A general partial Cram\'{e}r's condition for Edgeworth expansion of a function of sample means with applications

TL;DR

This paper introduces a general partial Cramér's condition (GPCC) to establish the validity of Edgeworth expansions for functions of sample means, including Pearson's correlation, with demonstrated higher accuracy through simulations.

Contribution

It proposes a new GPCC framework and applies it to prove Edgeworth expansion validity for a broad class of statistics, including Pearson's correlation.

Findings

Validated Edgeworth expansion for Pearson's correlation with mixed data types

Demonstrated higher accuracy of Edgeworth approximation in simulations

Extended the applicability of Edgeworth expansions to new statistical functions

Abstract

A large class of statistics can be formulated as smooth functions of sample means of random vectors. In this paper, we propose a general partial Cram\'{e}r's condition (GPCC) and apply it to establish the validity of the Edgeworth expansion for the distribution function of these functions of sample means. Additionally, we apply the proposed theorems to several specific statistics. In particular, by verifying the GPCC, we demonstrate for the first time the validity of the formal Edgeworth expansion of Pearson's correlation coefficient between random variables with absolutely continuous and discrete components. Furthermore, we conduct a series of simulation studies that show the Edgeworth expansion has higher accuracy.