A Note on Admissibility of Improved Unbiased Estimators in Two Variance Component Models

Abstract

The fact that multiples of unbiased quadratic estimators may have uniformly smaller mean squared error risks received some attention in statistics. Hodges and Lehmann (1951) improved the sample variance by multiplying it by (n-1)/(n+1), where n is the sample size. Perlman (1972) did a more comprehensive study of this phenomenon and LaMotte (1979) proved that for a normal random model each invariant quadratic form, as an estimator of its expectation, is dominated by an appriopriate scalar multiple of itself. These modified estimators, however, may not be admissible in the class of invariant quadratic estimators without the unbiasedness condition. A discussion of the problem for the balanced one-way ANOVA model was given by LaMotte (1979). Generally it is not known how to improve quadratic unbiased estimators by admissible quadratic estimators. In this paper we make a step in this direction and examine admissibility of multiples of admissible invariant quadratic unbiased estimators for a mixed linear model with two variance components only. The results are based on an explicit description of the class of admissible invariant quadratic estimators given by Gnot and Kleffe (1983). The above paper also helps to follow the technicalities presented in this note. Special attention receives estimation of the individual variance components.