Generalized least-squares estimator for known covariance

Let $X\in {\mathbb{R}}^{n\times p}$ have full column rank in a fixed-design first-moment regression model, so ${\mathbb{E}}_{\beta }Y=X\beta$ under each law $P_{\beta }$ (Fixed-design linear regression model). Suppose the same known symmetric positive-definite matrix $V$ is the covariance of $Y$ under every $P_{\beta }$. Put $C=(X^T{V}^{-1}X{)}^{-1}\text{and}{\widehat{\beta }}_{\mathrm{G}\mathrm{L}\mathrm{S}}:=C{X}^T{V}^{-1}Y\mathrm{.}$ This is a statistic and estimator of the identity target $\beta$ under Statistics and estimators and Estimands on a statistical model: for fixed $X$ and $V$, the displayed expression is a linear, hence Borel-measurable, function of the response. The positive definiteness of $X^T{V}^{-1}X$ and existence of its inverse are established by The GLS information matrix is positive definite ↗.

This definition uses the true covariance as known. Replacing $V$ by an estimate defines a plug-in procedure; the theorem below does not assign that procedure the same exact covariance or BLUE guarantee.

The marginal-covariance lemma later identifies $V=ZG{Z}^T+R$ for the mixed model. The GLS definition itself applies to any fixed-design first-moment model with the stated common covariance.