Existence of maximum likelihood estimates in normal variance-components models

Under the usual nonnegativity constraints on the variance components, a maximum likelihood estimate (MLE) of the parameter vector in a normal variance-components model is known to exist. We investigate the question of existence under more general types of constraints and, in particular, under the constraint that requires only that the variance–covariance matrix be positive definite. Attention is restricted to models in which all the possible variance–covariance matrices commute with one another. It is found that in some models, such as all random oneway models with a single group having the largest size and all balanced random two-way models, the likelihood becomes infinite under the positive definiteness constraints, so that no MLE exists. In (practically) all normal balanced mixed-effects classification models, a residual maximum likelihood estimate (REMLE) exists.