A Robust Test for Elliptical Symmetry

TL;DR

This paper introduces a new robust goodness-of-fit test for elliptical symmetry based on Tyler's estimator, which is rigorously analyzed and shown to have higher statistical power than existing tests.

Contribution

The paper develops and analyzes a novel robust GoF test for elliptical symmetry using Tyler's estimator, with a new theoretical framework for its analysis.

Findings

The proposed test is consistent against all alternatives to ellipticity.

Numerical simulations show the test has higher statistical power than existing methods.

The test is based on easily computable data statistics and does not require known covariance.

Abstract

Most signal processing and statistical applications heavily rely on specific data distribution models. The Gaussian distributions, although being the most common choice, are inadequate in most real world scenarios as they fail to account for data coming from heavy-tailed populations or contaminated by outliers. Such problems call for the use of Robust Statistics. The robust models and estimators are usually based on elliptical populations, making the latter ubiquitous in all methods of robust statistics. To determine whether such tools are applicable in any specific case, goodness-of-fit (GoF) tests are used to verify the ellipticity hypothesis. Ellipticity GoF tests are usually hard to analyze and often their statistical power is not particularly strong. In this work, assuming the true covariance matrix is unknown we design and rigorously analyze a robust GoF test consistent against all alternatives to ellipticity on the unit sphere. The proposed test is based on Tyler's estimator and is formulated in terms of easily computable statistics of the data. For its rigorous analysis, we develop a novel framework based on the exchangeable random variables calculus introduced by de Finetti. Our findings are supported by numerical simulations comparing them to other popular GoF tests and demonstrating the significantly higher statistical power of the suggested technique.