Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings

Abstract

In the high-dimensional setting, this article considers three interrelated problems: (a) testing the equality of two covariance matrices and ; (b) recovering the support of ; and (c) testing the equality of and row by row. We propose a new test for testing the hypothesis H ₀: and investigate its theoretical and numerical properties. The limiting null distribution of the test statistic is derived and the power of the test is studied. The test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives. The simulation results show that the test significantly outperforms the existing methods both in terms of size and power. Analysis of a prostate cancer dataset is carried out to demonstrate the application of the testing procedures. When the null hypothesis of equal covariance matrices is rejected, it is often of significant interest to further investigate how they differ from each other. Motivated by applications in genomics, we also consider recovering the support of and testing the equality of the two covariance matrices row by row. New procedures are introduced and their properties are studied. Applications to gene selection are also discussed. Supplementary materials for this article are available online.

Keywords:

• Extreme value Type I distribution
• Gene selection
• Hypothesis testing
• Sparsity

SUPPLEMENTARY MATERIALS

In the supplement we prove Proposition 1–3 and the technical results, Lemmas 3, 4, and 5, which are used in the proofs of the main results. We also present more extensive simulation results comparing the numerical performance of the proposed test with that of other tests.

The research of Tony Cai and Yin Xia was supported in part by NSF FRG grant DMS-0854973. Weidong Liu’s research was supported by NSFC, Grant No. 11201298, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Foundation for the Author of National Excellent Doctoral Dissertation of PR China and the startup fund from Shanghai Jiao Tong University (SJTU).