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

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. 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.