A0721
Title: A CLT for LSS of large dimensional sample covariance matrices with diverging spikes
Authors: Zhijun Liu - Northeast Normal University (China) [presenting]
Abstract: The central limit theorem (CLT) is established for linear spectral statistics (LSSs) of a large-dimensional sample covariance matrix when the population covariance matrices are involved with diverging spikes. This constitutes a nontrivial extension of the Bai-Silverstein theorem (BST), a theorem that has strongly influenced the development of high-dimensional statistics, especially in the applications of random matrix theory to statistics. The new CLT accommodates spiked eigenvalues, which may either be bounded or tend to infinity. The new CLT for LSS is then applied to test the hypothesis that the population covariance matrix is the identity matrix or a generalized spiked model. The asymptotic distributions of the corrected likelihood ratio test statistic and the corrected Nagao's trace test statistic are derived under the alternative hypothesis. Moreover, power comparisons are presented between these two LSSs and Roy's largest root test. In particular, it is demonstrated that except for the case in which there is only one spike, the LSSs could exhibit higher asymptotic power than Roy's largest root test.
