Two-sample tests for high-dimensional covariance matrices using both difference and ratio

By borrowing strengths from the difference and ratio between two sample covariance matrices, we propose three tests for testing the equality of two high-dimensional population covariance matrices. One test is shown to be powerful against dense alternatives, and the other two tests are suitable for general cases, including dense and sparse alternatives, or the mixture of the two. Based on random matrix theory, we investigate the asymptotical properties of these three tests under the null hypothesis as the sample size and the dimension tend to infinity proportionally. Limiting behaviors of the new tests are also studied under various local alternatives. Extensive simulation studies demonstrate that the proposed methods out-perform or perform equally well compared with the existing tests.