Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions II. submitted

For random samples of size n obtained from p-variate normal distribu-tions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the high-dimensional setting. These test statistics have been extensively studied in multivariate analysis, and their limiting dis-tributions under the null hypothesis were proved to be chi-square distributions as n goes to infinity and p remains fixed. In this paper, we consider the high-dimensional case where both p and n go to infinity with p/n → y ∈ (0,1]. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and vari-ances. We perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chi-square approximations for analyzing high-dimensional data