The difficulties of estimation of dispersion parameters in linear models—An illustration

Abstract

The point estimation of the parameter$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta }$ of a dispersion matrix$\sum {\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } } \right)}$ is illustrated by considering two linear models for observations with a common scalar mean. In the first model$\sum {\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } } \right)}$ has the structure which assures the validity of a univariate ANOVA with repeated measures; in the second,$\sum {\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } } \right)}$ corresponds to a permutationally invariant distribution. In both cases, results are presented about the estimability of$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta }$, thus obtaining the explicit form of the MINQE estimators introduced by Rao and Kleffe (1980). Finally, the consequences of the normality assumption are considered. The estimation methods based on the maximization of the complete and restricted likelihood functions are applied.