The Covariances of Quadratic Forms of Random Variables and Their Applications in Linear Mixed Models

An improved ANOVA estimator is obtained in this paper. For the linear mixed model, the improved estimator can be obtained by this idea and dominates ANOVA estimator. For random effects model of one-way classification, the estimator considered by Kelly and Mathew¹ is a special case. This idea is also used to improve spectral decomposition estimator. This paper applys a simple property of variance-covariance to prove that the spectral decomposition estimator uniformly dominates ANOVA estimator for random effects model with one variance component when the covariance matrix of random effects only has one eigenvalue. Finally, this result is extended and we obtain a better feasible generalized spectral decomposition estimator.