Projectors, generalized inverses and the BLUE's

Abstract

It is well known that in the Gauss-Markov model (Y, Xβ, σ²V) with |V| ≠ 0, the BLUE (best linear unbiased estimator) of Xβ is Y₁, the orthogonal projection of Y on M(X), the space spanned by the columns of X, with inner product defined as (x, y)=x'V⁻¹y. A quadratic function of Y₂, the projection of Y on the orthogonal complement of M(X), provides an estimate of σ². It may be seen that Y=Y₁+Y₂. When V is singular, the inner product definition as in non-singular case is not possible. In this paper a suitable theory of projection operators is developed for the case |V|=0, and a decomposition Y=Y₁+Y₂ is obtained such that Y₁ is the BLUE of Xβ and a quadratic function of Y₂ is the MINQUE (Minimum Norm Quadratic Unbiased Estimator) of σ² in the sense of Rao (1972).