Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects

The general mixed linear model can be written y = Xα + Zb, where α is a vector of fixed effects and b is a vector of random variables. Assume that $E(b) = 0$ and that Var (b) = σ²D with D known. Consider the estimation of λ₁'α + λ₂'β, where λ₁'α is estimable and β is the realized, though unobservable, value of b. Among linear estimators $c + r'y$ having E(c + r'y) ≡ E(λ₁'α + λ₂'b), mean squared error E(c + r'y - λ₁'α - λ₂'b)² is minimized by λ₁'α̂ + λ₂'β̂, where β̂ = DZ'V#(y - Xα̂), α̂ = (X'V#X) - X'V#y, and V# is any generalized inverse of $V = ZDZ'$ belonging to the Zyskind-Martin class. It is shown that α̂ and β̂ can be computed from the solution to any of a certain class of linear systems, and that doing so facilitates the exploitation, for computational purposes, of the kind of structure associated with ANOVA models. These results extend the Gauss-Markov theorem. The results can also be applied in a certain Bayesian setting.