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

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