  This paper considers the optimal modification of the likelihood ratio test
(LRT) for the equality of two high-dimensional covariance matrices. The
classical LRT is not well defined when the dimensions are larger than or equal
to one of the sample sizes. In this paper, an optimally modified test that
works well in cases where the dimensions may be larger than the sample sizes is
proposed. In addition, the test is established under the weakest conditions on
the moments and the dimensions of the samples. We also present weakly
consistent estimators of the fourth moments, which are necessary for the
proposed test, when they are not equal to 3. From the simulation results and
real data analysis, we find that the performances of the proposed statistics
are robust against affine transformations.
