Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model

Abstract

The weighted least-squares estimator of parametric functions K β under a general linear regression model ${\{ {\bf y},\,{\bf X \beta}, \sigma^2{\bf \Sigma} \}}$ is defined to be ${{\bf K}{\hat{\bf {\beta}}}}$, where ${\hat{{\bf \beta}}}$ is a vector that minimizes (y − X β)′V(y − X β) for a given nonnegative definite weight matrix V. In this paper, we study some algebraic and statistical properties of ${{\bf K}\hat{{\bf \beta}}}$ and the projection matrix associated with the estimator, such as, their ranks, unbiasedness, uniqueness, as well as equalities satisfied by the projection matrices.