A Bootstrap Hypothesis Test for High-Dimensional Mean Vectors

TL;DR

This paper introduces a new bootstrap hypothesis testing method for high-dimensional mean vectors that does not require strong independence or moment assumptions, with improved power and bias correction techniques.

Contribution

It proposes a novel class of bootstrap tests based on ll_p-statistics that relaxes traditional assumptions and includes bias correction procedures for better finite-sample performance.

Findings

The tests are asymptotically valid and unbiased.

The modified bootstrap test has enhanced power.

Bias correction procedures improve finite-sample accuracy.

Abstract

This paper is concerned with testing global null hypotheses about population mean vectors of high-dimensional data. Current tests require either strong mixing (independence) conditions on the individual components of the high-dimensional data or high-order moment conditions. In this paper, we propose a novel class of bootstrap hypothesis tests based on $\ell_p$-statistics with $p \in [1, \infty]$ which requires neither of these assumptions. We study asymptotic size, unbiasedness, consistency, and Bahadur slope of these tests. Capitalizing on these theoretical insights, we develop a modified bootstrap test with improved power properties and a self-normalized bootstrap test for elliptically distributed data. We then propose two novel bias correction procedures to improve the accuracy of the bootstrap test in finite samples, which leverage measure concentration and hypercontractivity properties of $\ell_p$-norms in high dimensions. Numerical experiments support our theoretical results in finite samples.