Likelihood Ratio Tests for High-Dimensional Normal Distributions

In the paper by Jiang and Yang (2013), six classical Likelihood Ratio Test (LRT) statistics are studied under high-dimensional settings. Assuming that a random sample of size n is observed from a p-dimensional normal population, they derive the central limit theorems (CLTs) when p/n → y ∈ (0, 1], which are different from the classical chisquare limits as n goes to infinity while p remains fixed. In this paper, by developing a new tool, we prove that the above six CLTs hold in a more applicable setting: p goes to infinity and p < n − c for some 1 ≤ c ≤ 4. This is an almost sufficient and necessary condition for the CLTs. Simulations of histograms, comparisons on sizes and powers with those in the classical chi-square approximations and discussions are presented afterwards.