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

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  • Alexis Derumigny
  • Lucas Girard
  • Yannick Guyonvarch

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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  • Alexis Derumigny & Lucas Girard & Yannick Guyonvarch, 2021. "Explicit non-asymptotic bounds for the distance to the first-order Edgeworth expansion," Papers 2101.05780, arXiv.org, revised Sep 2022.
    • Handle: RePEc:arx:papers:2101.05780
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    References listed on IDEAS

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    1. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, January.
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