Measuring deviations from spherical symmetry

TL;DR

This paper introduces a new measure for quantifying how much a distribution deviates from perfect spherical symmetry, based on minimum distance estimators and validated through simulations and real data.

Contribution

It proposes a novel deviation measure from spherical symmetry and develops estimators with asymptotic guarantees, expanding beyond traditional hypothesis testing.

Findings

The measure effectively quantifies deviations from spherical symmetry.

Estimators show good performance in simulations.

Application to real data demonstrates practical utility.

Abstract

Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector $\big (\|Y\|, Y/ \|Y\| )^\top $ and its best approximation by a distribution of a vector $\big (\|Y_s\|, Y_s/ \|Y_s \| )^\top $ corresponding to a random vector $Y_s$ with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.