Likelihood Ratio Tests for Covariance Matrices of High-Dimensional Normal Distributions

For a random sample of size n obtained from a p-variate normal population, the likelihood ratio test (LRT) for the covariance matrix equal to a given matrix is considered. By using the Selberg integral, we prove that the LRT statistic converges to a normal distribution under the assumption p/n → y ∈ (0, 1]. The result for y = 1 is much different from the case for y ∈ (0, 1). Another test is studied: given two sets of random observations of sample size n1 and n2 from two p-variate normal distributions, we study the LRT for testing the two normal distributions having equal covariance matrices. It is shown through a corollary of the Selberg integral that the LRT statistic has an aymptotic normal distribution under the assumption p/n1 → y1 ∈ (0, 1] and p/n2 → y2 ∈ (0, 1]. The case for max{y1, y2} = 1 is much different from the case max{y1, y2} < 1.