Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotics of higher criticism via Gaussian approximation

Jingkun Qiu

TL;DR

This work develops a unified Gaussian-approximation framework for high-dimensional testing statistics under dependent, non-Gaussian setting, focusing on the higher criticism (HC) and multi-level thresholding (MT) statistics built from dependent $t$-statistics with finite $(2+\delta)$-th moments. The authors represent HC and MT as suprema of empirical processes indexed by normalized indicator or thresholding function classes and prove sharp Gaussian coupling results via a big-block-small-block scheme and Berbee coupling, establishing asymptotic distributions that match the independent case under suitable mixing and moment conditions. Key contributions include (i) a general GA theorem for empirical processes with dependent observations, (ii) explicit HC and MT limit theorems allowing dependent $t$-statistics, and (iii) extended results for multilevel thresholding with a related Gaussian process $\mathbb{B}_p$ and a Husimi-type limit. The results enable robust detection boundaries for sparse signals in dependent high-dimensional data and provide a principled, moment-condition-based path beyond Gaussian i.i.d. assumptions. The framework broadens applicability to dependent Gaussian and non-Gaussian statistics and highlights the practical relevance of the HC and MT statistics in complex, large-scale inference tasks.

Abstract

Higher criticism is a large-scale testing procedure that can attain the optimal detection boundary for sparse and faint signals. However, there has been a lack of knowledge in most existing works about its asymptotic distribution for more realistic settings other than the independent Gaussian assumption while maintaining the power performance as much as possible. In this paper, we develop a unified framework to analyze the asymptotic distributions of the higher criticism statistic and the more general multi-level thresholding statistic when the individual test statistics are dependent $t$-statistics under a finite ($2+δ$)-th moment condition, $0<δ\leq1$. The key idea is to approximate the global test statistic by the supremum of an empirical process indexed by a normalized class of indicator or thresholding functions, respectively. A new Gaussian approximation theorem for suprema of empirical processes with dependent observations is established to derive the explicit asymptotic distributions.

Asymptotics of higher criticism via Gaussian approximation

TL;DR

This work develops a unified Gaussian-approximation framework for high-dimensional testing statistics under dependent, non-Gaussian setting, focusing on the higher criticism (HC) and multi-level thresholding (MT) statistics built from dependent -statistics with finite -th moments. The authors represent HC and MT as suprema of empirical processes indexed by normalized indicator or thresholding function classes and prove sharp Gaussian coupling results via a big-block-small-block scheme and Berbee coupling, establishing asymptotic distributions that match the independent case under suitable mixing and moment conditions. Key contributions include (i) a general GA theorem for empirical processes with dependent observations, (ii) explicit HC and MT limit theorems allowing dependent -statistics, and (iii) extended results for multilevel thresholding with a related Gaussian process and a Husimi-type limit. The results enable robust detection boundaries for sparse signals in dependent high-dimensional data and provide a principled, moment-condition-based path beyond Gaussian i.i.d. assumptions. The framework broadens applicability to dependent Gaussian and non-Gaussian statistics and highlights the practical relevance of the HC and MT statistics in complex, large-scale inference tasks.

Abstract

Higher criticism is a large-scale testing procedure that can attain the optimal detection boundary for sparse and faint signals. However, there has been a lack of knowledge in most existing works about its asymptotic distribution for more realistic settings other than the independent Gaussian assumption while maintaining the power performance as much as possible. In this paper, we develop a unified framework to analyze the asymptotic distributions of the higher criticism statistic and the more general multi-level thresholding statistic when the individual test statistics are dependent -statistics under a finite ()-th moment condition, . The key idea is to approximate the global test statistic by the supremum of an empirical process indexed by a normalized class of indicator or thresholding functions, respectively. A new Gaussian approximation theorem for suprema of empirical processes with dependent observations is established to derive the explicit asymptotic distributions.

Paper Structure

This paper contains 21 sections, 20 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gaussian approximation theorems
  3. Proof sketch of \ref{['thm:GA-mixing']}
  4. Higher criticism asymptotics
  5. Higher criticism $t$-statistic
  6. Extension for multi-level thresholding $t$-statistic
  7. Proof of Theorems \ref{['thm:GA-HC']} and \ref{['thm:HC-limit']}
  8. Proof of \ref{['thm:GA-HC-t']}
  9. Proof of \ref{['thm:GA-MT-t']}
  10. Proof of \ref{['thm:CCK14']}
  11. Proof of \ref{['thm:GA-mixing']}
  12. Proof of \ref{['thm:GA-mixing']}
  13. Proof of \ref{['lem:maximal-inequality-naive']}
  14. Proof of \ref{['lem:GA-blocking']}
  15. Proof of \ref{['lem:comparison-process']}
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that there are constants $v,b\in(0,\infty)$, $q\in[4,\infty]$, and $\delta_{3},\delta_{4}\in(0,1)$ such that $\mathop{\mathrm{vc}}\nolimits(\mathcal{F})\leq v$, $\|F\|_{P,q}\leq b$, and $\sup\{\|f\|_{P,k}:f\in\mathcal{F}\}\leq\delta_{k}\|F\|_{P,k}$ for $k=3,4$. Then for every $\gamma\in(0,1) where $K=c_{q}v\log(An/\delta_{3}\delta_{4})$, $c_{q}$ and $C_{q}$ are positive constants dependin

Figures (1)

  • Figure 1: Graphical illustration of $\mathop{\mathrm{vc}}\nolimits(\mathcal{F})=3$ for the particular choice of $\mathcal{F}$ in \ref{['lem:VC']}(ii), given the three fixed points $(11/4,9/4)$, $(15/4,13/4)$, and $(25/4,21/4)$. Each panel represents the selection of the subgraph of $f(\cdot,s)\in\mathcal{F}$ at a particular value of $s$, where the red squares indicate the points selected by the subgraph while the blue triangles are not selected.

Theorems & Definitions (23)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Lemma 1
  • Theorem 3
  • Remark 3
  • Theorem 4
  • Theorem 5
  • Lemma 2
  • ...and 13 more