Tests for high-dimensional covariance matrices

In this paper, we propose some new tests for high-dimensional covariance matrices that are applicable to generally distributed populations with finite fourth moments. The proposed test statistics are the maximum of the likelihood ratio test statistic and the statistic based on the Frobenius norm. The advantage of the new tests is the good performance in terms of power for both the traditional case, in which the dimension is much smaller than the sample size, and the high-dimensional case, in which the dimension is large compared to the sample size. In the one-sample case, the new test is proposed for testing the hypothesis that the high-dimensional covariance matrix equals an identity matrix. In the two-sample case, the new test is developed for testing the equality of two high-dimensional covariance matrices. By using the random matrix theory, the asymptotic distributions of the proposed new tests are derived under the assumption that the dimension and the sample size proportionally tend toward infinity. Finally, numerical studies are conducted to investigate the finite sample performance of the proposed new tests.