Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion

TL;DR

This paper derives explicit, non-asymptotic bounds on the distance between the distribution of a standardized sum of independent variables and its first-order Edgeworth expansion, valid for any sample size and improved under certain conditions.

Contribution

It provides new explicit bounds with different rates depending on moment and tail conditions, enhancing understanding of distribution approximation accuracy.

Findings

Bounds are valid for any sample size with $n^{-1/2}$ rate.

Bounds are sharper with unskewed variables.

New Berry-Esseen-type bounds are derived.

Abstract

In this article, we obtain explicit bounds on the uniform distance between the cumulative distribution function of a standardized sum $S_n$ of $n$ independent centered random variables with moments of order four and its first-order Edgeworth expansion. Those bounds are valid for any sample size with $n^{-1/2}$ rate under moment conditions only and $n^{-1}$ rate under additional regularity constraints on the tail behavior of the characteristic function of $S_n$. In both cases, the bounds are further sharpened if the variables involved in $S_n$ are unskewed. We also derive new Berry-Esseen-type bounds from our results and discuss their links with existing ones. We finally apply our results to illustrate the lack of finite-sample validity of one-sided tests based on the normal approximation of the mean.