On Structure Testing for Component Covariance Matrices of a High-Dimensional Mixture

• Activity Number: 388 - Random Matrices and Applications
• Type: Invited
• Date/Time: Tuesday, August 1, 2017 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Jianfeng YAO* and Weiming Li
• Companies: The University of Hong Kong and Shanghai University of Finance and Economics
• Keywords: high-dimensional mixture ; structure testing ; Sphericity test ; Large covariance matrix ; Marcenko-Pastur law

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. (This is a joint work with Weiming Li, Shanghai University of Finance and Economics).