On Asymptotic Optimality of Likelihood Ratio Tests for Multivariate Normal Distributions

Abstract

In multivariate analysis under normality assumptions, many likelihood ratio criteria $(\lambda^{(n)})$ are distributed as $k\prod^m_{i=1} Z^a_{li}(1 - Z_{li})^{b_i}\prod^{m'}_{j=1} Z^{c_j}_{2j}$ for some constants, $k, m, m', a_i, b_i,$ and $c_j$ when their associated null hypotheses are true, where $Z_{ij}$ are independently distributed beta variates. Let $T^{(n)} = -n^{-1} \ln \lambda^{(n)}$. This paper shows that a sequence $\{T^{(n)}\}$ of this kind is asymptotically optimal in the sense of exact slopes. Explicit forms of the exact slopes are obtained.