Some nonasymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests

TL;DR

This paper develops nonasymptotic methods for constructing confidence regions and performing multiple tests in high-dimensional settings, focusing on bootstrap techniques with concentration inequalities and resampled quantiles.

Contribution

It introduces novel nonasymptotic bounds for bootstrap confidence regions and multiple testing procedures in high dimensions, using concentration principles and Rademacher weights.

Findings

Nonasymptotic control of confidence levels in high dimensions

Effective bootstrap methods for large-dimensional mean vectors

Analysis of Monte Carlo approximation accuracy

Abstract

We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities.