Optimal tests for elliptical symmetry: specified and unspecified location

TL;DR

This paper introduces new optimal tests for elliptical symmetry in multivariate analysis, addressing both specified and unspecified location, with strong theoretical backing and broad applicability.

Contribution

It proposes novel, asymptotically optimal tests for elliptical symmetry that are affine-invariant, computationally efficient, and outperform existing methods.

Findings

Tests have a simple asymptotic null distribution.

They are affine-invariant and computationally fast.

They outperform existing competitors in power.

Abstract

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses location rather than non-elliptical alternatives. In this paper, we are proposing new classes of testing procedures, both for specified and unspecified location. The backbone of our construction is Le Cam's asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are meeting all the desired properties of a``good'' test of elliptical symmetry: they have a simple asymptotic distribution under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors.