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

Understand

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.