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 n ₁ and n ₂ 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 asymptotic normal distribution under the assumption p/n ₁→y ₁∈(0, 1] and p/n ₂→y ₂∈(0, 1]. The case for max{y ₁, y ₂}=1 is much different from the case max{y ₁, y ₂}<1.