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

TL;DR

This paper develops advanced bootstrap methods for the $k$th largest coordinate in high-dimensional data, extending second-order Gaussian approximation theory from maxima to order statistics.

Contribution

It introduces a factorial moments and inclusion-exclusion approach enabling second-order bootstrap inference for the $k$th order statistic in high dimensions.

Findings

Third-moment matching wild bootstrap achieves $n^{-1}$ coverage error.

Second-order accuracy obtained for double wild bootstrap.

Results extend Gaussian and bootstrap approximation theory from maxima to order statistics.

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.