Conditional tests for elliptical symmetry using robust estimators

TL;DR

This paper develops a robust testing procedure for elliptical symmetry in data distributions, replacing traditional estimators with robust ones, and evaluates its performance through theoretical analysis and numerical experiments.

Contribution

It introduces a new test for elliptical symmetry using robust estimators and analyzes its asymptotic properties and power compared to existing tests.

Findings

Robust estimators improve test performance under heavy-tailed distributions

The proposed test has favorable size and power properties in simulations

Comparison shows the new test outperforms some existing methods in certain scenarios

Abstract

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behaviour under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behaviour of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. This comparison was done looking not only at the observed level and power but we rather use the size-corrected relative exact power which provides a tool to assess the test statistic skill to detect alternatives. We also provide a numerical comparison with other competing tests.