Moderate deviation principles for classical likelihood ratio tests of high-dimensional normal distributions

Let x(1), ..., x(n), be a random sample from a Gaussian random vector of dimension p < n with mean mu and covariance matrix Sigma. Based on this sample, we consider the moderate deviation principle of the modified likelihood ratio test (LRT) for the testing problem H-0 : Sigma = lambda I-p versus H-1 : Sigma not equal lambda I-p, in the high-dimensional setting, where lambda is some unknown constant (Jiang and Yang (2013)). We assume that both the dimension p and sample size n go to infinity in such a way that p/n -> y is an element of(0, 1]. Under H-0, our results give the exponential convergence rate of the LRT statistic to the corresponding asymptotic distribution. (C) 2017 Elsevier Inc. All rights reserved.