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

Abstract

The general mixed linear model can be written $y = X\\alpha + Zb$, where $\\alpha$ is a vector of fixed effects and $b$ is a vector of random variables. Assume that $E(b) = 0$ and that $\\operatorname{Var} (b) = \\sigma^2D$ with $D$ known. Consider the estimation of $\\lambda_1'\\alpha + \\lambda_2'\\beta$, where $\\lambda_1'\\alpha$ is estimable and $\\beta$ is the realized, though unobservable, value of $b$. Among linear estimators $c + r'y$ having $E(c + r'y) \\equiv E(\\lambda_1'\\alpha + \\lambda_2'b)$, mean squared error $E(c + r'y - \\lambda_1'\\alpha - \\lambda_2'b)^2$ is minimized by $\\lambda_1'\\hat{\\alpha} + \\lambda_2'\\hat{\\beta}$, where $\\hat{\\beta} = DZ'V^{\\tt\\#}(y - X\\hat{\\alpha}), \\hat{\\alpha} = (X'V^{\\tt\\#}X) - X'V^{\\tt\\#}y$, and $V^{\\tt\\#}$ is any generalized inverse of $V = ZDZ'$ belonging to the Zyskind-Martin class. It is shown that $\\hat{\\alpha}$ and $\\hat{\\beta}$ 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.

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