Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series A, Pt. 1, pp.118--127

THE EQUALITY OF LINEAR TRANSFORMS OF THE ORDINARY LEAST SQUARES ESTIMATOR AND THE BEST LINEAR UNBIASED ESTIMATOR

By

JÜRGEN GROSS & GÖTZ TRENKLER, University of Dortmund, Dortmund, Germany

and

HANS JOACHIM WERNER, University of Bonn, Bonn, Germany

SUMMARY. We consider the equality of linear transforms of the ordinary least squares estimator (OLSE) and the {\it traditional} best linear unbiased estimator (BLUE) of Xb in the Gauss-Markov linear model l:= {y, Xb ,V}, where y is an observable random vector with expectation vector e (y)=Xb and dispersion matrix 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 COLSE(Xb )=CBLUE(Xb ). (2) For given X and C, the set of all those dispersion matrices V with COLSE(Xb )=CBLUE(Xb ). (3) For given X, V and C, the event of all appropiate (consistent) realizations of y under l on which COLSE(Xb coincides with CBLUE(Xb . Some special cases are also considered.

AMS (1991) subject classification. Primary 62J05.

Key words and phrases. Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), traditional BLUE, wider definition BLUE, BLUE conditions, linearly unbiasedly estimable function, natural restrictions, general Gauss-Markov model, generalized inverse (g-inverse), orthogonal projector.

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