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A0298
Title: On structure testing for component covariance matrices of a high-dimensional mixture Authors:  Weiming Li - Shanghai University of Finance and Economics (China) [presenting]
Abstract: By studying the family of $p$-dimensional scaled mixtures, a non trivial example is shown for the first time 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. The reasons of failure of the Marcenko 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. It is shown that the traditional Johns 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. Anew test using this CLT is constructed afterwards for the sphericity hypothesis.