A Uniform-in-$P$ Edgeworth Expansion under Weak Cram\'{e}r Conditions

TL;DR

This paper establishes a finite sample error bound for the Edgeworth expansion of sums of independent random vectors under a uniform Cramér condition, enabling uniform higher-order approximations for resampling distributions.

Contribution

It introduces a finite sample bound for the Edgeworth expansion error under a uniform Cramér condition, extending the applicability to a broader class of distributions.

Findings

Finite sample error bound for Edgeworth expansion under weak Cramér conditions

Uniform higher order expansion for resampling-based distributions

Applicable to potentially discrete, nonlattice random vectors

Abstract

This paper provides a finite sample bound for the error term in the Edgeworth expansion for a sum of independent, potentially discrete, nonlattice random vectors, using a uniform-in-$P$ version of the weaker Cram\'{e}r condition in Angst and Poly (2017). This finite sample bound can be used to derive an Edgeworth expansion that is uniform over the distributions of the random vectors. Using this result, we derive a uniform-in-$P$ higher order expansion of resampling-based distributions.