Testing and support recovery of multiple high-dimensional covariance matrices with false discovery rate control

Motivated by applications in genomics, we study in this paper four interrelated high-dimensional hypothesis testing problems on dependence structures among multiple populations. A new test statistic is constructed for testing the global hypothesis that multiple covariance matrices are equal, and its limiting null distribution is established. Correction methods are introduced to improve the accuracy of the test for finite samples. It is shown that the proposed tests are powerful against sparse alternatives and enjoy certain optimality properties. We then propose a multiple testing procedure for simultaneously testing the equality of the entries of the covariance matrices across multiple populations. The proposed method is shown to control the false discovery rate. A simulation study demonstrates that the proposed tests maintain the desired error rates under the null and have good power under the alternative. The methods are also applied to a Novartis multi-tissue analysis. In addition, testing and support recovery of submatrices of multiple covariance matrices are studied.