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

The general mixed linear model can be written y = X + Zb, where is a vector of fixed effects and b is a vector of random variables. Assume that E (b) = 0 and that Var (b) = ²D with D known. Consider the estimation of ₁' + ₂', where ₁' is estimable and is the realized, though unobservable, value of b. Among linear estimators c + r'y having E (c + r'y) E (₁' + ₂'b), mean squared error E (c + r'y - ₁' - ₂'b) ² is minimized by ₁' + ₂', where = DZ'V^\# (y - X), = (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 and 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.