Tests for large-dimensional shape matrices via Tyler's M estimators

Tyler's M estimator, as a robust alternative to the sample covariance matrix, has been widely applied in robust statistics. However, classical theory on Tyler's M estimator is mainly developed in the low-dimensional regime for elliptical populations. It remains largely unknown when the parameter of dimension $p$ grows proportionally to the sample size $n$ for general populations. By utilizing the eigenvalues of Tyler's M estimator, this article develops tests for the identity and equality of shape matrices in a large-dimensional framework where the dimension-to-sample size ratio $p/n$ has a limit in (0,1). The proposed tests can be applied to a broad class of multivariate distributions including the family of elliptical distributions (see model (2.1) below for details). To analyze both the null and alternative distributions of the proposed tests, we provide a unified theory on the spectrum of a large-dimensional Tyler's M estimator when the underlying population is general. Simulation results demonstrate good performance and robustness of our tests. An empirical analysis of the Fama-French 49 industrial portfolios is carried out to demonstrate the shape of the portfolios varying.