Self-normalization:Taming a wild population in a heavy-tailed world  

作　　者：SHAO Qi-man ZHOU Wen-xin 

机构地区：[1]Department of Statistics, The Chinese University of Hong Kong [2]Department of Mathematics, University of California

出　　处：《Applied Mathematics(A Journal of Chinese Universities)》2017年第3期253-269,共17页高校应用数学学报（英文版）（B辑）

基　　金：Supported by Hong Kong RGC GRF 14302515

摘　　要：The past two decades have witnessed the active development of a rich probability theory of Studentized statistics or self-normalized processes, typified by Student’s t-statistic as introduced by W. S. Gosset more than a century ago, and their applications to statistical problems in high dimensions, including feature selection and ranking, large-scale multiple testing and sparse, high dimensional signal detection. Many of these applications rely on the robustness property of Studentization/self-normalization against heavy-tailed sampling distributions. This paper gives an overview of the salient progress of self-normalized limit theory, from Student’s t-statistic to more general Studentized nonlinear statistics. Prototypical examples include Studentized one- and two-sample U-statistics. Furthermore, we go beyond independence and glimpse some very recent advances in self-normalized moderate deviations under dependence.The past two decades have witnessed the active development of a rich probability theory of Studentized statistics or self-normalized processes, typified by Student’s t-statistic as introduced by W. S. Gosset more than a century ago, and their applications to statistical problems in high dimensions, including feature selection and ranking, large-scale multiple testing and sparse, high dimensional signal detection. Many of these applications rely on the robustness property of Studentization/self-normalization against heavy-tailed sampling distributions. This paper gives an overview of the salient progress of self-normalized limit theory, from Student’s t-statistic to more general Studentized nonlinear statistics. Prototypical examples include Studentized one- and two-sample U-statistics. Furthermore, we go beyond independence and glimpse some very recent advances in self-normalized moderate deviations under dependence.

关 键 词：Berry-Esseen inequality Hotelling’s T ²-statistic large deviation moderate deviation SELF-NORMALIZATION Student’s t-statistic U-STATISTIC 

分 类 号：O21[理学—概率论与数理统计]