An elementary development of the equation characterizing best linear unbiased estimators

Abstract

Puntanen et al. [J. Statist. Plann. Inference 88 (2000) 173] provided two matrix-based proofs of the result stating that a linear estimator By represents the best linear unbiased estimator (BLUE) of the expectation vector X β under the general Gauss–Markov model M={ y , X β,σ 2 V } if and only if B ( X : VX ⊥)=( X : 0 ) , where X ⊥ is any matrix whose columns span the orthogonal complement to the column space of X. In this note, still another development of such a characterization is proposed with reference to the BLUE of any vector of estimable parametric functions K β . From the algebraic point of view, the present development seems to be the simplest from among all accessible in the literature till now.