Maximum likelihood estimators of the variance components based on theQ-reduced model

Summary

In a variance component model,$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Y} \sim \left( {\chi \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\beta } ,\sum\limits_{j = 1}^C {\sigma _j^2 } V_j } \right)$, Pukelsheim (1981) proved that the non-negative and unbiased estimation of the variance componentsσ ² _j ,j=1, …,c, entails a transformation of the original model toQ $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Y}$ (calledQ-reduced model). The maximum likelihood (ML) approach based on the likelihood ofQ $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{Y}$ (denotedQ-ML) is considered and applied to an incomplete block design (IBD) model. TheQ-ML estimators of variance components and are shown to be more efficient in the mean squared error sense than the non-negative MINQUE’s (minimum norm quadratic unbiased estimators) in the IBD. The effect of usingQ-ML estimators of the variance components to estimate the variance ratio in the combined estimator of the treatment contrast is also considered.