## The Annals of Statistics

- Ann. Statist.
- Volume 4, Number 4 (1976), 779-787.

### Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations

#### Abstract

In this note we discuss multivariate linear models from the coordinate-free point of view, as earlier done by Eaton (1970). We generalize the result of Eaton by allowing for missing observations. This leads to models of the kind $EY \in L, Cov Y \in\{P(I \otimes \sum)P'\}$ where $P$ is a diagonal mapping. The paper starts by deriving the conditions for existence of Gauss-Markov estimators (GME) of $EY$ in models where the covariance-mappings are not necessarily nonsingular. These conditions are then applied to the above models if $\Sigma$ runs either over all PSD-mappings or over all diagonal PSD-mappings. In the latter case $L$ must be of the form $L = L_1 \times \cdots \times L_p$ while in the general case some further conditions on the $L_i$ must be met. (If $P = I$, then $L_i = L_j$ must hold for all $i, j$; this is equivalent to the result obtained by Eaton). Examples show that these conditions are satisfied only under rather exceptional conditions.

#### Article information

**Source**

Ann. Statist., Volume 4, Number 4 (1976), 779-787.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343551

**Digital Object Identifier**

doi:10.1214/aos/1176343551

**Mathematical Reviews number (MathSciNet)**

MR411060

**Zentralblatt MATH identifier**

0336.62052

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F10: Point estimation

Secondary: 62J05: Linear regression

**Keywords**

Multivariate statistics linear models regression analysis Gauss-Markov estimation missing observations

#### Citation

Drygas, Hilmar. Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations. Ann. Statist. 4 (1976), no. 4, 779--787. doi:10.1214/aos/1176343551. https://projecteuclid.org/euclid.aos/1176343551