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

Yixi Ding, Qizhai Li, Yuke Shi, Liuquan Sun, Luobin Zhang

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.

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

Abstract

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

Paper Structure

This paper contains 25 sections, 25 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Smooth function
  3. Theoretical preparations
  4. Distributional approximation
  5. Anti-concentration inequality
  6. Gaussian approximation via Stein kernels
  7. Kolmogorov-Smirnov distances for $T_{(k)}$ and $T_{[k]}$
  8. Kolmogorov-Smirnov distances for $T_\psi$
  9. Combine several sets of top order statistics
  10. Numerical Studies
  11. Simulation studies
  12. Application to the urban traffic speed data
  13. A Concluding Remark
  14. Technique details
  15. Proof of Lemma \ref{['Derivatives of m']}
  16. ...and 10 more sections

Key Result

Lemma 2.1

For $j_1,j_2,j_3,j_4,j_5=1,\ldots,p$, there exist functions $U_{j_1j_2}$, $U_{j_1j_2j_3}$, $U_{j_1j_2j_3j_4}$, and $U_{j_1j_2j_3j_4j_5}$ such that (i) For $\boldsymbol{w}\in\mathbb R^p$, (ii) For $\boldsymbol{w}\in\mathbb R^p$, (iii) For $\boldsymbol{z}\in\mathbb R^p$ and $\boldsymbol{w}\in\mathbb R^p$ satisfying $\beta\|\boldsymbol{w}\|_{\infty}\leq 1$,

Figures (3)

  • Figure 1: The theoretical tail probabilities against the sample tail probabilities based on the Gaussian multiplier bootstrap procedure for $T_{(k)}$ when $k= 1,\ldots,6$ with $\rho=0.2$ or $0.8$.
  • Figure 2: The theoretical tail probabilities against the sample tail probabilities based on the Gaussian multiplier bootstrap procedure for $T_{\psi_{k}}$ when $k= 2,\ldots,5$ with $\rho=0.2$ or $0.8$.
  • Figure 3: The theoretical tail probabilities against the sample tail probabilities based on the Gaussian multiplier bootstrap procedure for $T_F$ with $\rho=0.2$ or $0.8$.

Theorems & Definitions (29)

  • Lemma 2.1
  • Proposition 3.1: Distributional approximation
  • Definition 1: Lévy concentration function
  • Proposition 3.2: Anti-concentration
  • Corollary 3.1
  • Definition 2
  • Proposition 3.3: Gaussian approximation via Stein kernels
  • Corollary 3.2: Multiplier bootstrap to Gaussian comparison
  • Remark 1
  • Theorem 4.1: Gaussian approximation for $T_{(k)}$
  • ...and 19 more