Estimators of Variance Components in the Balanced Incomplete Block Design

Abstract

Estimation of variance components when the design is a balanced incomplete block, the model is additive, and error, block and treatment components are normal, independent, with zero expectations, is considered using variance as a criterion.

The sections are arranged in the following manner: 1—The results are summarized. 2—The notation, model, statistics and analysis of variance tables are introduced. 3—A 1–1 (linear) transformation of the sufficient statistics is performed, and a number of ‘reasonable’ estimators are constructed. 4—The model and notation are redefined in terms of matrices; and variances and covariances of the sufficient statistics are obtained. 5—A method for obtaining variances and covariances of estimators is given. 6—Variances of some of the estimators are compared, and some are seen to be inadmissible. 7—Variances of the usual variance component estimators are obtained. 8—The variances of two well-known unbiased estimators of σ² are compared, but neither is uniformly better than the other, although one is easily shown to be admissible; two others are shown to be inadmissible. 9—The block and treatment component estimators are investigated.