Comments on maximum likelihood estimation and projections under multivariate statistical models

Under the multivariate model with linearly structured covariance matrix with unknown variance components and known mean parameters (Szatrowski, Ann Stat 8:802-810, 1980) showed that the maximum likelihood estimators of variance components have explicit representation if and only if the space of covariance matrix is a quadratic subspace. The aim of this paper is to rewrite these results for models with unknown expectation and to give sufficient conditions for maximum likelihood estimator of covariance matrix to be a projection of the maximum likelihood estimator of unstructured covariance onto the space of structured matrices. The results will be illustrated by examples of structures suitable for multivariate models with general mean, growth curve models as well as doubly multivariate models.