General order adjusted Edgeworth expansions for generalized $t$-tests

TL;DR

This paper introduces a generalized method for deriving Edgeworth expansions for various $t$-statistics, enabling more accurate inference in statistical testing through computer algebra and combinatorial algorithms.

Contribution

It develops Adjusted Edgeworth expansions that depend on sample size, applicable to multiple $t$-statistics, and provides a software package for broad research use.

Findings

Valid up to 5th order for various $t$-statistics

Includes software for practical implementation

Enhances accuracy of statistical inference

Abstract

We develop generalized approach to obtaining Edgeworth expansions for $t$-statistics of an arbitrary order using computer algebra and combinatorial algorithms. To incorporate various versions of mean-based statistics, we introduce Adjusted Edgeworth expansions that allow polynomials in the terms to depend on a sample size in a specific way and prove their validity. Provided results up to 5th order include one and two-sample ordinary $t$-statistics with biased and unbiased variance estimators, Welch $t$-test, and moderated $t$-statistics based on empirical Bayes method, as well as general results for any statistic with available moments of the sampling distribution. These results are included in a software package that aims to reach a broad community of researchers and serve to improve inference in a wide variety of analytical procedures; practical considerations of using such expansions are discussed.