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Concentration of the bootstrap empirical process, with applications to statistical inference

Guillaume Maillard, Adrien Saumard

Abstract

Considering a general framework of bootstrap with exchangeable weights, we show some concentration inequalities for the supremum of the bootstrap empirical process. On the one hand, we discuss the concentration of the bootstrap empirical process around its conditional expectation with respect to the original data, and on the other hand, the concentration of the latter quantity around its mean. For the concentration conditional on data, we build on Chatterjee's exchangeable pairs approach to concentration. To attain optimal concentration rates, we develop some refined arguments for the convergence of transposition walks on the symmetric group. The conditional expectation of the bootstrap empirical process is proved to be self-bounding, thus extending a well-known property for conditional Rademacher averages. To illustrate the interest of these concentration inequalities, we provide some new results pertaining to confidence regions for the estimation of a mean vector, as well as non-asymptotic bounds for the two-sample permutation test.

Concentration of the bootstrap empirical process, with applications to statistical inference

Abstract

Considering a general framework of bootstrap with exchangeable weights, we show some concentration inequalities for the supremum of the bootstrap empirical process. On the one hand, we discuss the concentration of the bootstrap empirical process around its conditional expectation with respect to the original data, and on the other hand, the concentration of the latter quantity around its mean. For the concentration conditional on data, we build on Chatterjee's exchangeable pairs approach to concentration. To attain optimal concentration rates, we develop some refined arguments for the convergence of transposition walks on the symmetric group. The conditional expectation of the bootstrap empirical process is proved to be self-bounding, thus extending a well-known property for conditional Rademacher averages. To illustrate the interest of these concentration inequalities, we provide some new results pertaining to confidence regions for the estimation of a mean vector, as well as non-asymptotic bounds for the two-sample permutation test.

Paper Structure

This paper contains 39 sections, 33 theorems, 440 equations.

Table of Contents

  1. Introduction
  2. The bootstrap empirical process
  3. Self-bounding property of the expectation conditioned on data
  4. Concentration conditional on data
  5. Proof steps for Theorem \ref{['thm_intro_concen_exchange']}
  6. Exponential moments and a decoupling inequality
  7. A covariance inequality through exchangeable pairs
  8. Implementing the exchangeable pair argument
  9. A general deviation bound
  10. Application I: Confidence regions for the mean in high-dimension
  11. Presentation
  12. New confidence bounds
  13. Application II: Permutation two-sample testing
  14. A general result
  15. Some examples
  16. ...and 24 more sections

Key Result

Theorem 1

If $X = (X_{i})_{1 \leq i \leq n}$ is a collection of $n$ independent random variables valued in $E$ and such that $\overline{g}(X)$ is measurable, then for any $x > 0$, with probability at least $1 - e^{-x}$. Moreover, with probability at least $1 - e^{-x}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (59)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Lemma 4
  • Definition 5
  • Theorem 6
  • Definition 7
  • Lemma 8
  • Theorem 9
  • Lemma 10
  • ...and 49 more