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

TL;DR

This paper establishes concentration inequalities for the bootstrap empirical process under exchangeable weights, with applications to statistical inference such as confidence regions and permutation tests.

Contribution

It introduces new concentration inequalities for the bootstrap empirical process, extending existing methods with refined arguments and applications to inference procedures.

Findings

Concentration around conditional expectation using exchangeable pairs.

Optimal concentration rates achieved via refined transposition walk analysis.

New bounds for confidence regions and permutation tests.

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.