Weighted Statistic in Detecting Faint and Sparse Alternatives for High-Dimensional Covariance Matrices

This article considers testing equality of two population covariance matrices when the data dimension p diverges with the sample size n (p/n → c > 0). We propose a weighted test statistic that is data-driven and powerful in both faint alternatives (many small disturbances) and sparse alternatives (several large disturbances). Its asymptotic null distribution is derived by large random matrix theory without assuming the existence of a limiting cumulative distribution function of the population covariance matrix. The simulation results confirm that our statistic is powerful against all alternatives, while other tests given in the literature fail in at least one situation. Supplementary materials for this article are available online.