A new approach to the estimation of variance components

Consideration is given to obtaining maximum-likelihood estimates of variance components for both balanced and unbalanced data. The tool which allows this is the inverse of the variance-covariance matrix which is given for all designs with equal cell sizes. The inverse is also obtained for several elementary models with unequal cell sizes. These inverses together with a new method for obtaining the likelihood equations are then used to consider the maximum-likelihood estimators. It is noted that in most cases these estimators are usually not linear combinations of the sufficient statistics. Possibilities for certain estimators which are closely kin to maximum-likelihood estimators are also considered