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.