High Dimensional Bootstrap and Asymptotic Expansion for the $k$-th Largest Coordinate

2026-04-06

AI Summary

This study addresses the lack of effective second-order self-normalized inference methods for the $k$-th largest coordinate of sums of high-dimensional independent random vectors, a challenge arising because classical extreme value theory does not readily extend to general order statistics. The authors innovatively reframe the problem as one of estimating rare orthant probabilities and, by integrating factorial moments with a weighted inclusion–exclusion principle, extend existing high-dimensional second-order Gaussian and bootstrap approximation theories—previously limited to maxima—to the $k$-th order statistic. They propose a third-moment-matching wild bootstrap and a pre-pivoted double wild bootstrap, which, under moment conditions, variance assumptions, and weak dependence, achieve a second-order Edgeworth expansion with coverage error of order $n^{-1}$, substantially improving inferential accuracy.

Abstract

We study bootstrap inference for the $k$th largest coordinate of a normalized sum of independent high-dimensional random vectors. Existing second-order theory for maxima does not directly extend to order statistics, because the event $\{T_{n,[k]}\le t\}$ is not a rectangle and its local structure is governed by exceedance counts rather than by a single boundary. We develop an approach based on factorial moments and weighted inclusion--exclusion that reduces the problem to a collection of rare-orthant probabilities and allows high-dimensional Edgeworth and Cornish--Fisher expansions to be transferred to the order-statistic setting. Under moment, variance, and weak-dependence conditions, we derive a second-order coverage expansion for wild-bootstrap critical values of the $k$th order statistic. In particular, a third-moment matching wild bootstrap achieves coverage error of order $n^{-1}$ up to logarithmic factors, and the same second-order accuracy is obtained for a prepivoted double wild bootstrap. We also show that the maximal-correlation condition can be replaced by a stationary Gaussian exponential-mixing assumption at the price of an explicit dependence remainder $r_d$, and this remainder can itself be of order $n^{-1}$ when the dimension is sufficiently large relative to the sample size. These results extend recent second-order Gaussian and bootstrap approximation theory from maxima to the $k$th order statistic in high dimension.

Problem

• high-dimensional
• bootstrap
• order statistic
• asymptotic expansion
• coverage error

Innovation

• high-dimensional bootstrap
• order statistics
• Edgeworth expansion
• wild bootstrap
• coverage error