolute-statistic limit’s scalar CDF with cdf ηAbs without requiring global strict monotonicity of that CDF.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ωlim}
  [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_7}
  [inst_2 : MeasurableSpace Ωstar] {η ηAbs : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] [MeasureTheory.IsProbabilityMeasure ηAbs] {νstar : MeasureTheory.Measure Ωstar}
  {Alim : Ωstar → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {Astar : Nat → Ω → Ωs → Real} {T : Nat → Ω → Real}
  {ξ : Ωlim → Real} {critLim α : Real},
  MeasureTheory.TendstoInDistribution T Filter.atTop ξ (fun x => μ) ν →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat) (ω : Ω), AEMeasurable (Astar n ω) (Pstar n ω)) →
        Real.instLT.lt 0 α →
          Real.instLT.lt α 1 →
            (∀ (ε : Real),
                Real.instLT.lt 0 ε →
                  Real.instLT.lt ((ProbabilityTheory.cdf ηAbs).toFun (instHSub.hSub critLim ε)) (instHSub.hSub 1 α)) →
              (∀ (ε : Real),
                  Real.instLT.lt 0 ε →
                    Real.instLT.lt (instHSub.hSub 1 α) ((ProbabilityTheory.cdf ηAbs).toFun (instHAdd.hAdd critLim ε))) →
                (HansenEconometrics.TendstoInBootstrapDistribution μ Pstar (fun n ω ωs x => Astar n ω ωs) νstar
                    fun ωstar x => Alim ωstar) →
                  ProbabilityTheory.HasLaw Alim ηAbs νstar →
                    (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf ηAbs).toFun y) x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Astar (instHSub.hSub 1 α) n) μ) →
                        ProbabilityTheory.HasLaw ξ η ν →
                          Real.instLE.le 0 critLim →
                            Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg critLim)) (instHDiv.hDiv α 2) →
                              Eq ((ProbabilityTheory.cdf η).toFun critLim) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                Filter.Tendsto
                                  (fun n =>
                                    MeasureTheory.Measure.instFunLike.coe μ
                                      (setOf fun ω =>
                                        HansenEconometrics.bootstrapAbsTestReject (T n ω)
                                          (HansenEconometrics.bootstrapScalarLowerQuantile Pstar Astar
                                            (instHSub.hSub 1 α) n ω)))
                                  Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantile

HansenEconometrics/Chapter10Bootstrap/Tests.lean:928

theorem HansenEconometrics.chapter10_indexed_abs_test_rejectionProb_law_quantile_prob

Indexed two-sided bootstrap-test calibration from a bootstrap distribution whose absolute-statistic scalar limit has law ηAbs.

This sample-size-dependent law-facing wrapper lets the absolute bootstrap statistic converge on an auxiliary probability space while HasLaw supplies the scalar CDF used by the lower critical-value quantile route.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} {Ωboot : Nat → Type u_7} [inst : (n : Nat) → MeasurableSpace (Ωboot n)]
  [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_8}
  [inst_3 : MeasurableSpace Ωstar] {η ηAbs : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] [MeasureTheory.IsProbabilityMeasure ηAbs] {νstar : MeasureTheory.Measure Ωstar}
  {Alim : Ωstar → Real} {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Astar : (n : Nat) → Ω → Ωboot n → Real} {T : Nat → Ω → Real} {ξ : Ωlim → Real} {critLim α : Real},
  MeasureTheory.TendstoInDistribution T Filter.atTop ξ (fun x => μ) ν →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat) (ω : Ω), AEMeasurable (Astar n ω) (Pstar n ω)) →
        Real.instLT.lt 0 α →
          Real.instLT.lt α 1 →
            (StrictMono fun x => (ProbabilityTheory.cdf ηAbs).toFun x) →
              Eq ((ProbabilityTheory.cdf ηAbs).toFun critLim) (instHSub.hSub 1 α) →
                (HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar (fun n ω ωs x => Astar n ω ωs) νstar
                    fun ωstar x => Alim ωstar) →
                  ProbabilityTheory.HasLaw Alim ηAbs νstar →
                    (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf ηAbs).toFun y) x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Astar (instHSub.hSub 1 α) n)
                            μ) →
                        ProbabilityTheory.HasLaw ξ η ν →
                          Real.instLE.le 0 critLim →
                            Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg critLim)) (instHDiv.hDiv α 2) →
                              Eq ((ProbabilityTheory.cdf η).toFun critLim) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                Filter.Tendsto
                                  (fun n =>
                                    MeasureTheory.Measure.instFunLike.coe μ
                                      (setOf fun ω =>
                                        HansenEconometrics.bootstrapAbsTestReject (T n ω)
                                          (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Astar
                                            (instHSub.hSub 1 α) n ω)))
                                  Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantileIndexed

HansenEconometrics/Chapter10Bootstrap/Tests.lean:1318

theorem HansenEconometrics.chapter10_indexed_abs_test_rejectionProb_law_quantile_prob_brackets

Indexed two-sided bootstrap-test calibration from an auxiliary absolute-statistic limit, retaining local CDF bracketing at the lower critical-value endpoint.

This sample-size-dependent law-facing wrapper is the indexed counterpart of chapter10_bootstrap_abs_test_rejectionProb_law_quantile_prob_brackets.

Formal statement

∀ {Ω : Type u_1} {Ωlim : Type u_3} {mΩ : MeasurableSpace Ω} {mΩlim : MeasurableSpace Ωlim} {μ : MeasureTheory.Measure Ω}
  {ν : MeasureTheory.Measure Ωlim} {Ωboot : Nat → Type u_7} [inst : (n : Nat) → MeasurableSpace (Ωboot n)]
  [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : MeasureTheory.IsProbabilityMeasure ν] {Ωstar : Type u_8}
  [inst_3 : MeasurableSpace Ωstar] {η ηAbs : MeasureTheory.Measure Real} [MeasureTheory.IsProbabilityMeasure η]
  [MeasureTheory.NoAtoms η] [MeasureTheory.IsProbabilityMeasure ηAbs] {νstar : MeasureTheory.Measure Ωstar}
  {Alim : Ωstar → Real} {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {Astar : (n : Nat) → Ω → Ωboot n → Real} {T : Nat → Ω → Real} {ξ : Ωlim → Real} {critLim α : Real},
  MeasureTheory.TendstoInDistribution T Filter.atTop ξ (fun x => μ) ν →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat) (ω : Ω), AEMeasurable (Astar n ω) (Pstar n ω)) →
        Real.instLT.lt 0 α →
          Real.instLT.lt α 1 →
            (∀ (ε : Real),
                Real.instLT.lt 0 ε →
                  Real.instLT.lt ((ProbabilityTheory.cdf ηAbs).toFun (instHSub.hSub critLim ε)) (instHSub.hSub 1 α)) →
              (∀ (ε : Real),
                  Real.instLT.lt 0 ε →
                    Real.instLT.lt (instHSub.hSub 1 α) ((ProbabilityTheory.cdf ηAbs).toFun (instHAdd.hAdd critLim ε))) →
                (HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar (fun n ω ωs x => Astar n ω ωs) νstar
                    fun ωstar x => Alim ωstar) →
                  ProbabilityTheory.HasLaw Alim ηAbs νstar →
                    (∀ (x : Real), ContinuousAt (fun y => (ProbabilityTheory.cdf ηAbs).toFun y) x) →
                      (∀ (n : Nat),
                          AEMeasurable
                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Astar (instHSub.hSub 1 α) n)
                            μ) →
                        ProbabilityTheory.HasLaw ξ η ν →
                          Real.instLE.le 0 critLim →
                            Eq ((ProbabilityTheory.cdf η).toFun (Real.instNeg.neg critLim)) (instHDiv.hDiv α 2) →
                              Eq ((ProbabilityTheory.cdf η).toFun critLim) (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                Filter.Tendsto
                                  (fun n =>
                                    MeasureTheory.Measure.instFunLike.coe μ
                                      (setOf fun ω =>
                                        HansenEconometrics.bootstrapAbsTestReject (T n ω)
                                          (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar Astar
                                            (instHSub.hSub 1 α) n ω)))
                                  Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantileIndexed

HansenEconometrics/Chapter10Bootstrap/Tests.lean:1383

Theorem 10.16 HC2/HC3 OLS wrappers4 endpoints

The bootstrap critical value satisfies q_{1-\alpha}^*\xrightarrow{p}q_{1-\alpha} and \Pr(\lvert T\rvert\gt q_{1-\alpha}^*\mid H_0)\to\alpha. Ordinary HC2/HC3 scalar OLS t-statistics can use the same two-sided bootstrap-test route once the absolute bootstrap t-statistic premise is supplied

theorem HansenEconometrics.chapter10_olsHC2_abs_test_rejectionProb_tendsto_alpha_of_bootstrap_regression_tstat

Theorem 10.16 with the actual statistic specialized to the ordinary HC2 OLS scalar t-statistic.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ critLim α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      Real.instLT.lt 0 seθ →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
            (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
            fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
          (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt
                              ((ProbabilityTheory.cdf
                                    (MeasureTheory.Measure.map (fun z => abs z)
                                      (ProbabilityTheory.gaussianReal 0 1))).toFun
                                (instHSub.hSub critLim ε))
                              (instHSub.hSub 1 α)) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt (instHSub.hSub 1 α)
                                ((ProbabilityTheory.cdf
                                      (MeasureTheory.Measure.map (fun z => abs z)
                                        (ProbabilityTheory.gaussianReal 0 1))).toFun
                                  (instHAdd.hAdd critLim ε))) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                  (instHSub.hSub 1 α) n)
                                μ) →
                            Real.instLE.le 0 critLim →
                              Eq
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (Real.instNeg.neg critLim))
                                  (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun critLim)
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.bootstrapAbsTestReject
                                            (HansenEconometrics.olsLinearTStatOrZero R
                                              (HansenEconometrics.olsHetCovHC2Star
                                                (HansenEconometrics.stackRegressors X n ω)
                                                (HansenEconometrics.stackOutcomes y n ω))
                                              (HansenEconometrics.stackRegressors X n ω)
                                              (HansenEconometrics.stackOutcomes y n ω) β n.cast.sqrt)
                                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                              (fun n ω ωs =>
                                                abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                              (instHSub.hSub 1 α) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (11)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsHetCovHC2Star
HansenEconometrics.olsLinearTStatOrZero
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Tests.lean:4428

theorem HansenEconometrics.chapter10_indexed_olsHC2_abs_test_rejectionProb_tendsto_alpha_of_bootstrap_regression_tstat

Indexed Theorem 10.16 with the actual statistic specialized to the ordinary HC2 OLS scalar t-statistic.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {TthetaStar seThetaStar : (n : Nat) → Ω → Ωboot n → Real}
  {seθ critLim α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      Real.instLT.lt 0 seθ →
        (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
            (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
            fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
          (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar seThetaStar fun x => seθ) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt
                              ((ProbabilityTheory.cdf
                                    (MeasureTheory.Measure.map (fun z => abs z)
                                      (ProbabilityTheory.gaussianReal 0 1))).toFun
                                (instHSub.hSub critLim ε))
                              (instHSub.hSub 1 α)) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt (instHSub.hSub 1 α)
                                ((ProbabilityTheory.cdf
                                      (MeasureTheory.Measure.map (fun z => abs z)
                                        (ProbabilityTheory.gaussianReal 0 1))).toFun
                                  (instHAdd.hAdd critLim ε))) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                  (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                  (instHSub.hSub 1 α) n)
                                μ) →
                            Real.instLE.le 0 critLim →
                              Eq
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (Real.instNeg.neg critLim))
                                  (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun critLim)
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.bootstrapAbsTestReject
                                            (HansenEconometrics.olsLinearTStatOrZero R
                                              (HansenEconometrics.olsHetCovHC2Star
                                                (HansenEconometrics.stackRegressors X n ω)
                                                (HansenEconometrics.stackOutcomes y n ω))
                                              (HansenEconometrics.stackRegressors X n ω)
                                              (HansenEconometrics.stackOutcomes y n ω) β n.cast.sqrt)
                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                              (fun n ω ωs =>
                                                abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                              (instHSub.hSub 1 α) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (11)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsHetCovHC2Star
HansenEconometrics.olsLinearTStatOrZero
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Tests.lean:4498

theorem HansenEconometrics.chapter10_olsHC3_abs_test_rejectionProb_tendsto_alpha_of_bootstrap_regression_tstat

Theorem 10.16 with the actual statistic specialized to the ordinary HC3 OLS scalar t-statistic.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {k : Type u_6} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs}
  {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ] [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ critLim α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      Real.instLT.lt 0 seθ →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
            (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
            fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
          (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt
                              ((ProbabilityTheory.cdf
                                    (MeasureTheory.Measure.map (fun z => abs z)
                                      (ProbabilityTheory.gaussianReal 0 1))).toFun
                                (instHSub.hSub critLim ε))
                              (instHSub.hSub 1 α)) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt (instHSub.hSub 1 α)
                                ((ProbabilityTheory.cdf
                                      (MeasureTheory.Measure.map (fun z => abs z)
                                        (ProbabilityTheory.gaussianReal 0 1))).toFun
                                  (instHAdd.hAdd critLim ε))) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                  (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                  (instHSub.hSub 1 α) n)
                                μ) →
                            Real.instLE.le 0 critLim →
                              Eq
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (Real.instNeg.neg critLim))
                                  (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun critLim)
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.bootstrapAbsTestReject
                                            (HansenEconometrics.olsLinearTStatOrZero R
                                              (HansenEconometrics.olsHetCovHC3Star
                                                (HansenEconometrics.stackRegressors X n ω)
                                                (HansenEconometrics.stackOutcomes y n ω))
                                              (HansenEconometrics.stackRegressors X n ω)
                                              (HansenEconometrics.stackOutcomes y n ω) β n.cast.sqrt)
                                            (HansenEconometrics.bootstrapScalarLowerQuantile Pstar
                                              (fun n ω ωs =>
                                                abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                              (instHSub.hSub 1 α) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (11)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantile
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsHetCovHC3Star
HansenEconometrics.olsLinearTStatOrZero
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Tests.lean:4568

theorem HansenEconometrics.chapter10_indexed_olsHC3_abs_test_rejectionProb_tendsto_alpha_of_bootstrap_regression_tstat

Indexed Theorem 10.16 with the actual statistic specialized to the ordinary HC3 OLS scalar t-statistic.

Formal statement

∀ {Ω : Type u_1} {k : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ωboot : Nat → Type u_7}
  [inst : (n : Nat) → MeasurableSpace (Ωboot n)] [inst_1 : MeasureTheory.IsProbabilityMeasure μ] [inst_2 : Fintype k]
  [inst_3 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {TthetaStar seThetaStar : (n : Nat) → Ω → Ωboot n → Real}
  {seθ critLim α : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    Real.instLT.lt 0 (HansenEconometrics.linearRestrictionStdError R (HansenEconometrics.heteroAsymCov μ X e)) →
      Real.instLT.lt 0 seθ →
        (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
            (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
            fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
          (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
            (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
                (HansenEconometrics.TendstoInBootstrapProbabilityIndexed μ Pstar seThetaStar fun x => seθ) →
                  Real.instLT.lt 0 α →
                    Real.instLT.lt α 1 →
                      (∀ (ε : Real),
                          Real.instLT.lt 0 ε →
                            Real.instLT.lt
                              ((ProbabilityTheory.cdf
                                    (MeasureTheory.Measure.map (fun z => abs z)
                                      (ProbabilityTheory.gaussianReal 0 1))).toFun
                                (instHSub.hSub critLim ε))
                              (instHSub.hSub 1 α)) →
                        (∀ (ε : Real),
                            Real.instLT.lt 0 ε →
                              Real.instLT.lt (instHSub.hSub 1 α)
                                ((ProbabilityTheory.cdf
                                      (MeasureTheory.Measure.map (fun z => abs z)
                                        (ProbabilityTheory.gaussianReal 0 1))).toFun
                                  (instHAdd.hAdd critLim ε))) →
                          (∀ (n : Nat),
                              AEMeasurable
                                (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                  (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                  (instHSub.hSub 1 α) n)
                                μ) →
                            Real.instLE.le 0 critLim →
                              Eq
                                  ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun
                                    (Real.instNeg.neg critLim))
                                  (instHDiv.hDiv α 2) →
                                Eq ((ProbabilityTheory.cdf (ProbabilityTheory.gaussianReal 0 1)).toFun critLim)
                                    (instHSub.hSub 1 (instHDiv.hDiv α 2)) →
                                  Filter.Tendsto
                                    (fun n =>
                                      MeasureTheory.Measure.instFunLike.coe μ
                                        (setOf fun ω =>
                                          HansenEconometrics.bootstrapAbsTestReject
                                            (HansenEconometrics.olsLinearTStatOrZero R
                                              (HansenEconometrics.olsHetCovHC3Star
                                                (HansenEconometrics.stackRegressors X n ω)
                                                (HansenEconometrics.stackOutcomes y n ω))
                                              (HansenEconometrics.stackRegressors X n ω)
                                              (HansenEconometrics.stackOutcomes y n ω) β n.cast.sqrt)
                                            (HansenEconometrics.bootstrapScalarLowerQuantileIndexed Pstar
                                              (fun n ω ωs =>
                                                abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                                              (instHSub.hSub 1 α) n ω)))
                                    Filter.atTop (nhds (ENNReal.ofReal α))

Direct statement dependencies (11)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapProbabilityIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.bootstrapScalarLowerQuantileIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.linearRestrictionStdError
HansenEconometrics.olsHetCovHC3Star
HansenEconometrics.olsLinearTStatOrZero
HansenEconometrics.stackOutcomes
HansenEconometrics.stackRegressors

HansenEconometrics/Chapter10Bootstrap/Tests.lean:4638

Theorem 10.176 of 12 linked endpoints

The bootstrap two-sided test has the higher-order refinement q_{1-\alpha}^*=z_{1-\alpha}+o_p(n^{-1}) and size error O(n^{-1}).

theorem HansenEconometrics.chapter10_abs_test_secondOrder_rejection_expansion

Hansen Theorem 10.17, fixed-critical Edgeworth component.

For a two-sided test using a fixed critical value c, the rejection probability 1 - (Fₙ(c) - Fₙ(-c)) inherits the symmetric second-order Edgeworth expansion. The bootstrap-quantile step of Theorem 10.17 supplies the additional critical-value transfer premise needed to turn this fixed-critical expansion into the o(n^{-1}) bootstrap-test refinement.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c alpha : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub 1 (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c)))) alpha →
            Filter.Tendsto
              (fun n =>
                instHMul.hMul n.cast
                  (instHAdd.hAdd
                    (instHSub.hSub
                      (instHSub.hSub 1
                        (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                          (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c))))
                      alpha)
                    (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
              Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:486

theorem HansenEconometrics.chapter10_abs_test_secondOrder_rejection_expansion_of_transfer

Hansen Theorem 10.17 transfer form.

Once the bootstrap critical-value argument supplies an o(n⁻¹) difference between the random-critical rejection probability and the fixed-critical rejection probability, the fixed-critical Edgeworth expansion transfers to the bootstrap test.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c alpha : Real} {randomReject : Nat → Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub 1 (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c)))) alpha →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub (randomReject n)
                      (instHSub.hSub 1
                        (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                          (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c))))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHAdd.hAdd (instHSub.hSub (randomReject n) alpha)
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:529

theorem HansenEconometrics.chapter10_abs_test_secondOrder_rejection_expansion_one_sub_alpha

Hansen Theorem 10.17 in the textbook central-coverage calibration form.

The fixed critical value is calibrated by F(c) - F(-c) = 1 - α, so the two-sided rejection probability has limiting size α with the same second-order correction as chapter10_abs_test_secondOrder_rejection_expansion.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c alpha : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) (instHSub.hSub 1 alpha) →
            Filter.Tendsto
              (fun n =>
                instHMul.hMul n.cast
                  (instHAdd.hAdd
                    (instHSub.hSub
                      (instHSub.hSub 1
                        (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                          (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c))))
                      alpha)
                    (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
              Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:603

theorem HansenEconometrics.chapter10_abs_test_secondOrder_rejection_event_expansion_of_transfer

Hansen Theorem 10.17 transfer form for the actual two-sided bootstrap-test event.

The replacement premise is the theorem’s higher-order bootstrap critical-value step: replacing the fixed critical value c by the random/bootstrap critical value changes the rejection probability by o(n⁻¹).

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {crit : Nat → Ω → Real} {c alpha : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub 1 (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c)))) alpha →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub
                      (MeasureTheory.Measure.instFunLike.coe μ
                          (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (T n ω) (crit n ω))).toReal
                      (instHSub.hSub 1
                        (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                          (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c))))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHAdd.hAdd
                      (instHSub.hSub
                        (MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (T n ω) (crit n ω))).toReal
                        alpha)
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (3)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:568

theorem HansenEconometrics.chapter10_abs_test_secondOrder_rejection_expansion_one_sub_alpha_of_transfer

Hansen Theorem 10.17 transfer form with textbook central coverage F(c) - F(-c) = 1 - α.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {c alpha : Real} {randomReject : Nat → Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) (instHSub.hSub 1 alpha) →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub (randomReject n)
                      (instHSub.hSub 1
                        (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                          (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c))))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHAdd.hAdd (instHSub.hSub (randomReject n) alpha)
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (2)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:630

theorem HansenEconometrics.chapter10_abs_test_secondOrder_rejection_event_expansion_one_sub_alpha_of_transfer

Event-probability form of Hansen Theorem 10.17 with central coverage F(c) - F(-c) = 1 - α.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {T : Nat → Ω → Real} {baseCDF density p1 p2 : Real → Real} {crit : Nat → Ω → Real} {c alpha : Real},
  HansenEconometrics.SecondOrderEdgeworthExpansion μ T baseCDF density p1 p2 →
    Eq (p1 (Real.instNeg.neg c)) (p1 c) →
      Eq (p2 (Real.instNeg.neg c)) (Real.instNeg.neg (p2 c)) →
        Eq (density (Real.instNeg.neg c)) (density c) →
          Eq (instHSub.hSub (baseCDF c) (baseCDF (Real.instNeg.neg c))) (instHSub.hSub 1 alpha) →
            Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHSub.hSub
                      (MeasureTheory.Measure.instFunLike.coe μ
                          (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (T n ω) (crit n ω))).toReal
                      (instHSub.hSub 1
                        (instHSub.hSub (HansenEconometrics.statisticCDFReal μ T n c)
                          (HansenEconometrics.statisticCDFReal μ T n (Real.instNeg.neg c))))))
                Filter.atTop (nhds 0) →
              Filter.Tendsto
                (fun n =>
                  instHMul.hMul n.cast
                    (instHAdd.hAdd
                      (instHSub.hSub
                        (MeasureTheory.Measure.instFunLike.coe μ
                            (setOf fun ω => HansenEconometrics.bootstrapAbsTestReject (T n ω) (crit n ω))).toReal
                        alpha)
                      (instHMul.hMul (Real.instInv.inv n.cast) (instHMul.hMul 2 (instHMul.hMul (p2 c) (density c))))))
                Filter.atTop (nhds 0)

Direct statement dependencies (3)

HansenEconometrics.SecondOrderEdgeworthExpansion
HansenEconometrics.bootstrapAbsTestReject
HansenEconometrics.statisticCDFReal

HansenEconometrics/Chapter10Bootstrap/HigherOrder.lean:665

Theorem 10.186 of 28 linked endpoints

For regression, \sqrt n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}N(0,V_\theta) and the studentized statistic satisfies T^*\xrightarrow{d^*}N(0,1).

theorem HansenEconometrics.chapter10_bootstrap_regression_theta_gaussian

Hansen Theorem 10.18, nonlinear-regression delta-method Gaussian wrapper.

If the bootstrap regression coefficient statistic converges weakly to N(0,Vβ), then the derivative-linearized statistic for a smooth transformation with Jacobian R converges weakly to N(0,R’ Vβ R). This is the regression surface of the bootstrap Delta method; the concrete OLS bootstrap constructor supplies the coefficient-level bootstrap CLT premise.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] [inst_2 : DecidableEq k]
  [inst_3 : DecidableEq q] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {Vβ : Matrix k k Real} (R : Matrix k q Real),
  Vβ.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
        (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
      HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
        (fun n ω ωs =>
          ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs))
        (ProbabilityTheory.multivariateGaussian 0
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R))
        fun z => z

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Regression.lean:1246

theorem HansenEconometrics.chapter10_bootstrap_regression_tstat_standardNormal

Hansen Theorem 10.18, regression bootstrap t-statistic standard-normal wrapper.

If the transformed regression numerator and feasible standard-error scale have joint bootstrap weak limit (s Z, s) with Z ~ N(0,1), and the scale itself converges to the positive constant s in bootstrap probability, then the studentized transformed statistic has standard-normal bootstrap weak limit. Concrete regression applications supply the joint numerator/scale limit and scale consistency from the model-specific covariance estimator.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ : Real},
  Real.instLT.lt 0 seθ →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
        (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
        fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
            (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
              HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
                (fun n ω ωs => instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs))
                (ProbabilityTheory.gaussianReal 0 1) fun z => z

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6535

theorem HansenEconometrics.chapter10_bootstrap_regression_theta_gaussian_distribution

Hansen Theorem 10.18, regression Gaussian CDF wrapper.

This is the Hansen Definition 10.2 face of chapter10_bootstrap_regression_theta_gaussian: after the coefficient-level bootstrap CLT and the delta-method linear map, coordinate CDF convergence follows at transformed Gaussian continuity points whose lower-orthant frontiers are null.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] [inst_2 : DecidableEq k]
  [inst_3 : DecidableEq q] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {Vβ : Matrix k k Real} (R : Matrix k q Real),
  Vβ.PosSemidef →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
        (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
          (∀ (x : q → Real),
              ContinuousAt
                  (fun y =>
                    HansenEconometrics.vectorCDF
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R))
                      (fun z => z.ofLp) y)
                  x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe
                    (MeasureTheory.Measure.map (fun z => z.ofLp)
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R)))
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                    (TbetaStar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R))
              fun z => z.ofLp

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:1271

theorem HansenEconometrics.chapter10_bootstrap_regression_abs_tstat_standardNormalAbs

Hansen Theorem 10.18, absolute regression bootstrap t-statistic absolute-standard-normal wrapper.

This is the weak bootstrap law for the absolute statistic used by the two-sided bootstrap-test critical-value route in Theorem 10.16.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TthetaStar seThetaStar : Nat → Ω → Ωs → Real} {seθ : Real},
  Real.instLT.lt 0 seθ →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
        (fun n ω ωs => { fst := TthetaStar n ω ωs, snd := seThetaStar n ω ωs }) (ProbabilityTheory.gaussianReal 0 1)
        fun z => { fst := instHMul.hMul seθ z, snd := seθ }) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (TthetaStar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (seThetaStar n ω)) →
            (HansenEconometrics.TendstoInBootstrapProbability μ Pstar seThetaStar fun x => seθ) →
              HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
                (fun n ω ωs => abs (instHDiv.hDiv (TthetaStar n ω ωs) (seThetaStar n ω ωs)))
                (MeasureTheory.Measure.map (fun z => abs z) (ProbabilityTheory.gaussianReal 0 1)) fun z => z

Direct statement dependencies (2)

HansenEconometrics.TendstoInBootstrapProbability
HansenEconometrics.TendstoInBootstrapWeakDistribution

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6634

theorem HansenEconometrics.chapter10_indexed_bootstrap_score_gaussian_finSucc_resampleMean

Hansen Theorem 10.18 score-level ordinary-bootstrap CLT.

The ordinary Fin (n+1) nonparametric bootstrap CLT applied to the regression score vectors e_i X_i gives a Euclidean score Gaussian with covariance scoreCovMat.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e : Nat → Ω → Real},
  HansenEconometrics.ScoreCLTConditions μ X e →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs => HansenEconometrics.regressionBootstrapScoreFinSucc X e n ω ωs)
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.scoreCovMat μ X e)) fun z => z

Direct statement dependencies (4)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.regressionBootstrapScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:578

theorem HansenEconometrics.chapter10_bootstrap_regression_theta_gaussian_distribution_posDef

Hansen Theorem 10.18, regression Gaussian CDF wrapper with positive definite transformed covariance.

When R’ Vβ R is positive definite, the transformed Gaussian lower-orthant null-frontier premise in chapter10_bootstrap_regression_theta_gaussian_distribution is automatic.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] [inst_2 : DecidableEq k]
  [inst_3 : DecidableEq q] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {Vβ : Matrix k k Real} (R : Matrix k q Real),
  Vβ.PosSemidef →
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R).PosDef →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
          (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                    (TbetaStar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R))
              fun z => z.ofLp

Direct statement dependencies (3)

HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Regression.lean:1310

Theorem 10.18 robust feasible HC Gaussian wrappers6 endpoints

For regression, \sqrt n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}N(0,V_\theta) and the studentized statistic satisfies T^*\xrightarrow{d^*}N(0,1). Robust feasible HC conditions discharge positive semidefiniteness of the heteroskedastic coefficient covariance in the regression Gaussian route

theorem HansenEconometrics.chapter10_bootstrap_regression_theta_gaussian_of_robustFeasibleHCMomentConditions

Hansen Theorem 10.18 regression Gaussian wrapper under the Chapter 7 robust feasible HC condition package.

This discharges positive semidefiniteness of the heteroskedastic coefficient covariance from RobustFeasibleHCMomentConditions. The coefficient-level bootstrap CLT remains the model-specific premise.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q]
  [inst_3 : DecidableEq k] [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} (β : k → Real)
  (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
      HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar
        (fun n ω ωs =>
          ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs))
        (ProbabilityTheory.multivariateGaussian 0
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e)) R))
        fun z => z

Direct statement dependencies (4)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Regression.lean:1343

theorem HansenEconometrics.chapter10_bootstrap_regression_theta_gaussian_distribution_of_robustFeasibleHCMomentConditions

Hansen Definition 10.2 face of chapter10_bootstrap_regression_theta_gaussian_of_robustFeasibleHCMomentConditions.

The transformed Gaussian frontier premise is left explicit; use the _posDef variant when R’ Vβ R is positive definite.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q]
  [inst_3 : DecidableEq k] [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} (β : k → Real)
  (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
          (∀ (x : q → Real),
              ContinuousAt
                  (fun y =>
                    HansenEconometrics.vectorCDF
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                            (HansenEconometrics.heteroAsymCov μ X e))
                          R))
                      (fun z => z.ofLp) y)
                  x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe
                    (MeasureTheory.Measure.map (fun z => z.ofLp)
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                            (HansenEconometrics.heteroAsymCov μ X e))
                          R)))
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                    (TbetaStar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                    (HansenEconometrics.heteroAsymCov μ X e))
                  R))
              fun z => z.ofLp

Direct statement dependencies (7)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:1374

theorem HansenEconometrics.chapter10_bootstrap_regression_theta_gaussian_distribution_posDef_of_robustFeasibleHCMomentConditions

Positive-definite transformed-covariance version of chapter10_bootstrap_regression_theta_gaussian_distribution_of_robustFeasibleHCMomentConditions.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q]
  [inst_3 : DecidableEq k] [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} (β : k → Real)
  (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e))
          R).PosDef →
      (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
          (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            HansenEconometrics.TendstoInBootstrapDistribution μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                    (TbetaStar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                    (HansenEconometrics.heteroAsymCov μ X e))
                  R))
              fun z => z.ofLp

Direct statement dependencies (5)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistribution
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Regression.lean:1417

theorem HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_of_robustFeasibleHCMomentConditions

Indexed Hansen Theorem 10.18 regression Gaussian wrapper under the Chapter 7 robust feasible HC condition package.

This discharges positive semidefiniteness of heteroAsymCov μ X e; the indexed coefficient-level bootstrap CLT remains explicit.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_5 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
      HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar
        (fun n ω ωs =>
          ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs))
        (ProbabilityTheory.multivariateGaussian 0
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e)) R))
        fun z => z

Direct statement dependencies (4)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6420

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_distribution_of_robustFeasibleHCMomentConditions

Indexed Hansen Definition 10.2 face of chapter10_indexed_bootstrap_regression_theta_gaussian_of_robustFeasibleHCMomentConditions.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_5 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
      (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
        (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
          (∀ (x : q → Real),
              ContinuousAt
                  (fun y =>
                    HansenEconometrics.vectorCDF
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                            (HansenEconometrics.heteroAsymCov μ X e))
                          R))
                      (fun z => z.ofLp) y)
                  x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe
                    (MeasureTheory.Measure.map (fun z => z.ofLp)
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                            (HansenEconometrics.heteroAsymCov μ X e))
                          R)))
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                    (TbetaStar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                    (HansenEconometrics.heteroAsymCov μ X e))
                  R))
              fun z => z.ofLp

Direct statement dependencies (7)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6449

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_distribution_posDef_of_robustFeasibleHCMomentConditions

Positive-definite transformed-covariance version of chapter10_indexed_bootstrap_regression_theta_gaussian_distribution_of_robustFeasibleHCMomentConditions.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_5 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e))
          R).PosDef →
      (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
          (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
        (∀ (n : Nat) (ω : Ω), MeasureTheory.IsFiniteMeasure (Pstar n ω)) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            HansenEconometrics.TendstoInBootstrapDistributionIndexed μ Pstar
              (fun n ω ωs =>
                (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                    (TbetaStar n ω ωs)).ofLp)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                    (HansenEconometrics.heteroAsymCov μ X e))
                  R))
              fun z => z.ofLp

Direct statement dependencies (5)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6493

Theorem 10.18 robust feasible HC finite score/base wrappers2 endpoints

For regression, \sqrt n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}N(0,V_\theta) and the studentized statistic satisfies T^*\xrightarrow{d^*}N(0,1). Robust feasible HC conditions discharge the condition-package conversion for the concrete Fin (n+1) ordinary-bootstrap score and linearized coefficient CLTs

theorem HansenEconometrics.chapter10_indexed_bootstrap_score_gaussian_finSucc_resampleMean_of_robustFeasibleHCMomentConditions

Robust-feasible HC face of the ordinary finite-resample regression score bootstrap CLT.

The Chapter 7 robust-feasible condition package supplies the score CLT conditions; positive definiteness of the score covariance remains explicit.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  (β : k → Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs => HansenEconometrics.regressionBootstrapScoreFinSucc X e n ω ωs)
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.scoreCovMat μ X e)) fun z => z

Direct statement dependencies (4)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.regressionBootstrapScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:716

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearizedScore_gaussian_finSucc_resampleMean_of_robustFeasibleHCMomentConditions

Robust-feasible HC face of the ordinary finite-resample linearized regression coefficient CLT.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  (β : k → Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs => HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs)
        (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z

Direct statement dependencies (5)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:739

Theorem 10.18 robust feasible HC finite ordinary-bootstrap wrappers6 of 7 linked endpoints

For regression, \sqrt n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}N(0,V_\theta) and the studentized statistic satisfies T^*\xrightarrow{d^*}N(0,1). Robust feasible HC conditions feed the concrete Fin (n+1) ordinary-bootstrap score route by projecting to the Chapter 10 score CLT package

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_distribution_finSucc_linearizedScore_of_robustFeasibleHCMomentConditions

Hansen Definition 10.2 robust-feasible HC face of the finite ordinary-bootstrap linearized score route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      (∀ (x : q → Real),
          ContinuousAt
              (fun y =>
                HansenEconometrics.vectorCDF
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                        (HansenEconometrics.heteroAsymCov μ X e))
                      R))
                  (fun z => z.ofLp) y)
              x →
            Eq
              (MeasureTheory.Measure.instFunLike.coe
                (MeasureTheory.Measure.map (fun z => z.ofLp)
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                        (HansenEconometrics.heteroAsymCov μ X e))
                      R)))
                (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
              0) →
        HansenEconometrics.TendstoInBootstrapDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
          (fun n ω ωs =>
            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs)).ofLp)
          (ProbabilityTheory.multivariateGaussian 0
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e))
              R))
          fun z => z.ofLp

Direct statement dependencies (8)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6069

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_distribution_finSucc_of_linearizedScore_tight_of_robustFeasibleHCMomentConditions

Hansen Definition 10.2 robust-feasible HC face of the finite ordinary-bootstrap nonlinear coefficient-transfer route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {TbetaStar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → EuclideanSpace Real k}
  (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
        (∀ (η : Real),
            Real.instLT.lt 0 η →
              Exists fun K =>
                And (IsCompact K)
                  (And
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Not
                              (Set.instMembership.mem K
                                (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                      Filter.atTop fun x => 0)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs => Not (Set.instMembership.mem K (TbetaStar n ω ωs))))
                      Filter.atTop fun x => 0))) →
          (∀ (δ : Real),
              Real.instLT.lt 0 δ →
                MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (ProbabilityTheory.uniformOn Set.univ).real
                      (setOf fun ωs =>
                        Real.instLE.le δ
                          ((PiLp.instDist 2 fun x => Real).dist (TbetaStar n ω ωs)
                            (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                  Filter.atTop fun x => 0) →
            (∀ (x : q → Real),
                ContinuousAt
                    (fun y =>
                      HansenEconometrics.vectorCDF
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                              (HansenEconometrics.heteroAsymCov μ X e))
                            R))
                        (fun z => z.ofLp) y)
                    x →
                  Eq
                    (MeasureTheory.Measure.instFunLike.coe
                      (MeasureTheory.Measure.map (fun z => z.ofLp)
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                              (HansenEconometrics.heteroAsymCov μ X e))
                            R)))
                      (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                    0) →
              HansenEconometrics.TendstoInBootstrapDistributionIndexed μ
                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                (fun n ω ωs =>
                  (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                      (TbetaStar n ω ωs)).ofLp)
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                      (HansenEconometrics.heteroAsymCov μ X e))
                    R))
                fun z => z.ofLp

Direct statement dependencies (8)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6237

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_finSucc_linearizedScore_of_robustFeasibleHCMomentConditions

Robust-feasible HC face of the finite ordinary-bootstrap linearized score route.

The Chapter 7 robust-feasible condition package supplies the score CLT conditions; positive definiteness of the score covariance remains explicit.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
        (fun n ω ωs =>
          ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
            (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
        (ProbabilityTheory.multivariateGaussian 0
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e)) R))
        fun z => z

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6045

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_beta_gaussian_finSucc_of_linearizedScore_tight_of_robustFeasibleHCMomentConditions

Robust-feasible HC face of the finite ordinary-bootstrap nonlinear coefficient-transfer route.

The robust-feasible condition package supplies the score CLT conditions; the model-specific nonlinear OLS work remains the conditional closeness and compact-tail premise for TbetaStar.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {TbetaStar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → EuclideanSpace Real k}
  (β : k → Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
        (∀ (η : Real),
            Real.instLT.lt 0 η →
              Exists fun K =>
                And (IsCompact K)
                  (And
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Not
                              (Set.instMembership.mem K
                                (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                      Filter.atTop fun x => 0)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs => Not (Set.instMembership.mem K (TbetaStar n ω ωs))))
                      Filter.atTop fun x => 0))) →
          (∀ (δ : Real),
              Real.instLT.lt 0 δ →
                MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (ProbabilityTheory.uniformOn Set.univ).real
                      (setOf fun ωs =>
                        Real.instLE.le δ
                          ((PiLp.instDist 2 fun x => Real).dist (TbetaStar n ω ωs)
                            (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                  Filter.atTop fun x => 0) →
            HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ
              (fun n x => ProbabilityTheory.uniformOn Set.univ) TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z

Direct statement dependencies (5)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6135

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_finSucc_of_linearizedScore_tight_of_robustFeasibleHCMomentConditions

Robust-feasible HC face of the finite ordinary-bootstrap transformed nonlinear coefficient-transfer route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {TbetaStar : (n : Nat) → Ω → (Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)) → EuclideanSpace Real k}
  (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
        (∀ (η : Real),
            Real.instLT.lt 0 η →
              Exists fun K =>
                And (IsCompact K)
                  (And
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Not
                              (Set.instMembership.mem K
                                (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                      Filter.atTop fun x => 0)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs => Not (Set.instMembership.mem K (TbetaStar n ω ωs))))
                      Filter.atTop fun x => 0))) →
          (∀ (δ : Real),
              Real.instLT.lt 0 δ →
                MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (ProbabilityTheory.uniformOn Set.univ).real
                      (setOf fun ωs =>
                        Real.instLE.le δ
                          ((PiLp.instDist 2 fun x => Real).dist (TbetaStar n ω ωs)
                            (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                  Filter.atTop fun x => 0) →
            HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ
              (fun n x => ProbabilityTheory.uniformOn Set.univ)
              (fun n ω ωs =>
                ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                  (TbetaStar n ω ωs))
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose
                    (HansenEconometrics.heteroAsymCov μ X e))
                  R))
              fun z => z

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6186

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_theta_gaussian_distribution_posDef_finSucc_linearizedScore_of_robustFeasibleHCMomentConditions

Positive-definite transformed-covariance robust-feasible HC CDF face of the finite ordinary-bootstrap linearized score route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : Fintype q] [inst_3 : DecidableEq k]
  [inst_4 : DecidableEq q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} (β : k → Real) (R : Matrix k q Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e))
            R).PosDef →
        HansenEconometrics.TendstoInBootstrapDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
          (fun n ω ωs =>
            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs)).ofLp)
          (ProbabilityTheory.multivariateGaussian 0
            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose (HansenEconometrics.heteroAsymCov μ X e))
              R))
          fun z => z.ofLp

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:6105

Theorem 10.18 scalar finite OLS Definition 10.2 wrappers6 of 8 linked endpoints

For regression, \sqrt n(\hat\theta^*-\hat\theta)\xrightarrow{d^*}N(0,V_\theta) and the studentized statistic satisfies T^*\xrightarrow{d^*}N(0,1). The concrete one-row ordinary-bootstrap OLS numerator has CDF convergence faces matching the transformed-statistic route

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearRestriction_gaussian_distribution_finSucc_olsBetaOrZero_of_gapEnvelope_tight

Hansen Definition 10.2 CDF face of the scalar one-row ordinary-bootstrap OLS restriction transfer from the explicit finite OLS-linearization gap envelope.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  (β : k → Real) (R : Matrix Unit k Real),
  (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
    HansenEconometrics.ScoreCLTConditions μ X e →
      (HansenEconometrics.scoreCovMat μ X e).PosDef →
        (∀ (η : Real),
            Real.instLT.lt 0 η →
              Exists fun K =>
                And (IsCompact K)
                  (And
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Not
                              (Set.instMembership.mem K
                                (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                      Filter.atTop fun x => 0)
                    (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Not
                              (Set.instMembership.mem K
                                (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))))
                      Filter.atTop fun x => 0))) →
          (∀ (δ : Real),
              Real.instLT.lt 0 δ →
                MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (ProbabilityTheory.uniformOn Set.univ).real
                      (setOf fun ωs =>
                        Real.instLE.le δ
                          (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                  Filter.atTop fun x => 0) →
            (∀ (x : Unit → Real),
                ContinuousAt
                    (fun y =>
                      HansenEconometrics.vectorCDF
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                              (HansenEconometrics.heteroAsymCov μ X e))
                            R.transpose))
                        (fun z => z.ofLp) y)
                    x →
                  Eq
                    (MeasureTheory.Measure.instFunLike.coe
                      (MeasureTheory.Measure.map (fun z => z.ofLp)
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                              (HansenEconometrics.heteroAsymCov μ X e))
                            R.transpose)))
                      (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                    0) →
              HansenEconometrics.TendstoInBootstrapDistributionIndexed μ
                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                (fun n ω ωs x => HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                    R.transpose))
                fun z => z.ofLp

Direct statement dependencies (10)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:5466

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearRestriction_gaussian_distribution_finSucc_olsBetaOrZero_of_gapEnvelope_bounds

Hansen Definition 10.2 CDF face of the bounded scalar one-row ordinary-bootstrap OLS gap-envelope transfer.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Clin Cbeta : Real} (β : k → Real) (R : Matrix Unit k Real),
  (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
    HansenEconometrics.ScoreCLTConditions μ X e →
      (HansenEconometrics.scoreCovMat μ X e).PosDef →
        Filter.Eventually
            (fun n =>
              ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                Real.instLE.le
                  ((PiLp.instNorm 2 fun x => Real).norm
                    (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                  Clin)
            Filter.atTop →
          Filter.Eventually
              (fun n =>
                ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                  Real.instLE.le
                    ((PiLp.instNorm 2 fun x => Real).norm
                      (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                    Cbeta)
              Filter.atTop →
            (∀ (δ : Real),
                Real.instLT.lt 0 δ →
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (ProbabilityTheory.uniformOn Set.univ).real
                        (setOf fun ωs =>
                          Real.instLE.le δ
                            (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                    Filter.atTop fun x => 0) →
              (∀ (x : Unit → Real),
                  ContinuousAt
                      (fun y =>
                        HansenEconometrics.vectorCDF
                          (ProbabilityTheory.multivariateGaussian 0
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                                (HansenEconometrics.heteroAsymCov μ X e))
                              R.transpose))
                          (fun z => z.ofLp) y)
                      x →
                    Eq
                      (MeasureTheory.Measure.instFunLike.coe
                        (MeasureTheory.Measure.map (fun z => z.ofLp)
                          (ProbabilityTheory.multivariateGaussian 0
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                                (HansenEconometrics.heteroAsymCov μ X e))
                              R.transpose)))
                        (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                      0) →
                HansenEconometrics.TendstoInBootstrapDistributionIndexed μ
                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                  (fun n ω ωs x => HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                      R.transpose))
                  fun z => z.ofLp

Direct statement dependencies (10)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:5714

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearRestriction_gaussian_distribution_posDef_finSucc_olsBetaOrZero_of_gapEnvelope_tight

Positive-definite Hansen Definition 10.2 CDF face of the scalar one-row ordinary-bootstrap OLS gap-envelope transfer.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  (β : k → Real) (R : Matrix Unit k Real),
  (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
    HansenEconometrics.ScoreCLTConditions μ X e →
      (HansenEconometrics.scoreCovMat μ X e).PosDef →
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
              R.transpose).PosDef →
          (∀ (η : Real),
              Real.instLT.lt 0 η →
                Exists fun K =>
                  And (IsCompact K)
                    (And
                      (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          (ProbabilityTheory.uniformOn Set.univ).real
                            (setOf fun ωs =>
                              Not
                                (Set.instMembership.mem K
                                  (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                        Filter.atTop fun x => 0)
                      (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          (ProbabilityTheory.uniformOn Set.univ).real
                            (setOf fun ωs =>
                              Not
                                (Set.instMembership.mem K
                                  (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))))
                        Filter.atTop fun x => 0))) →
            (∀ (δ : Real),
                Real.instLT.lt 0 δ →
                  MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (ProbabilityTheory.uniformOn Set.univ).real
                        (setOf fun ωs =>
                          Real.instLE.le δ
                            (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                    Filter.atTop fun x => 0) →
              HansenEconometrics.TendstoInBootstrapDistributionIndexed μ
                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                (fun n ω ωs x => HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                    R.transpose))
                fun z => z.ofLp

Direct statement dependencies (8)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:5546

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearRestriction_gaussian_distribution_posDef_finSucc_olsBetaOrZero_of_gapEnvelope_bounds

Positive-definite Hansen Definition 10.2 CDF face of the bounded scalar one-row ordinary-bootstrap OLS gap-envelope transfer.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Clin Cbeta : Real} (β : k → Real) (R : Matrix Unit k Real),
  (∀ (i : Nat) (ω : Ω), Eq (y i ω) (instHAdd.hAdd (dotProduct (X i ω) β) (e i ω))) →
    HansenEconometrics.ScoreCLTConditions μ X e →
      (HansenEconometrics.scoreCovMat μ X e).PosDef →
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
              (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
              R.transpose).PosDef →
          Filter.Eventually
              (fun n =>
                ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                  Real.instLE.le
                    ((PiLp.instNorm 2 fun x => Real).norm
                      (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                    Clin)
              Filter.atTop →
            Filter.Eventually
                (fun n =>
                  ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                    Real.instLE.le
                      ((PiLp.instNorm 2 fun x => Real).norm
                        (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                      Cbeta)
                Filter.atTop →
              (∀ (δ : Real),
                  Real.instLT.lt 0 δ →
                    MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        (ProbabilityTheory.uniformOn Set.univ).real
                          (setOf fun ωs =>
                            Real.instLE.le δ
                              (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                      Filter.atTop fun x => 0) →
                HansenEconometrics.TendstoInBootstrapDistributionIndexed μ
                  (fun n x => ProbabilityTheory.uniformOn Set.univ)
                  (fun n ω ωs x => HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                  (ProbabilityTheory.multivariateGaussian 0
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                      R.transpose))
                  fun z => z.ofLp

Direct statement dependencies (8)

HansenEconometrics.ScoreCLTConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:5785

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearRestriction_gaussian_distribution_finSucc_olsBetaOrZero_of_gapEnvelope_tight_of_robustFeasibleHCMomentConditions

Robust-feasible HC Hansen Definition 10.2 CDF face of the scalar one-row ordinary-bootstrap OLS gap-envelope transfer.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      (∀ (η : Real),
          Real.instLT.lt 0 η →
            Exists fun K =>
              And (IsCompact K)
                (And
                  (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (ProbabilityTheory.uniformOn Set.univ).real
                        (setOf fun ωs =>
                          Not
                            (Set.instMembership.mem K
                              (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))))
                    Filter.atTop fun x => 0)
                  (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (ProbabilityTheory.uniformOn Set.univ).real
                        (setOf fun ωs =>
                          Not
                            (Set.instMembership.mem K
                              (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))))
                    Filter.atTop fun x => 0))) →
        (∀ (δ : Real),
            Real.instLT.lt 0 δ →
              MeasureTheory.TendstoInMeasure μ
                (fun n ω =>
                  (ProbabilityTheory.uniformOn Set.univ).real
                    (setOf fun ωs =>
                      Real.instLE.le δ
                        (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                Filter.atTop fun x => 0) →
          (∀ (x : Unit → Real),
              ContinuousAt
                  (fun y =>
                    HansenEconometrics.vectorCDF
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                          R.transpose))
                      (fun z => z.ofLp) y)
                  x →
                Eq
                  (MeasureTheory.Measure.instFunLike.coe
                    (MeasureTheory.Measure.map (fun z => z.ofLp)
                      (ProbabilityTheory.multivariateGaussian 0
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                          R.transpose)))
                    (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                  0) →
            HansenEconometrics.TendstoInBootstrapDistributionIndexed μ (fun n x => ProbabilityTheory.uniformOn Set.univ)
              (fun n ω ωs x => HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
              (ProbabilityTheory.multivariateGaussian 0
                (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                  R.transpose))
              fun z => z.ofLp

Direct statement dependencies (10)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:5600

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_linearRestriction_gaussian_distribution_finSucc_olsBetaOrZero_of_gapEnvelope_bounds_of_robustFeasibleHCMomentConditions

Robust-feasible HC Hansen Definition 10.2 CDF face of the bounded scalar one-row ordinary-bootstrap OLS gap-envelope transfer.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} [inst_1 : Fintype k] [inst_2 : DecidableEq k] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Clin Cbeta : Real} (β : k → Real) (R : Matrix Unit k Real),
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (HansenEconometrics.scoreCovMat μ X e).PosDef →
      Filter.Eventually
          (fun n =>
            ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
              Real.instLE.le
                ((PiLp.instNorm 2 fun x => Real).norm
                  (HansenEconometrics.regressionLinearizedScoreFinSucc μ X e n ω ωs))
                Clin)
          Filter.atTop →
        Filter.Eventually
            (fun n =>
              ∀ (ω : Ω) (ωs : Fin (instHAdd.hAdd n 1) → Fin (instHAdd.hAdd n 1)),
                Real.instLE.le
                  ((PiLp.instNorm 2 fun x => Real).norm
                    (HansenEconometrics.regressionBootstrapBetaStatisticFinSucc X y n ω ωs))
                  Cbeta)
            Filter.atTop →
          (∀ (δ : Real),
              Real.instLT.lt 0 δ →
                MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (ProbabilityTheory.uniformOn Set.univ).real
                      (setOf fun ωs =>
                        Real.instLE.le δ
                          (HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc μ X e β n ω ωs)))
                  Filter.atTop fun x => 0) →
            (∀ (x : Unit → Real),
                ContinuousAt
                    (fun y =>
                      HansenEconometrics.vectorCDF
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                              (HansenEconometrics.heteroAsymCov μ X e))
                            R.transpose))
                        (fun z => z.ofLp) y)
                    x →
                  Eq
                    (MeasureTheory.Measure.instFunLike.coe
                      (MeasureTheory.Measure.map (fun z => z.ofLp)
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R
                              (HansenEconometrics.heteroAsymCov μ X e))
                            R.transpose)))
                      (frontier (setOf fun z => HansenEconometrics.coordinateLE z x)))
                    0) →
              HansenEconometrics.TendstoInBootstrapDistributionIndexed μ
                (fun n x => ProbabilityTheory.uniformOn Set.univ)
                (fun n ω ωs x => HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc R X y n ω ωs)
                (ProbabilityTheory.multivariateGaussian 0
                  (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                    (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R (HansenEconometrics.heteroAsymCov μ X e))
                    R.transpose))
                fun z => z.ofLp

Direct statement dependencies (10)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapDistributionIndexed
HansenEconometrics.coordinateLE
HansenEconometrics.heteroAsymCov
HansenEconometrics.regressionBootstrapBetaLinearizedGapEnvelopeFinSucc
HansenEconometrics.regressionBootstrapBetaStatisticFinSucc
HansenEconometrics.regressionBootstrapLinearRestrictionStatisticFinSucc
HansenEconometrics.regressionLinearizedScoreFinSucc
HansenEconometrics.scoreCovMat
HansenEconometrics.vectorCDF

HansenEconometrics/Chapter10Bootstrap/Regression.lean:5830

Theorem 10.196 of 20 linked endpoints

The trimmed bootstrap covariance for the transformed regression statistic satisfies \hat V_{\theta}^{\mathrm{boot},\tau}\xrightarrow{p^*}V_\theta.

theorem HansenEconometrics.chapter10_bootstrap_regression_trimmedVariance_tendsto_of_linearization_normFourth

Hansen Theorem 10.19, regression-facing trimmed covariance consistency from coefficient-level Gaussian bootstrap convergence and norm-fourth control.

This specializes the smooth exact-linearization trimmed covariance route to the regression transform Rᵀ Tβ*, so callers do not have to separately provide the trimmed conditional mean and cross-moment convergence premises.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] [inst_2 : DecidableEq k]
  [inst_3 : DecidableEq q] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real} {Vβ : Matrix k k Real} (R : Matrix k q Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R.transpose.transpose))]
  {B : Real},
  Vβ.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
            (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω) (a : q),
                MeasureTheory.MemLp
                  (fun ωs =>
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp
                      a)
                  2 (Pstar n ω)) →
              (∀ (a : q),
                  MeasureTheory.MemLp (fun z => z.ofLp a) 2
                    (ProbabilityTheory.multivariateGaussian 0
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R.transpose.transpose))) →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                          (setOf fun ωs =>
                            Real.instLT.lt (τ n)
                              (Pi.normedRing.norm
                                (ContinuousLinearMap.funLike.coe
                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                    (TbetaStar n ω ωs)).ofLp))).toReal)
                    Filter.atTop fun x => 0) →
                  Real.instLE.le 0 B →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                        Filter.atTop fun x => B) →
                      (∀ (n : Nat) (ω : Ω),
                          MeasureTheory.Integrable
                            (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                            (Pstar n ω)) →
                        MeasureTheory.TendstoInMeasure μ
                          (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                            (fun n ω ωs =>
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                            τ)
                          Filter.atTop fun x => HansenEconometrics.smoothFunctionVarianceFunctional R Vβ

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:7505

theorem HansenEconometrics.chapter10_bootstrap_regression_trimmedVariance_tendsto_of_linearization_secondMoment

Hansen Theorem 10.19 regression-facing trimmed covariance route with the trimming-tail probability discharged by conditional second moments and a diverging threshold.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] [inst_2 : DecidableEq k]
  [inst_3 : DecidableEq q] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real} {Vβ : Matrix k k Real} (R : Matrix k q Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R.transpose.transpose))]
  {Bsecond Bfourth : Real},
  Vβ.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.MemLp
                    (fun ωs =>
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp)
                    2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : q),
                    MeasureTheory.MemLp
                      (fun ωs =>
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                              (TbetaStar n ω ωs)).ofLp
                          a)
                      2 (Pstar n ω)) →
                  (∀ (a : q),
                      MeasureTheory.MemLp (fun z => z.ofLp a) 2
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ)
                            R.transpose.transpose))) →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow
                              (Pi.normedRing.norm
                                (ContinuousLinearMap.funLike.coe
                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                              2)
                        Filter.atTop fun x => Bsecond) →
                      Real.instLE.le 0 Bfourth →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                            Filter.atTop fun x => Bfourth) →
                          (∀ (n : Nat) (ω : Ω),
                              MeasureTheory.Integrable
                                (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                                (Pstar n ω)) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                (fun n ω ωs =>
                                  (ContinuousLinearMap.funLike.coe
                                      (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                      (TbetaStar n ω ωs)).ofLp)
                                τ)
                              Filter.atTop fun x => HansenEconometrics.smoothFunctionVarianceFunctional R Vβ

Direct statement dependencies (4)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:7661

theorem HansenEconometrics.chapter10_regression_finiteReplicationTrimmedVariance_l2_of_linearization_normFourth

Hansen Theorem 10.19 finite-replication regression trimmed covariance route from coefficient-level Gaussian bootstrap convergence, norm-fourth control, and coordinatewise L² simulation-error bounds.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q]
  [inst_2 : DecidableEq k] [inst_3 : DecidableEq q] {Zsim : Nat → Nat → Ω → q → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real}
  {Vβ : Matrix k k Real} (R : Matrix k q Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R.transpose.transpose))]
  {B : Real} {Cfinite : q → q → Real},
  Vβ.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
            (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω) (a : q),
                MeasureTheory.MemLp
                  (fun ωs =>
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp
                      a)
                  2 (Pstar n ω)) →
              (∀ (a : q),
                  MeasureTheory.MemLp (fun z => z.ofLp a) 2
                    (ProbabilityTheory.multivariateGaussian 0
                      (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R.transpose.transpose))) →
                (MeasureTheory.TendstoInMeasure μ
                    (fun n ω =>
                      (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                          (setOf fun ωs =>
                            Real.instLT.lt (τ n)
                              (Pi.normedRing.norm
                                (ContinuousLinearMap.funLike.coe
                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                    (TbetaStar n ω ωs)).ofLp))).toReal)
                    Filter.atTop fun x => 0) →
                  Real.instLE.le 0 B →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                        Filter.atTop fun x => B) →
                      (∀ (n : Nat) (ω : Ω),
                          MeasureTheory.Integrable
                            (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                            (Pstar n ω)) →
                        (∀ (a c : q) (n : Nat),
                            MeasureTheory.Integrable
                              (fun ω =>
                                instHPow.hPow
                                  (Real.norm.norm
                                    (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                      (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                        (fun n ω ωs =>
                                          (ContinuousLinearMap.funLike.coe
                                              (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                              (TbetaStar n ω ωs)).ofLp)
                                        τ n ω)
                                      a c))
                                  2)
                              μ) →
                          (∀ (a c : q),
                              Filter.Eventually
                                (fun n =>
                                  Real.instLE.le
                                    (MeasureTheory.integral μ fun ω =>
                                      instHPow.hPow
                                        (Real.norm.norm
                                          (instHSub.hSub
                                            (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                            (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                              (fun n ω ωs =>
                                                (ContinuousLinearMap.funLike.coe
                                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                    (TbetaStar n ω ωs)).ofLp)
                                              τ n ω)
                                            a c))
                                        2)
                                    (instHDiv.hDiv (Cfinite a c) n.cast))
                                Filter.atTop) →
                            MeasureTheory.TendstoInMeasure μ
                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                              HansenEconometrics.smoothFunctionVarianceFunctional R Vβ

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:12558

theorem HansenEconometrics.chapter10_regression_finiteReplicationTrimmedVariance_l2_of_linearization_secondMoment

Hansen Theorem 10.19 finite-replication regression trimmed covariance route with trimming-tail negligibility discharged by conditional second moments and a diverging threshold.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q]
  [inst_2 : DecidableEq k] [inst_3 : DecidableEq q] {Zsim : Nat → Nat → Ω → q → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real}
  {Vβ : Matrix k k Real} (R : Matrix k q Real)
  [MeasureTheory.IsFiniteMeasure
      (ProbabilityTheory.multivariateGaussian 0
        (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ) R.transpose.transpose))]
  {Bsecond Bfourth : Real} {Cfinite : q → q → Real},
  Vβ.PosSemidef →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 Vβ) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.MemLp
                    (fun ωs =>
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp)
                    2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : q),
                    MeasureTheory.MemLp
                      (fun ωs =>
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                              (TbetaStar n ω ωs)).ofLp
                          a)
                      2 (Pstar n ω)) →
                  (∀ (a : q),
                      MeasureTheory.MemLp (fun z => z.ofLp a) 2
                        (ProbabilityTheory.multivariateGaussian 0
                          (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul
                            (Matrix.instHMulOfFintypeOfMulOfAddCommMonoid.hMul R.transpose Vβ)
                            R.transpose.transpose))) →
                    (MeasureTheory.TendstoInMeasure μ
                        (fun n ω =>
                          MeasureTheory.integral (Pstar n ω) fun ωs =>
                            instHPow.hPow
                              (Pi.normedRing.norm
                                (ContinuousLinearMap.funLike.coe
                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                              2)
                        Filter.atTop fun x => Bsecond) →
                      Real.instLE.le 0 Bfourth →
                        (MeasureTheory.TendstoInMeasure μ
                            (fun n ω =>
                              MeasureTheory.integral (Pstar n ω) fun ωs =>
                                instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                            Filter.atTop fun x => Bfourth) →
                          (∀ (n : Nat) (ω : Ω),
                              MeasureTheory.Integrable
                                (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                                (Pstar n ω)) →
                            (∀ (a c : q) (n : Nat),
                                MeasureTheory.Integrable
                                  (fun ω =>
                                    instHPow.hPow
                                      (Real.norm.norm
                                        (instHSub.hSub
                                          (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                          (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                            (fun n ω ωs =>
                                              (ContinuousLinearMap.funLike.coe
                                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                  (TbetaStar n ω ωs)).ofLp)
                                            τ n ω)
                                          a c))
                                      2)
                                  μ) →
                              (∀ (a c : q),
                                  Filter.Eventually
                                    (fun n =>
                                      Real.instLE.le
                                        (MeasureTheory.integral μ fun ω =>
                                          instHPow.hPow
                                            (Real.norm.norm
                                              (instHSub.hSub
                                                (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                                (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                                  (fun n ω ωs =>
                                                    (ContinuousLinearMap.funLike.coe
                                                        (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                        (TbetaStar n ω ωs)).ofLp)
                                                  τ n ω)
                                                a c))
                                            2)
                                        (instHDiv.hDiv (Cfinite a c) n.cast))
                                    Filter.atTop) →
                                MeasureTheory.TendstoInMeasure μ
                                  (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                                  HansenEconometrics.smoothFunctionVarianceFunctional R Vβ

Direct statement dependencies (5)

HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:12728

theorem HansenEconometrics.chapter10_bootstrap_regression_trimmedVariance_tendsto

Hansen Theorem 10.19, regression-facing trimmed bootstrap variance bridge.

For the transformed regression statistic, if the trimmed conditional mean converges to zero and the trimmed conditional cross moment converges to the delta-method covariance R’ Vβ R, then the trimmed bootstrap covariance estimator converges to R’ Vβ R. The concrete regression proof supplies these moment premises from Theorems 10.11 and 10.12.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  {k : Type u_7} {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {ZthetaStar : Nat → Ω → Ωs → q → Real} {τ : Nat → Real} {Vβ : Matrix k k Real} (R : Matrix k q Real),
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : q),
        MeasureTheory.MemLp (fun ωs => HansenEconometrics.trimmedBootstrapStatistic ZthetaStar τ n ω ωs a) 2
          (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ
          (HansenEconometrics.bootstrapMeanVec Pstar (HansenEconometrics.trimmedBootstrapStatistic ZthetaStar τ))
          Filter.atTop fun x => 0) →
        (MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapCrossMomentMat Pstar
              (HansenEconometrics.trimmedBootstrapStatistic ZthetaStar τ))
            Filter.atTop fun x => HansenEconometrics.smoothFunctionVarianceFunctional R Vβ) →
          MeasureTheory.TendstoInMeasure μ (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar ZthetaStar τ)
            Filter.atTop fun x => HansenEconometrics.smoothFunctionVarianceFunctional R Vβ

Direct statement dependencies (5)

HansenEconometrics.bootstrapCrossMomentMat
HansenEconometrics.bootstrapMeanVec
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat
HansenEconometrics.trimmedBootstrapStatistic

HansenEconometrics/Chapter10Bootstrap/Regression.lean:7442

theorem HansenEconometrics.chapter10_indexed_regression_finiteReplicationTrimmedVariance_l2

Indexed finite-replication version of Hansen Theorem 10.19 for sample-size-dependent bootstrap spaces.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {k : Type u_7}
  {q : Type u_8} [inst : Fintype k] [inst_1 : Fintype q] {Ωboot : Nat → Type u_9}
  [inst_2 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → q → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)} {ZthetaStar : (n : Nat) → Ω → Ωboot n → q → Real}
  {τ : Nat → Real} {Vβ : Matrix k k Real} (R : Matrix k q Real) {Cfinite : q → q → Real},
  (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
    (∀ (n : Nat) (ω : Ω) (a : q),
        MeasureTheory.MemLp (fun ωs => HansenEconometrics.trimmedBootstrapStatisticIndexed ZthetaStar τ n ω ωs a) 2
          (Pstar n ω)) →
      (MeasureTheory.TendstoInMeasure μ
          (HansenEconometrics.bootstrapMeanVecIndexed Pstar
            (HansenEconometrics.trimmedBootstrapStatisticIndexed ZthetaStar τ))
          Filter.atTop fun x => 0) →
        (MeasureTheory.TendstoInMeasure μ
            (HansenEconometrics.bootstrapCrossMomentMatIndexed Pstar
              (HansenEconometrics.trimmedBootstrapStatisticIndexed ZthetaStar τ))
            Filter.atTop fun x => HansenEconometrics.smoothFunctionVarianceFunctional R Vβ) →
          (∀ (a c : q) (n : Nat),
              MeasureTheory.Integrable
                (fun ω =>
                  instHPow.hPow
                    (Real.norm.norm
                      (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                        (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar ZthetaStar τ n ω) a c))
                    2)
                μ) →
            (∀ (a c : q),
                Filter.Eventually
                  (fun n =>
                    Real.instLE.le
                      (MeasureTheory.integral μ fun ω =>
                        instHPow.hPow
                          (Real.norm.norm
                            (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                              (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar ZthetaStar τ n ω) a c))
                          2)
                      (instHDiv.hDiv (Cfinite a c) n.cast))
                  Filter.atTop) →
              MeasureTheory.TendstoInMeasure μ (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim)
                Filter.atTop fun x => HansenEconometrics.smoothFunctionVarianceFunctional R Vβ

Direct statement dependencies (6)

HansenEconometrics.bootstrapCrossMomentMatIndexed
HansenEconometrics.bootstrapMeanVecIndexed
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed
HansenEconometrics.trimmedBootstrapStatisticIndexed

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:12505

Theorem 10.19 robust feasible HC trimmed covariance wrappers4 endpoints

The trimmed bootstrap covariance for the transformed regression statistic satisfies \hat V_{\theta}^{\mathrm{boot},\tau}\xrightarrow{p^*}V_\theta. Robust feasible HC conditions discharge positive semidefiniteness of the heteroskedastic coefficient covariance in the norm-fourth and second-moment trimmed covariance routes

theorem

HansenEconometrics.chapter10_bootstrap_regression_trimmedVariance_normFourth_gaussianLimit_of_robustFeasibleHCMomentConditions

Robust-feasible HC specialization of the Theorem 10.19 norm-fourth trimmed covariance route.

This fixes Vβ = heteroAsymCov μ X e and discharges positive semidefiniteness from the Chapter 7 robust feasible HC condition package. The coefficient-level bootstrap weak convergence and norm-fourth/trimming premises remain explicit.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] [inst_3 : Fintype q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real}
  (β : k → Real) (R : Matrix k q Real) {B : Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
            (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω) (a : q),
                MeasureTheory.MemLp
                  (fun ωs =>
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp
                      a)
                  2 (Pstar n ω)) →
              (MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                        (setOf fun ωs =>
                          Real.instLT.lt (τ n)
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                  (TbetaStar n ω ωs)).ofLp))).toReal)
                  Filter.atTop fun x => 0) →
                Real.instLE.le 0 B →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                      Filter.atTop fun x => B) →
                    (∀ (n : Nat) (ω : Ω),
                        MeasureTheory.Integrable
                          (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          (Pstar n ω)) →
                      MeasureTheory.TendstoInMeasure μ
                        (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                          (fun n ω ωs =>
                            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                (TbetaStar n ω ωs)).ofLp)
                          τ)
                        Filter.atTop fun x =>
                        HansenEconometrics.smoothFunctionVarianceFunctional R (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:8050

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_trimmedVariance_normFourth_gaussianLimit_of_robustFeasibleHCMomentConditions

Indexed robust-feasible HC specialization of the Theorem 10.19 norm-fourth trimmed covariance route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : DecidableEq k] [inst_3 : Fintype q]
  {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} {τ : Nat → Real} (β : k → Real) (R : Matrix k q Real)
  {B : Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
            (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω) (a : q),
                MeasureTheory.MemLp
                  (fun ωs =>
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp
                      a)
                  2 (Pstar n ω)) →
              (MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                        (setOf fun ωs =>
                          Real.instLT.lt (τ n)
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                  (TbetaStar n ω ωs)).ofLp))).toReal)
                  Filter.atTop fun x => 0) →
                Real.instLE.le 0 B →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                      Filter.atTop fun x => B) →
                    (∀ (n : Nat) (ω : Ω),
                        MeasureTheory.Integrable
                          (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          (Pstar n ω)) →
                      MeasureTheory.TendstoInMeasure μ
                        (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                          (fun n ω ωs =>
                            (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                (TbetaStar n ω ωs)).ofLp)
                          τ)
                        Filter.atTop fun x =>
                        HansenEconometrics.smoothFunctionVarianceFunctional R (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/Regression.lean:8108

theorem

HansenEconometrics.chapter10_bootstrap_regression_trimmedVariance_secondMoment_gaussianLimit_of_robustFeasibleHCMomentConditions

Robust-feasible HC specialization of the Theorem 10.19 second-moment/diverging-threshold trimmed covariance route.

The conditional second-moment and norm-fourth premises remain explicit; this wrapper only supplies the heteroskedastic covariance positive-semidefinite premise from RobustFeasibleHCMomentConditions.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] [inst_3 : Fintype q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Pstar : Nat → Ω → MeasureTheory.Measure Ωs} {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real}
  (β : k → Real) (R : Matrix k q Real) {Bsecond Bfourth : Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.MemLp
                    (fun ωs =>
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp)
                    2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : q),
                    MeasureTheory.MemLp
                      (fun ωs =>
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                              (TbetaStar n ω ωs)).ofLp
                          a)
                      2 (Pstar n ω)) →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                            2)
                      Filter.atTop fun x => Bsecond) →
                    Real.instLE.le 0 Bfourth →
                      (MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          Filter.atTop fun x => Bfourth) →
                        (∀ (n : Nat) (ω : Ω),
                            MeasureTheory.Integrable
                              (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                              (Pstar n ω)) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                              (fun n ω ωs =>
                                (ContinuousLinearMap.funLike.coe
                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                              τ)
                            Filter.atTop fun x =>
                            HansenEconometrics.smoothFunctionVarianceFunctional R
                              (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/Regression.lean:8171

theorem

HansenEconometrics.chapter10_indexed_bootstrap_regression_trimmedVariance_secondMoment_gaussianLimit_of_robustFeasibleHCMomentConditions

Indexed robust-feasible HC specialization of the Theorem 10.19 second-moment/diverging-threshold trimmed covariance route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : DecidableEq k] [inst_3 : Fintype q]
  {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} {τ : Nat → Real} (β : k → Real) (R : Matrix k q Real)
  {Bsecond Bfourth : Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.MemLp
                    (fun ωs =>
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp)
                    2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : q),
                    MeasureTheory.MemLp
                      (fun ωs =>
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                              (TbetaStar n ω ωs)).ofLp
                          a)
                      2 (Pstar n ω)) →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                            2)
                      Filter.atTop fun x => Bsecond) →
                    Real.instLE.le 0 Bfourth →
                      (MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          Filter.atTop fun x => Bfourth) →
                        (∀ (n : Nat) (ω : Ω),
                            MeasureTheory.Integrable
                              (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                              (Pstar n ω)) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                              (fun n ω ωs =>
                                (ContinuousLinearMap.funLike.coe
                                    (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                              τ)
                            Filter.atTop fun x =>
                            HansenEconometrics.smoothFunctionVarianceFunctional R
                              (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (6)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/Regression.lean:8236

Theorem 10.19 robust feasible HC finite-replication trimmed covariance wrappers4 endpoints

The trimmed bootstrap covariance for the transformed regression statistic satisfies \hat V_{\theta}^{\mathrm{boot},\tau}\xrightarrow{p^*}V_\theta. Robust feasible HC conditions also discharge positive semidefiniteness for the finite-replication Gaussian-limit L² routes

theorem

HansenEconometrics.chapter10_regression_finiteReplicationTrimmedVariance_normFourth_gaussianLimit_l2_of_robustFeasibleHCMomentConditions

Robust-feasible HC specialization of the finite-replication Theorem 10.19 norm-fourth route.

This fixes Vβ = heteroAsymCov μ X e in the centered finite-replication trimmed covariance estimator and discharges covariance positive semidefiniteness from the Chapter 7 robust feasible HC condition package.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] [inst_3 : Fintype q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Zsim : Nat → Nat → Ω → q → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real} (β : k → Real) (R : Matrix k q Real) {B : Real}
  {Cfinite : q → q → Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
            (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω) (a : q),
                MeasureTheory.MemLp
                  (fun ωs =>
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp
                      a)
                  2 (Pstar n ω)) →
              (MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                        (setOf fun ωs =>
                          Real.instLT.lt (τ n)
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                  (TbetaStar n ω ωs)).ofLp))).toReal)
                  Filter.atTop fun x => 0) →
                Real.instLE.le 0 B →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                      Filter.atTop fun x => B) →
                    (∀ (n : Nat) (ω : Ω),
                        MeasureTheory.Integrable
                          (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          (Pstar n ω)) →
                      (∀ (a c : q) (n : Nat),
                          MeasureTheory.Integrable
                            (fun ω =>
                              instHPow.hPow
                                (Real.norm.norm
                                  (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                    (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                      (fun n ω ωs =>
                                        (ContinuousLinearMap.funLike.coe
                                            (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                            (TbetaStar n ω ωs)).ofLp)
                                      τ n ω)
                                    a c))
                                2)
                            μ) →
                        (∀ (a c : q),
                            Filter.Eventually
                              (fun n =>
                                Real.instLE.le
                                  (MeasureTheory.integral μ fun ω =>
                                    instHPow.hPow
                                      (Real.norm.norm
                                        (instHSub.hSub
                                          (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                          (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                            (fun n ω ωs =>
                                              (ContinuousLinearMap.funLike.coe
                                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                  (TbetaStar n ω ωs)).ofLp)
                                            τ n ω)
                                          a c))
                                      2)
                                  (instHDiv.hDiv (Cfinite a c) n.cast))
                              Filter.atTop) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                            HansenEconometrics.smoothFunctionVarianceFunctional R
                              (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (7)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:13228

theorem

HansenEconometrics.chapter10_indexed_regression_finiteReplicationTrimmedVariance_normFourth_gaussianLimit_l2_of_robustFeasibleHCMomentConditions

Indexed robust-feasible HC specialization of the finite-replication Theorem 10.19 norm-fourth route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : DecidableEq k] [inst_3 : Fintype q]
  {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → q → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} {τ : Nat → Real} (β : k → Real) (R : Matrix k q Real)
  {B : Real} {Cfinite : q → q → Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLE.le 0 (τ n)) →
        (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
            (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
          (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
            (∀ (n : Nat) (ω : Ω) (a : q),
                MeasureTheory.MemLp
                  (fun ωs =>
                    (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp
                      a)
                  2 (Pstar n ω)) →
              (MeasureTheory.TendstoInMeasure μ
                  (fun n ω =>
                    (MeasureTheory.Measure.instFunLike.coe (Pstar n ω)
                        (setOf fun ωs =>
                          Real.instLT.lt (τ n)
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                  (TbetaStar n ω ωs)).ofLp))).toReal)
                  Filter.atTop fun x => 0) →
                Real.instLE.le 0 B →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                      Filter.atTop fun x => B) →
                    (∀ (n : Nat) (ω : Ω),
                        MeasureTheory.Integrable
                          (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          (Pstar n ω)) →
                      (∀ (a c : q) (n : Nat),
                          MeasureTheory.Integrable
                            (fun ω =>
                              instHPow.hPow
                                (Real.norm.norm
                                  (instHSub.hSub (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                    (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                                      (fun n ω ωs =>
                                        (ContinuousLinearMap.funLike.coe
                                            (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                            (TbetaStar n ω ωs)).ofLp)
                                      τ n ω)
                                    a c))
                                2)
                            μ) →
                        (∀ (a c : q),
                            Filter.Eventually
                              (fun n =>
                                Real.instLE.le
                                  (MeasureTheory.integral μ fun ω =>
                                    instHPow.hPow
                                      (Real.norm.norm
                                        (instHSub.hSub
                                          (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                          (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                                            (fun n ω ωs =>
                                              (ContinuousLinearMap.funLike.coe
                                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                  (TbetaStar n ω ωs)).ofLp)
                                            τ n ω)
                                          a c))
                                      2)
                                  (instHDiv.hDiv (Cfinite a c) n.cast))
                              Filter.atTop) →
                          MeasureTheory.TendstoInMeasure μ
                            (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                            HansenEconometrics.smoothFunctionVarianceFunctional R
                              (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (7)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:13305

theorem

HansenEconometrics.chapter10_regression_finiteReplicationTrimmedVariance_secondMoment_gaussianLimit_l2_of_robustFeasibleHCMomentConditions

Robust-feasible HC specialization of the finite-replication Theorem 10.19 second-moment/diverging-threshold route.

The coefficient-level bootstrap weak convergence, conditional second-moment, norm-fourth, and coordinatewise finite-replication L² simulation-error premises remain explicit.

Formal statement

∀ {Ω : Type u_1} {Ωs : Type u_2} {mΩ : MeasurableSpace Ω} {mΩs : MeasurableSpace Ωs} {μ : MeasureTheory.Measure Ω}
  [inst : MeasureTheory.IsProbabilityMeasure μ] {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k]
  [inst_2 : DecidableEq k] [inst_3 : Fintype q] {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real}
  {Zsim : Nat → Nat → Ω → q → Real} {Pstar : Nat → Ω → MeasureTheory.Measure Ωs}
  {TbetaStar : Nat → Ω → Ωs → EuclideanSpace Real k} {τ : Nat → Real} (β : k → Real) (R : Matrix k q Real)
  {Bsecond Bfourth : Real} {Cfinite : q → q → Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistribution μ Pstar TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.MemLp
                    (fun ωs =>
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp)
                    2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : q),
                    MeasureTheory.MemLp
                      (fun ωs =>
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                              (TbetaStar n ω ωs)).ofLp
                          a)
                      2 (Pstar n ω)) →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                            2)
                      Filter.atTop fun x => Bsecond) →
                    Real.instLE.le 0 Bfourth →
                      (MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          Filter.atTop fun x => Bfourth) →
                        (∀ (n : Nat) (ω : Ω),
                            MeasureTheory.Integrable
                              (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                              (Pstar n ω)) →
                          (∀ (a c : q) (n : Nat),
                              MeasureTheory.Integrable
                                (fun ω =>
                                  instHPow.hPow
                                    (Real.norm.norm
                                      (instHSub.hSub
                                        (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                        (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                          (fun n ω ωs =>
                                            (ContinuousLinearMap.funLike.coe
                                                (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                (TbetaStar n ω ωs)).ofLp)
                                          τ n ω)
                                        a c))
                                    2)
                                μ) →
                            (∀ (a c : q),
                                Filter.Eventually
                                  (fun n =>
                                    Real.instLE.le
                                      (MeasureTheory.integral μ fun ω =>
                                        instHPow.hPow
                                          (Real.norm.norm
                                            (instHSub.hSub
                                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                              (HansenEconometrics.trimmedBootstrapCovarianceMat Pstar
                                                (fun n ω ωs =>
                                                  (ContinuousLinearMap.funLike.coe
                                                      (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                      (TbetaStar n ω ωs)).ofLp)
                                                τ n ω)
                                              a c))
                                          2)
                                      (instHDiv.hDiv (Cfinite a c) n.cast))
                                  Filter.atTop) →
                              MeasureTheory.TendstoInMeasure μ
                                (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                                HansenEconometrics.smoothFunctionVarianceFunctional R
                                  (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (7)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistribution
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMat

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:13387

theorem

HansenEconometrics.chapter10_indexed_regression_finiteReplicationTrimmedVariance_secondMoment_gaussianLimit_l2_of_robustFeasibleHCMomentConditions

Indexed robust-feasible HC specialization of the finite-replication Theorem 10.19 second-moment/diverging-threshold route.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : MeasureTheory.IsProbabilityMeasure μ]
  {k : Type u_7} {q : Type u_8} [inst_1 : Fintype k] [inst_2 : DecidableEq k] [inst_3 : Fintype q]
  {X : Nat → Ω → k → Real} {e y : Nat → Ω → Real} {Ωboot : Nat → Type u_9}
  [inst_4 : (n : Nat) → MeasurableSpace (Ωboot n)] {Zsim : Nat → Nat → Ω → q → Real}
  {Pstar : (n : Nat) → Ω → MeasureTheory.Measure (Ωboot n)}
  {TbetaStar : (n : Nat) → Ω → Ωboot n → EuclideanSpace Real k} {τ : Nat → Real} (β : k → Real) (R : Matrix k q Real)
  {Bsecond Bfourth : Real} {Cfinite : q → q → Real},
  HansenEconometrics.RobustFeasibleHCMomentConditions μ X e y β →
    (∀ (n : Nat) (ω : Ω), MeasureTheory.IsProbabilityMeasure (Pstar n ω)) →
      (∀ (n : Nat), Real.instLT.lt 0 (τ n)) →
        Filter.Tendsto (fun n => Real.instInv.inv (instHPow.hPow (τ n) 2)) Filter.atTop (nhds 0) →
          (HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed μ Pstar TbetaStar
              (ProbabilityTheory.multivariateGaussian 0 (HansenEconometrics.heteroAsymCov μ X e)) fun z => z) →
            (∀ (n : Nat) (ω : Ω), Measurable (TbetaStar n ω)) →
              (∀ (n : Nat) (ω : Ω),
                  MeasureTheory.MemLp
                    (fun ωs =>
                      (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                          (TbetaStar n ω ωs)).ofLp)
                    2 (Pstar n ω)) →
                (∀ (n : Nat) (ω : Ω) (a : q),
                    MeasureTheory.MemLp
                      (fun ωs =>
                        (ContinuousLinearMap.funLike.coe (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                              (TbetaStar n ω ωs)).ofLp
                          a)
                      2 (Pstar n ω)) →
                  (MeasureTheory.TendstoInMeasure μ
                      (fun n ω =>
                        MeasureTheory.integral (Pstar n ω) fun ωs =>
                          instHPow.hPow
                            (Pi.normedRing.norm
                              (ContinuousLinearMap.funLike.coe
                                  (HansenEconometrics.matrixContinuousLinearMap R.transpose) (TbetaStar n ω ωs)).ofLp)
                            2)
                      Filter.atTop fun x => Bsecond) →
                    Real.instLE.le 0 Bfourth →
                      (MeasureTheory.TendstoInMeasure μ
                          (fun n ω =>
                            MeasureTheory.integral (Pstar n ω) fun ωs =>
                              instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                          Filter.atTop fun x => Bfourth) →
                        (∀ (n : Nat) (ω : Ω),
                            MeasureTheory.Integrable
                              (fun ωs => instHPow.hPow ((PiLp.instNorm 2 fun x => Real).norm (TbetaStar n ω ωs)) 4)
                              (Pstar n ω)) →
                          (∀ (a c : q) (n : Nat),
                              MeasureTheory.Integrable
                                (fun ω =>
                                  instHPow.hPow
                                    (Real.norm.norm
                                      (instHSub.hSub
                                        (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                        (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                                          (fun n ω ωs =>
                                            (ContinuousLinearMap.funLike.coe
                                                (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                (TbetaStar n ω ωs)).ofLp)
                                          τ n ω)
                                        a c))
                                    2)
                                μ) →
                            (∀ (a c : q),
                                Filter.Eventually
                                  (fun n =>
                                    Real.instLE.le
                                      (MeasureTheory.integral μ fun ω =>
                                        instHPow.hPow
                                          (Real.norm.norm
                                            (instHSub.hSub
                                              (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim n ω)
                                              (HansenEconometrics.trimmedBootstrapCovarianceMatIndexed Pstar
                                                (fun n ω ωs =>
                                                  (ContinuousLinearMap.funLike.coe
                                                      (HansenEconometrics.matrixContinuousLinearMap R.transpose)
                                                      (TbetaStar n ω ωs)).ofLp)
                                                τ n ω)
                                              a c))
                                          2)
                                      (instHDiv.hDiv (Cfinite a c) n.cast))
                                  Filter.atTop) →
                              MeasureTheory.TendstoInMeasure μ
                                (HansenEconometrics.finiteReplicationCovarianceCenteredMat Zsim) Filter.atTop fun x =>
                                HansenEconometrics.smoothFunctionVarianceFunctional R
                                  (HansenEconometrics.heteroAsymCov μ X e)

Direct statement dependencies (7)

HansenEconometrics.RobustFeasibleHCMomentConditions
HansenEconometrics.TendstoInBootstrapWeakDistributionIndexed
HansenEconometrics.finiteReplicationCovarianceCenteredMat
HansenEconometrics.heteroAsymCov
HansenEconometrics.matrixContinuousLinearMap
HansenEconometrics.smoothFunctionVarianceFunctional
HansenEconometrics.trimmedBootstrapCovarianceMatIndexed

HansenEconometrics/Chapter10Bootstrap/FiniteReplication.lean:13470

Theorem 10.206 endpoints

If u_i are independent and uniformly integrable, then for every r\gt 1, n^{-r}\sum_{i=1}^n\lvert u_i\rvert^r\xrightarrow{p}0.

theorem HansenEconometrics.chapter10_marcinkiewicz_wlln_rpow_of_uniformIntegrable

Hansen Theorem 10.20, Marcinkiewicz WLLN.

If uᵢ is uniformly integrable, then for every real r > 1, n^{-r} ∑ |uᵢ|^r ->p 0. Hansen states the theorem with independence as a sufficient condition for the ordinary WLLN step; this formulation is slightly stronger because Mathlib’s probability-theory uniform integrability already provides the Oₚ(1) absolute-mean factor, and Chapter 6’s maximum theorem provides the oₚ(1) scaled-maximum factor.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {u : Nat → Ω → Real} {r : Real},
  Real.instLT.lt 1 r →
    MeasureTheory.UniformIntegrable u 1 μ →
      MeasureTheory.TendstoInMeasure μ (HansenEconometrics.marcinkiewiczWLLNStatisticRpow u r) Filter.atTop fun x => 0

Direct statement dependencies (1)

HansenEconometrics.marcinkiewiczWLLNStatisticRpow

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:572

theorem HansenEconometrics.marcinkiewiczWLLNStatisticRpow_succ_tendsto_zero_of_uniformIntegrable

Shifted real-exponent version of Hansen Theorem 10.20.

Ordinary nonparametric bootstrap support uses Fin (n+1) to avoid the empty sample at n = 0; this is the corresponding shifted form of Hansen’s stated real-exponent Marcinkiewicz WLLN.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {u : Nat → Ω → Real} {r : Real},
  Real.instLT.lt 1 r →
    MeasureTheory.UniformIntegrable u 1 μ →
      MeasureTheory.TendstoInMeasure μ
        (fun n ω => HansenEconometrics.marcinkiewiczWLLNStatisticRpow u r (instHAdd.hAdd n 1) ω) Filter.atTop fun x => 0

Direct statement dependencies (1)

HansenEconometrics.marcinkiewiczWLLNStatisticRpow

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:587

theorem HansenEconometrics.chapter10_marcinkiewicz_wlln_natPower_of_uniformIntegrable

Hansen Theorem 10.20, natural-power uniformly-integrable wrapper.

For natural p ≥ 2, uniform integrability of the real sequence uᵢ implies n^{-p} ∑ |uᵢ|^p ->p 0. The textbook states the same argument for every real r > 1; this wrapper records the integer-power surface needed by the Chapter 10 bootstrap variance and Lindeberg proofs.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {u : Nat → Ω → Real} {p : Nat},
  instLENat.le 2 p →
    MeasureTheory.UniformIntegrable u 1 μ →
      MeasureTheory.TendstoInMeasure μ (HansenEconometrics.marcinkiewiczWLLNStatisticNat u p) Filter.atTop fun x => 0

Direct statement dependencies (1)

HansenEconometrics.marcinkiewiczWLLNStatisticNat

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:418

theorem HansenEconometrics.marcinkiewiczWLLNStatisticNat_succ_tendsto_zero_of_uniformIntegrable

Shifted Fin (n+1) version of Hansen Theorem 10.20.

Ordinary nonparametric bootstrap support uses Fin (n+1) to avoid the empty sample at n = 0; this is the corresponding shifted Marcinkiewicz WLLN.

Formal statement

∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
  {u : Nat → Ω → Real} {p : Nat},
  instLENat.le 2 p →
    MeasureTheory.UniformIntegrable u 1 μ →
      MeasureTheory.TendstoInMeasure μ
        (fun n ω => HansenEconometrics.marcinkiewiczWLLNStatisticNat u p (instHAdd.hAdd n 1) ω) Filter.atTop fun x => 0

Direct statement dependencies (1)

HansenEconometrics.marcinkiewiczWLLNStatisticNat

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:432

theorem HansenEconometrics.marcinkiewiczWLLNStatisticRpow_le_max_mul_sampleAbsMean

Deterministic inequality in Hansen’s proof of Theorem 10.20 for real exponents r > 1.

This is the textbook display n^{-r} ∑ |uᵢ|^r ≤ (n^{-1} max |uᵢ|)^{r-1} (n^{-1} ∑ |uᵢ|).

Formal statement

∀ {Ω : Type u_1} {u : Nat → Ω → Real} {r : Real} {n : Nat} {ω : Ω},
  Real.instLT.lt 1 r →
    Real.instLE.le (HansenEconometrics.marcinkiewiczWLLNStatisticRpow u r n ω)
      (instHMul.hMul (instHPow.hPow (HansenEconometrics.scaledMaxNNNorm u n ω) (instHSub.hSub r 1))
        (HansenEconometrics.sampleAbsMean u n ω))

Direct statement dependencies (3)

HansenEconometrics.marcinkiewiczWLLNStatisticRpow
HansenEconometrics.sampleAbsMean
HansenEconometrics.scaledMaxNNNorm

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:256

theorem HansenEconometrics.marcinkiewiczWLLNStatisticNat_le_max_mul_sampleAbsMean

Deterministic inequality in Hansen’s proof of Theorem 10.20.

For natural powers p ≥ 2, n^{-p} ∑ |uᵢ|^p is bounded by (n^{-1} max |uᵢ|)^{p-1} (n^{-1} ∑ |uᵢ|).

Formal statement

∀ {Ω : Type u_1} {u : Nat → Ω → Real} {p n : Nat} {ω : Ω},
  instLENat.le 2 p →
    Real.instLE.le (HansenEconometrics.marcinkiewiczWLLNStatisticNat u p n ω)
      (instHMul.hMul (instHPow.hPow (HansenEconometrics.scaledMaxNNNorm u n ω) (instHSub.hSub p 1))
        (HansenEconometrics.sampleAbsMean u n ω))

Direct statement dependencies (3)

HansenEconometrics.marcinkiewiczWLLNStatisticNat
HansenEconometrics.sampleAbsMean
HansenEconometrics.scaledMaxNNNorm

HansenEconometrics/Chapter10Bootstrap/WLLN.lean:197