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 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 .