Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings

TL;DR

This paper develops non-asymptotic strong approximation bounds for the supremum of empirical processes indexed by complex, VC-type classes, using Gaussian couplings based on Slepian-Stein methods, with applications in nonparametric statistics.

Contribution

It introduces a novel coupling approach for empirical process suprema that handles increasing complexity and non-centrality, extending Gaussian approximation techniques.

Findings

Provides non-asymptotic bounds for empirical process suprema

Enables analysis of nonparametric test power with bootstrap methods

Applicable to classes with growing complexity as sample size increases

Abstract

We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian-Stein methods and Gaussian comparison inequalities. The increasing complexity of classes of functions and non-centrality of the processes make the results useful for applications in modern nonparametric statistics (Gin\'{e} and Nickl, 2015), in particular allowing us to study the power properties of nonparametric tests using Gaussian and bootstrap approximations.