On Best Linear Estimation and General Gauss-Markov Theorem in Linear Models with Arbitrary Nonnegative Covariance Structure

Abstract

Given the general linear model $y = X\beta + e$ having the covariance matrix $\sigma ^2 V$ of the errors, with $\sigma ^2 > 0$, V known and nonnegative (possibly singular), we specify the complete nonempty class $\mathcal{V}$ of conditional inverses of V such that, for any estimable parametric function $\lambda '\beta $ and any $V^ * $ in $\mathcal{V}$, a best linear unbiased estimator of $\lambda '\beta $ is given by $\lambda '\hat \beta $, where $\hat \beta $ is any solution to the general normal equations $X'V^ * X\beta = X'V^ * y$. Properties of the solutions $\hat \beta $ are presented. It is further verified that if y is distributed as a multivariate normal variable then $\lambda '\hat \beta $ is the maximum likelihood estimator of $\lambda '\beta $. A procedure for testing hypotheses, using solutions to the general normal equations, is also presented.