Normal approximation for nonlinear statistics using a concentration inequality approach

TL;DR

This paper develops optimal Berry-Esseen bounds for a wide class of nonlinear sampling statistics using a concentration inequality approach, with applications to various statistical measures.

Contribution

It introduces a new method to derive sharp Berry-Esseen bounds for nonlinear statistics, improving upon existing results.

Findings

Established best possible bounds for many known statistics

Extended results to U-statistics, L-statistics, and nonlinear functions

Demonstrated applicability to random sums and multisample statistics

Abstract

Let $T$ be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry--Esseen type bounds for $T$ are obtained. The bounds are the best possible for many known statistics. Applications to U-statistics, multisample U-statistics, L-statistics, random sums and functions of nonlinear statistics are discussed.