The Annals of Mathematical Statistics

On Canonical Forms, Non-Negative Covariance Matrices and Best and Simple Least Squares Linear Estimators in Linear Models

George Zyskind

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Abstract

Aspects of best linear estimation are explored for the model $y = X\beta + e$ with arbitrary non-negative (possibly singular) covariance matrix $\sigma^2V$. Alternative necessary and sufficient conditions for all simple least squares estimators to be also best linear unbiased estimators (blue's) are presented. Further, it is shown that a linear function $w'y$ is blue for its expectation if and only if $Vw \epsilon \mathscr{C} (X)$, the column space of $X$. Conditions on the equality of subsets of blue's and simple least squares estimators are explored. Applications are made to the standard linear model with covariance matrix $\sigma^2I$ and with additional known and consistent equality constraints on the parameters. Formulae for blue's and their variances are presented in terms of adjustments to the corresponding expressions for the case of the unrestricted standard linear model with covariance matrix $\sigma^2I$.

Article information

Source
Ann. Math. Statist. Volume 38, Number 4 (1967), 1092-1109.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aoms/1177698779

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aoms/1177698779

Mathematical Reviews number (MathSciNet)
MR214237

Zentralblatt MATH identifier
0171.17103

Citation

Zyskind, George. On Canonical Forms, Non-Negative Covariance Matrices and Best and Simple Least Squares Linear Estimators in Linear Models. Ann. Math. Statist. 38 (1967), no. 4, 1092--1109. doi:10.1214/aoms/1177698779. http://projecteuclid.org/euclid.aoms/1177698779.


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