Properties of the matrix V + XTX' in linear statistical models

It is well known, due originally to C.R. Rao in early 1970s, that the best linear unbiased estimator, BLUE, of X— in the linear model M = {y, X—, V} can be expressed in the form X (XÕW≠X)≠XÕW≠y,whereW is a specific matrix of the form W = V+XTXÕ withTsatisfying the column space condition C(W)=C(X : V). Wedenote this class of matrices as W. Choice of T as an identity matrix gives an obvious member W = V + XXÕ œ W. The matrices belonging to the class W have several interesting mathematical properties. In particular, the use of matrix W œ W appears to be surprisingly handy and helpful tool when dealing with the linear statistical models. Our aim is to review and collect together some essential features of W and its use in linear statistical models. While doing this, we go through some related basic properties of the best linear unbiased estimation.