High Dimensional Bootstrap and Asymptotic Expansion for the $k$-th Largest Coordinate
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917386323492864 |
|---|---|
| author | Feng, Long |
| author_facet | Feng, Long |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04785 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High Dimensional Bootstrap and Asymptotic Expansion for the $k$-th Largest Coordinate Feng, Long Statistics Theory Methodology 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. |
| title | High Dimensional Bootstrap and Asymptotic Expansion for the $k$-th Largest Coordinate |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2604.04785 |