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
*X*b in the Gauss-Markov linear model * l*:= {*y*,* X*b ,*V*}, where *y* is an observable random vector with expectation vector
e (y)=*X*b 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 *C*OLSE(Xb )=*C*BLUE(Xb ). (2)
For given *X* and *C*, the set of all those dispersion matrices
* V* with *C*OLSE(Xb )=*C*BLUE(Xb ). (3) For given *X*, *V*
and *C*, the event of all *appropiate* (consistent) realizations
of *y* under *l* on which *C*OLSE(Xb coincides with
*C*BLUE(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.