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]p/n→y∈(0,1]. The result for y  =1 is much different from the case for y∈(0,1)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/n1→y1∈(0,1]p/n1→y1∈(0,1] and p/n2→y2∈(0,1]p/n2→y2∈(0,1]. The case for max{y1,y2}=1max{y1,y2}=1 is much different from the case max{y1,y2}<1.