Gaussian Multiplier Bootstrap Procedure for the $k$th Largest Coordinate of High-Dimensional Statistics

TL;DR

This paper develops Gaussian multiplier bootstrap methods for approximating the distribution of the $k$th largest statistic in high-dimensional data, extending previous work from the maximum to general order statistics.

Contribution

It provides error bounds for Gaussian approximations of the $k$th largest statistics, applicable when the dimension exceeds the sample size.

Findings

Error bounds for Gaussian multiplier bootstrap approximations.

Method effective for high-dimensional data where p > n.

Validated through simulations and real data analysis.

Abstract

We consider the problem of Gaussian multiplier bootstrap procedures for the $k$th largest statistics and functions of the top $k$ order statistics, which are commonly encountered in high-dimensional statistical inference. Such a problem has been studied previously for $k=1$ (i.e., maxima). However, in many applications, a general $k$ ($k\geq 1$) is of great interest. We provide the upper bounds for the errors between Gaussian approximations and Gaussian multiplier approximations. The dimension $p$ is allowed to be larger than the sample size $n$. The effectiveness of the proposed methods is demonstrated via the computer numerical results and a real-world data analysis.