Edgeworth approximations for distributions of symmetric statistics

TL;DR

This paper develops Edgeworth approximations for the distribution of symmetric, asymptotically linear statistics based on independent observations, providing a refined asymptotic expansion with a small remainder.

Contribution

It introduces a new Edgeworth expansion for symmetric statistics using Hoeffding's decomposition, with proven validity under specific moment and dimensionality conditions.

Findings

Edgeworth expansion with remainder o(N^{-1}) established

Validity proven under Cramér's condition and moment assumptions

Applicable to a broad class of symmetric statistics

Abstract

We study the distribution of a general class of asymptoticallylinear statistics which are symmetric functions of $N$ independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order $o(N^{-1})$. The Edgeworth expansion is based on Hoeffding's decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cram\'er's condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.