A class of optimal tests for symmetry based on local Edgeworth approximations

TL;DR

This paper develops a framework for optimal symmetry tests in univariate data by leveraging local Edgeworth approximations, ensuring classical skewness tests are optimal near Gaussian distributions.

Contribution

It introduces a new class of local asymmetric alternatives using Edgeworth expansions, establishing optimality of classical skewness tests in the Gaussian vicinity.

Findings

Classical skewness tests are optimal near Gaussian densities.

The proposed local Edgeworth-based alternatives effectively embed symmetry testing.

The framework separates the roles of location, scale, and skewness.

Abstract

The objective of this paper is to provide, for the problem of univariate symmetry (with respect to specified or unspecified location), a concept of optimality, and to construct tests achieving such optimality. This requires embedding symmetry into adequate families of asymmetric (local) alternatives. We construct such families by considering non-Gaussian generalizations of classical first-order Edgeworth expansions indexed by a measure of skewness such that (i) location, scale and skewness play well-separated roles (diagonality of the corresponding information matrices) and (ii) the classical tests based on the Pearson--Fisher coefficient of skewness are optimal in the vicinity of Gaussian densities.