ML estimation for the multivariate normal distribution with general linear model mea1 and linear-structure covariance matrix; one-population, complete-data case

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Abstract
In this study we investigate maximum likelihood (ML) estimation for the multivariate oormal distribution with general linear model mean vector, , and linear-structure covariance matrix , where and the are properly defined,known matrices, and are vectors of unknown parameters,Likelihood equations are obtained for the cases where the parameters are subject to three types of constraints: where are known matrices and vectors respectively. Kecessary and sufficient conditions for an explicit solution to these equations and examples of practical situations where these conditions are satisfied are presented, it is shown that these conditions cannot be met for the cases where is constrained ((2) and (3)) and is totally reducible, unless . Two iterative procedures, the Method of Scoring and the EM algorithm, are suggested to solve che equations when they do not have explicit solution, Finally, asymptotic joint null and nonnull distributions of the ML estimators of and are given. and also one iteration of the Method of Scoring is shown to produce asyniptotically efficient estimates of the parameters provided we have a consistent estimate of .