The moderate deviation principles of likelihood ratio tests under alternative hypothesis

Abstract

Let [Formula: see text] be independent and identically distributed (i.i.d.) real-valued random vectors from distribution [Formula: see text], where the sample size [Formula: see text] and the vector dimension [Formula: see text] satisfy [Formula: see text]. We are interested in the exponential convergence rate of the likelihood ratio test (LRT) statistics for testing [Formula: see text] equal to a given matrix and [Formula: see text] equal to a given pair. In traditional statistical theory, the LRT statistics have been studied under the null hypothesis and finite-dimensional conditions. In this paper, we prove the moderate deviation principle (MDP) under the high-dimensional conditions for the two LRT statistics. We show that our results hold under the null hypothesis and the alternative hypothesis as well. Some numerical simulations indicate that our conclusions have good performance.