Invariant test based on the modified correction to LRT for the equality of two high-dimensional covariance matrices

In this paper, we propose an invariant test based on the modified correction to the likelihood ratio test (LRT) of the equality of two high-dimensional covariance matrices. It is well-known that the classical log-LRT is not well defined when the dimension is larger than or equal to one of the sample sizes. Or even the log-LRT is well-defined, it is usually perceived as a bad statistic in the high-dimensional cases because of their low powers under some alternatives. In this paper, we will justify the usefulness of the modified log-LRT, and an invariant test that works well in cases where the dimension is larger than the sample sizes. Besides, the test is established under the weakest conditions on the dimensions and the moments of the samples. The asymptotic distribution of the proposed test statistic is also obtained under the null hypothesis. What is more, we also propose a lite version of the modified LRT in the paper. A simulation study and a real data analysis show that the performances of the two proposed statistics are invariant under affine transformations.