## The Annals of Statistics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343414

**Digital Object Identifier**

doi:10.1214/aos/1176343414

**Mathematical Reviews number (MathSciNet)**

MR398007

**Zentralblatt MATH identifier**

0323.62043

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J05: Linear regression

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176343414