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

Let be a -dimensional normal distribution. Testing equal to a given matrix or equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension is fixed, it is known that the LRT statistics go to -distributions. When 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 and are proportional to each other. The condition suffices in our results. [ABSTRACT FROM AUTHOR]