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In this work, we are motivated by the recent work of Zhang et al. (2019) andstudy a new invariant test for equality of two large scale covariance matrices.Two modified likelihood ratio tests (LRTs) by Zhang et al. (2019) are based onthe sum of log of eigenvalues (or 1- eigenvalues) of the Beta-matrix. However,as the dimension increases, many eigenvalues of the Beta-matrix are close to 0or 1 and the modified LRTs are greatly influenced by them. In this work,instead, we consider the simple sum of the eigenvalues (of the Beta-matrix) andcompute its asymptotic normality when all $n_1, n_2, p$ increase at the samerate. We numerically show that our test has higher power than two modifiedlikelihood ratio tests by Zhang et al. (2019) in all cases both we and theyconsider.

An Invariant Test for Equality of Two Large Scale Covariance Matrices