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

Alexis Derumigny, Lucas Girard, Yannick Guyonvarch

arXiv 14 Jan 2021 · Mathematics — Probability · publishedSankhya A (2023)

arXiv:2101.05780 · PDF · DOI · OpenAlex · Extracted main text

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.

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