A study of two high-dimensional likelihood ratio tests under alternative hypotheses

Let Np(μ,∑) be a p-dimensional normal distribution. Testing ∑ equal to a given matrix or (μ,∑) equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension p is fixed, it is known that the LRT statistics go to 2-distributions. When p is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that p and n are proportional to each other. The condition n - 1 > p ∞ suffices in our results.