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The equality of linear transforms of the ordinary least squares estimator and the best linear unbiased estimator. (English) Zbl 1004.62056
Summary: We consider the equality of linear transforms of the ordinary least squares estimator (OLSE) and the traditional best linear unbiased estimator (BLUE) of $$X\beta$$ in the Gauss-Markov linear model $${\mathfrak L}:= \{y,X \beta,V\}$$, where $$y$$ is an observable random vector with expectation vector $${\mathcal E}(y)=X\beta$$ and dispersion matrix $${\mathcal D}(y)=V$$. Of much interest to us are explicit parametric representations of the following three sets:
(1) For given $$X$$ and $$V$$, the set of all those matrices $$C$$ with $$C$$ OLSE$$(X\beta)= C$$ BLUE$$(X,\beta)$$. (2) For given $$X$$ and $$C$$, the set of all those dispersion matrices $$V$$ with $$C$$ Olse$$(X\beta) =C$$ BLUE$$(X\beta)$$. (3) For given $$X,V$$ and $$C$$, the event of all appropriate (consistent) realizations of $$y$$ under $${\mathfrak L}$$ on which $$C$$ OLSE$$(X,\beta)$$ coincides with $$C$$ BLUE$$(X,\beta)$$. Some special cases are also considered.

##### MSC:
 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models