Projectors, Generalized Inverses and the Blue'S
C. Radhakrishna Rao · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 1974
Summary It is well known that in the Gauss–Markov model (Y, Xβ, σ2 V) 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-1 y. 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).