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High-dimensional bootstrap and asymptotic expansion

Yuta Koike

Abstract

The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that bootstrap approximation for the maximum of a sum of independent random vectors is justified even when the dimension is much larger than the sample size. In this context, numerical experiments suggest that third-moment matching bootstrap approximations would outperform normal approximation even without studentization, but the existing theoretical results cannot explain this phenomenon. In this paper, we develop an asymptotic expansion formula for the bootstrap coverage probability and show that it can give an explanation for the above phenomenon. In particular, we find the following interesting blessing of dimensionality phenomenon: The third-moment matching wild bootstrap is second-order accurate in high dimensions even without studentization if the covariance matrix has identical diagonal entries and bounded eigenvalues. We also show that a double wild bootstrap method is second-order accurate regardless of the covariance structure. The validity of these results is established under the assumption that the underlying distributions admit Stein kernels.

High-dimensional bootstrap and asymptotic expansion

Abstract

The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that bootstrap approximation for the maximum of a sum of independent random vectors is justified even when the dimension is much larger than the sample size. In this context, numerical experiments suggest that third-moment matching bootstrap approximations would outperform normal approximation even without studentization, but the existing theoretical results cannot explain this phenomenon. In this paper, we develop an asymptotic expansion formula for the bootstrap coverage probability and show that it can give an explanation for the above phenomenon. In particular, we find the following interesting blessing of dimensionality phenomenon: The third-moment matching wild bootstrap is second-order accurate in high dimensions even without studentization if the covariance matrix has identical diagonal entries and bounded eigenvalues. We also show that a double wild bootstrap method is second-order accurate regardless of the covariance structure. The validity of these results is established under the assumption that the underlying distributions admit Stein kernels.
Paper Structure (32 sections, 41 theorems, 192 equations, 1 figure, 2 tables)

This paper contains 32 sections, 41 theorems, 192 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Valid Edgeworth expansion in high dimensions
  5. Asymptotic expansion of coverage probability
  6. Double wild bootstrap
  7. Simulation study
  8. Proofs for Section \ref{['sec:2nd-main']}
  9. General error bounds via approximate Stein kernel
  10. Proof of Theorem \ref{['coro1']}
  11. Proof of Theorem \ref{['prop:boot']}
  12. Proof of Proposition \ref{['prop:copula']}
  13. Cheeger constant of the gamma distribution
  14. Proof of Corollary \ref{['coro:lb-ga']}
  15. Proofs for Sections \ref{['sec:coverage']} and \ref{['sec:db']}
  16. ...and 17 more sections

Key Result

Theorem 2.1

Suppose that $X_i$ has a Stein kernel $\tau^X_i$ for every $i=1,\dots,n$. Suppose also that there exists a constant $b>0$ such that and Further, assume $\log^3d\leq n$. Then,

Figures (1)

  • Figure 1: P--P plots for the rejection rate $P(T_n\geq \hat{c}_{1-\alpha})$ against the nominal significance level $\alpha$ when $n=200$ and $d=400$. The rejection rate is evaluated based on 20,000 Monte Carlo iterations. The critical value $\hat{c}_{1-\alpha}$ is computed by the Gaussian wild bootstrap for the left panel and the wild bootstrap with $w_1$ generated from the standardized beta distribution with parameters $\alpha,\beta$ given by \ref{['beta-param']} with $\nu=0.1$ for the right panel, respectively. The number of bootstrap replications is 499. $X_1,\dots,X_n$ are generated from a Gaussian copula model with gamma marginals as in the simulation study of \ref{['sec:simulate']}. The parameter matrix is $R=(0.2^{|j-k|})_{1\leq j,k\leq d}$.

Theorems & Definitions (59)

  • Definition 2.1: Stein kernel
  • Remark 2.1: Alternative definition
  • Theorem 2.1: Edgeworth expansion for $S_n$
  • Remark 2.2
  • Remark 2.3: Proof strategy
  • Example 2.1: Log-concave distribution
  • Lemma 2.1: Stein kernel of log-concave distribution
  • Example 2.2: Gaussian copula model
  • Proposition 2.1: Stein kernel of Gaussian copula model
  • Example 2.3: Affine transformation
  • ...and 49 more