Fisher transformation via Edgeworth expansion

TL;DR

This paper introduces a method using Edgeworth expansion to improve Fisher's z transformation for better approximation of the correlation coefficient's distribution, with practical implementation in Mathematica.

Contribution

It provides a novel approach to refine Fisher's z transformation by eliminating skewness through Edgeworth expansion, enhancing accuracy for sample statistics.

Findings

Enhanced approximation accuracy of correlation distribution

Method to eliminate skewness in distribution approximation

Flexible code for various sample statistics

Abstract

We show how to calculate individual terms of the Edgeworth series to approximate the distribution of the Pearson correlation coefficient with the help of a simple Mathematica program. We also demonstrate how to eliminate the corresponding skewness, thus making the approximation substantially more accurate. This leads, in a rather natural way, to deriving a superior (in terms of its accuracy) version of Fisher's z transformation. The code can be easily modified to deal with any sample statistics defined as a function of several sample means, based on a random independent sample from a multivariate distribution.