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

TLDR

The study concerns high‑dimensional random vectors that are either Gaussian or symmetric and bounded, with dimensionality potentially exceeding the sample size, and aims for nonasymptotic confidence‑level control inspired by learning‑theory results. The paper investigates generalized bootstrap confidence regions for the mean of a random vector with unknown coordinate dependence. Two methods are examined: a concentration‑principle approach applicable to many resampling weights, and a resampled‑quantile method using Rademacher weights. The concentration‑principle approach yields several noteworthy intermediate results, and the paper also evaluates the accuracy of Monte‑Carlo approximations for the resampled quantities.

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.