Asymptotic Nonnull Distributions for Likelihood Ratio Statistics in the Multivariate Normal Patterned Mean and Covariance Matrix Testing Problem

Abstract

The multivariate normal patterned mean and covariance matrix testing problem is studied for general one and $k$-population hypotheses. T. W. Anderson's iterative algorithm for finding the maximum likelihood estimates, the forms of the likelihood ratio tests, and asymptotic chi-square distributions of these tests under the null hypothesis are given. The nonnull asymptotic normal distribution is derived using the standard delta method. This derivation involves using several extensions of matrix identities given in Anderson, matrix derivatives and asymptotic likelihood equations. The forms of the variances are greatly simplified using a result of Szatrowski when the maximum likelihood estimates under the null hypothesis have explicit representations.