Minimum Variance Quadratic Unbiased Estimation (MIVQUE) of Variance Components

Abstract

Minimum variance quadratic unbiased estimators (MIVQUE's) of variance components from unbalanced data are obtained for the one-way classification random model under normality. Explicit, computable expressions are given for the estimators, their variances, and their covariance. The variance expressions provide readily-calculated lower bounds for the variances of any quadratic unbiased estimators of the variance components. For unbalanced data, the estimators are functions of the data and of constants σ _ao ² and σ _e ², taken as a priori estimates of the variance components σ _a ² and σ _e ². The estimators are, for unbalanced data, only locally minimum variance, i.e., they are only minimum variance when σ _ao ² = σ _a ² and σ _eo ² = σ _e ². However, numerical results suggest that the “MIVQUE” of σ _a ² may have much smaller variance than the usual ANOVA estimator with unbalanced data, even when σ _ao ² and σ _eo ² deviate considerably from σ _a ² and σ _e ² respectively. In contrast, the ANOVA estimator of σ _e ² seems to have smaller variance than the “MIVQUE” unless σ _ao ² and σ _eo ² are choices close to σ _a ² and σ _e ².