Bootstrapping $\ell_p$-Statistics in High Dimensions

TL;DR

This paper introduces a bootstrap method for estimating the distribution of high-dimensional $ ext{ell}_p$-statistics, enabling accurate inference in settings where the dimension exceeds the sample size, with proven theoretical guarantees and practical applications.

Contribution

It develops a non-asymptotic Gaussian approximation for $ ext{ell}_p$-statistics and proposes a consistent bootstrap procedure for high-dimensional inference.

Findings

Bootstrap procedure is consistent under mild covariance conditions.

The proposed test maintains asymptotic correctness and high power.

Numerical simulations validate the theoretical results.

Abstract

This paper considers a new bootstrap procedure to estimate the distribution of high-dimensional $\ell_p$-statistics, i.e. the $\ell_p$-norms of the sum of $n$ independent $d$-dimensional random vectors with $d \gg n$ and $p \in [1, \infty]$. We provide a non-asymptotic characterization of the sampling distribution of $\ell_p$-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test for simultaneous inference on high-dimensional mean vectors. We establish its asymptotic correctness and consistency under high-dimensional alternatives, and discuss the power of the test as well as the size of associated confidence sets. We illustrate the bootstrap and testing procedure numerically on simulated data.