Asymptotic Properties of the Estimators for Multivariate Components of Variance

Estimation of the covariance matrices in the multivariate balanced one-way random effect model is discussed. The rank of the between-group covariance matrix plays a large role in model building as well as in assessing asymptotic properties of the estimated covariance matrices. The restricted (residual) maximum likelihood estimators derived under a rank condition are considered. Asymptotic properties of the estimators are derived for a possibly incorrectly specified rank and under either the number of groups, the number of replicates, or both, tending to infinity. A higher order expansion covering various cases leads to a common approximate inference procedure which can be used in a wide range of practical situations. A simulation study is also presented.