Optimal rank-based tests for homogeneity of scatter

TL;DR

This paper introduces rank-based tests for homogeneity of scatter matrices in elliptical populations that are robust, efficient, and outperform traditional methods across various distributions.

Contribution

It develops a class of tests based on multivariate ranks and signs that are robust to heavy tails and achieve semiparametric efficiency, improving over existing parametric procedures.

Findings

Tests are valid without moment assumptions.

Normal-score version outperforms Gaussian likelihood ratio tests.

Effective across a broad range of non-Gaussian distributions.

Abstract

We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in $m$ elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavy-tailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normal-score version outperforms traditional Gaussian likelihood ratio tests and their pseudo-Gaussian robustifications under a very broad range of non-Gaussian densities including, for instance, all multivariate Student and power-exponential distributions.