Combined unbiased estimators based onQ-reduced model

Abstract

In the variance component model$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Y} \sim \left( {X\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\beta } ,\sum\limits_{j = 1}^c {\sigma _j^2 v_j } } \right)$, Pukelsheim (1981,Anal. of Statist.) proved that the non-negative and unbiased estimation of the variance component σ ² _j ,j=1, 2, …,c entails a transformation of the original model to$Q\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Y}$ (called theQ-reduced model). Lee and Kapadia (1988,Metrika), considered the maximum likelihood approach based on the likelihood of theQY (denoted byQ — ML) and applied to an incomplete block design model. In this note, the results given in their paper are used to show that the variance of the combined estimator of treatment contrast proposed by Graybill and Deal (1959,Biometrics) can be reduced if theQ — ML estimators of the variance components are used instead ofAOV estimators as is done in Graybill and Deal’s paper.