Projected tests for high-dimensional covariance matrices

Abstract

The classic likelihood ratio test for testing the equality of two covariance matrices breakdowns due to the singularity of the sample covariance matrices when the data dimension is larger than the sample size. In this paper, we present a conceptually simple method using random matrices to project the data onto a one-dimensional random subspace so that conventional methods can be applied. Both one-sample and two-sample tests for high-dimensional covariance matrices are considered. A transformation using the precision matrix is used to help maintain the information on the off-diagonal elements of the covariance matrices. Multiple projections are used to improve the performance of the proposed tests. An extremal type theorem is established and used to estimate the significance level. Simulations and an application to the Acute Lymphoblastic Leukemia (ALL) data are given to illustrate our method.