Stein’s method for nonlinear statistics: A brief survey and recent progress

Abstract

Stein’s method is a powerful tool for proving central limit theorems along with explicit error bounds in probability theory, where uniform and non-uniform Berry–Esseen bounds spark general interest. Nonlinear statistics, typified by Hoeffding’s class of U-statistics, L-statistics, random sums and functions of nonlinear statistics, are building blocks in various statistical inference problems. However, because the standardized statistics often involve unknown nuisance parameters, the Studentized analogues are most commonly used in practice. This paper begins with a brief review of some standard techniques in Stein’s method, and their applications in deriving Berry–Esseen bounds and Cramér moderate deviations for nonlinear statistics, and then using the concentration inequality approach, establishes Berry–Esseen bounds for Studentized nonlinear statistics in a general framework. As direct applications, sharp Berry–Esseen bounds for Studentized U- and L- statistics are obtained.