Nonnegative estimation of variance components in unbalanced mixed models with two variance components

An unbalanced mixed linear model with two variance components is considered, one variance component (say [sigma]12 >= 0) corresponding to a random effect (treatments) and a second variance component (say [sigma]2 > 0) corresponding to the experimental errors. Sufficient conditions are obtained under which there will exist a nonnegative invariant quadratic estimator (IQE) having a uniformly smaller mean squared error (MSE) than every unbiased IQE of [sigma]12. In particular, for the one-way unbalanced ANOVA model, necessary and sufficient conditions are also obtained for a multiple of the usual treatment sum of squares to uniformly dominate the ANOVA estimator of [sigma]12. For estimating [sigma]2, it is shown that the best multiple of the residual sum of squares can be improved by using nonquadratic estimators. One such estimator is obtained using the idea of a testimator ([13], Ann. Inst. Statist. Math.16 155-160). A second estimator is obtained following the approach in [14], Ann. Statist.2 190-198). Numerical results regarding the performance of the proposed estimators of [sigma]12 are also reported.