Explicit Non-Asymptotic Bounds for the Distance to the First-Order Edgeworth Expansion

Limites non asymptotiques explicites pour la distance à l'expansion de Edgeworth du premier ordre

Alexis Derumigny, Lucas Girard and Yannick Guyonvarch
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Alexis Derumigny: TU Delft - Delft University of Technology
Lucas Girard: CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Yannick Guyonvarch: UMR PSAE - Paris-Saclay Applied Economics - AgroParisTech - Université Paris-Saclay - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement

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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 Sn. In both cases, the bounds are further sharpened if the variables involved in Sn are unskewed. We also derive new Berry-Esseen-type bounds from our results and discuss their links with existing ones. Following these theoretical results, we discuss the practical use of our bounds, which depend on possibly unknown moments of the distribution of S-n. Finally, we apply our bounds to investigate several aspects of the non-asymptotic behavior of one-sided tests: informativeness, sufficient sample size in experimental design, distortions in terms of levels and p-values.

Keywords: Berry-Esseen bound; Edgeworth expansion; Normal approximation; Central limit theorem; Non-asymptotic tests (search for similar items in EconPapers)
Date: 2024
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Published in Sankhya A, 2024, 86, pp.261-336. ⟨10.1007/s13171-023-00320-y⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-04344911

DOI: 10.1007/s13171-023-00320-y

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