The Annals of Statistics

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

David Harville

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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.

Article information

Ann. Statist., Volume 4, Number 2 (1976), 384-395.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62J05: Linear regression
Secondary: 62F10: Point estimation 62F15: Bayesian inference 15A09: Matrix inversion, generalized inverses 15A24: Matrix equations and identities

Gauss-Markov theorem linear models mixed analysis-of-variance models best linear unbiased estimation singular covariance matrices Bayesian inference


Harville, David. Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects. Ann. Statist. 4 (1976), no. 2, 384--395. doi:10.1214/aos/1176343414.

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