On structure testing for component covariance matrices of a high-dimensional mixture

TL;DR

This paper investigates the spectral properties of sample covariance matrices in high-dimensional mixtures, revealing non-convergence to the Marčenko-Pastur law due to dependence, and proposes a new sphericity test with higher power for mixture models.

Contribution

It introduces a novel limit law for eigenvalues in high-dimensional mixtures and develops a new sphericity test that outperforms existing methods in mixture analysis.

Findings

Eigenvalue distribution does not converge to Marčenko-Pastur law in certain mixtures.

Traditional sphericity tests fail for high-dimensional mixtures.

The new test effectively detects non-spherical covariance structures.

Abstract

By studying the family of $p$-dimensional scale mixtures, this paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix {\em does not converge} to the celebrated Mar\v{c}enko-Pastur law. A different and new limit is found and characterized. The reasons of failure of the Mar\v{c}enko-Pastur limit in this situation are found to be a strong dependence between the $p$-coordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analize the traditional John's type test we establish a novel and general CLT for linear statistics of eigenvalues of the sample covariance matrix. It is shown that the John's test and its recent high-dimensional extensions both fail for high-dimensional mixtures, precisely due to the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used ICL and BIC criteria in detecting non spherical component covariance matrices of a high-dimensional mixture.