Likelihood ratio tests for high-dimensional normal distributions.

Abstract

For a random sample of size n obtained from p-variate normal distributions, we consider the likelihood ratio tests (LRT) for their means and covariance matrices. Most of these test statistics have been extensively studied in the classical multivariate analysis and their limiting distributions under the null hypothesis were proved to be a Chi-Square distribution under the assumption that n goes to infinity while p remains fixed. In our research, we consider the high-dimensional case where both p and n go to infinity and their ratio p/n converges to a constant y in (0, 1]. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to a normal random variable and we also give the explicit forms of its mean and variance. We run simulation study to show that the likelihood ratio test using this new central limit theorem outperforms the one using the traditional Chi-square approximation for analyzing high-dimensional data.