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

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 $\sigma^2I$.