## The Annals of Statistics

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

Hilmar Drygas

#### 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

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