Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances

The problem of finding maximum likelihood estimates for patterned means and covariance matrices in multivariate analysis is considered. Necessary and sufficient conditions are presented for the existence of explicit solutions and the obtaining of these explicit solutions in one iteration of the scoring equations from any positive definite starting point. Cases in which averaging yields the explicit maximum likelihood estimates are discussed. These results can be applied to the problems of finding maximum likelihood estimates for the parameters in the complete, compound and circular symmetry patterns; mixed models in the analysis of variance; and for finding asymptotic distributions of likelihood ratio statistics when the parameters under the null hypothesis have explicit maximum likelihood estimates.