Unified theory of least squares

Let (Y, Xβ, σ2I) where E(Y)=Xβ and D(Y) = E(Y→Xβ)'=σ2G, be the Gauss-Markoff model, where A' denotes the transpose of the matrix A. Further let β^ be astationary point (supposed to exist for all Y) of Y - Xβ)' M(Y-Xβ); i.e., where its derivative with respect to β is the zero vector. It is shown that if β^; is the BLUE of p'β for every P∈S(X'), the linear space generated by the columns of X', and an unbiased estimator of σ2 is ƒ-1(Y-Xβ^)' M(Y-Xβ^) f=R(G:X)-R(X), where R(V) denotes the rank of V, then it is necessary and sufficient that M is a symmetric g-inverse of (G+X∪X') where U is any summarice matrix such that S(G:X) = S(G + X∪X'). The method is valid whether G is singular or not and R(X) is full or not. A simple choice of U is always U=k2I, K¬0