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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
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