Gram Charlier and Edgeworth expansion for sample variance

TL;DR

This paper develops Edgeworth expansions for the sample variance in strongly mixing processes, broadening theoretical understanding and providing practical tools for dependent data analysis.

Contribution

It derives valid Edgeworth expansions for the empirical variance under minimal assumptions for strongly mixing processes, including non-normal cases.

Findings

Edgeworth expansions are valid for strongly mixing processes.

Results apply to non-normal and dependent data.

Provides an alternative to existing methods for variance approximation.

Abstract

In this paper, we derive a valid Edgeworth expansions for the Bessel corrected empirical variance when data are generated by a strongly mixing process whose distribution can be arbitrarily. The constraint of strongly mixing process makes the problem not easy. Indeed, even for a strongly mixing normal process, the distribution is unknown. Here, we do not assume any other assumption than a sufficiently fast decrease of the underlying distribution to make the Edgeworth expansion convergent. This results can obviously apply to strongly mixing normal process and provide an alternative to the work of Moschopoulos (1985) and Mathai (1982).