We show that in the mixed model of the analysis of variance, there is a sequence of roots of the likelihood equations which is consistent, asymptotically normal, and efficient in the sense of attaining the Cramer-Rao lower bound for the covariance matrix. These results follow directly by an application of a general result of Weiss (1971, 1973) concerning maximum likelihood estimates. This problem differs from standard problems in that we do not have independent, identically distributed observations and that estimates of different parameters may require normalizing sequences of different orders of magnitude. We give some examples and comment briefly on likelihood ratio tests for these models.
"Asymptotic Properties of Maximum Likelihood Estimates in the Mixed Model of the Analysis of Variance." Ann. Statist. 5 (4) 746 - 762, July, 1977. https://doi.org/10.1214/aos/1176343897