The Annals of Statistics

Gauss-Markov Estimation for Multivariate Linear Models with Missing Observations

Hilmar Drygas

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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

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

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 62J05: Linear regression

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


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.

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