## The Annals of Statistics

- Ann. Statist.
- Volume 9, Number 3 (1981), 633-637.

### Unbiased and Minimum-Variance Unbiased Estimation of Estimable Functions for Fixed Linear Models with Arbitrary Covariance Structure

#### Abstract

Consider a general linear model for a column vector $y$ of data having $E(y) = X \alpha$ and $\operatorname{Var}(y) = \sigma^2H$, where $\alpha$ is a vector of unknown parameters and $X$ and $H$ are given matrices that are possibly deficient in rank. Let $b = Ty$, where $T$ is any matrix of maximum rank such that $TH = \phi$. The estimation of a linear function of $\alpha$ by functions of the form $c + a'y$, where $c$ and $a$ are permitted to depend on $b$, is investigated. Allowing $c$ and $a$ to depend on $b$ expands the class of unbiased estimators in a nontrivial way; however, it does not add to the class of linear functions of $\alpha$ that are estimable. Any minimum-variance unbiased estimator is identically [for $y$ in the column space of $(X, H)$] equal to the estimator that has minimum variance among strictly linear unbiased estimators.

#### Article information

**Source**

Ann. Statist., Volume 9, Number 3 (1981), 633-637.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345467

**Digital Object Identifier**

doi:10.1214/aos/1176345467

**Mathematical Reviews number (MathSciNet)**

MR615439

**Zentralblatt MATH identifier**

0477.62053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J05: Linear regression

**Keywords**

Linear models best linear unbiased estimation singular covariance matrices Gauss-Markov theorem

#### Citation

Harville, David A. Unbiased and Minimum-Variance Unbiased Estimation of Estimable Functions for Fixed Linear Models with Arbitrary Covariance Structure. Ann. Statist. 9 (1981), no. 3, 633--637. doi:10.1214/aos/1176345467. https://projecteuclid.org/euclid.aos/1176345467