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.
Publié le : 1976-07-14
Classification:
Multivariate statistics,
linear models,
regression analysis,
Gauss-Markov estimation,
missing observations,
62F10,
62J05
@article{1176343551,
author = {Drygas, Hilmar},
title = {Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations},
journal = {Ann. Statist.},
volume = {4},
number = {1},
year = {1976},
pages = { 779-787},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343551}
}
Drygas, Hilmar. Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations. Ann. Statist., Tome 4 (1976) no. 1, pp. 779-787. http://gdmltest.u-ga.fr/item/1176343551/