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

Abstract

By studying the family of p-dimensional scaled mixtures, this paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated Marcenko-Pastur law. A different and new limit is found and characterized. We also address the problem of testing whether the mixture has a spherical covariance matrix. It is shown that the traditional John's test and its recent high-dimensional extensions both fail for high-dimensional mixtures, precisely due to the different spectral limit above. In order to find a remedy, we establish a novel and general CLT for linear statistics of eigenvalues of the sample covariance matrix. A new test using this CLT is constructed afterwards for the sphericity hypothesis.