A study of the equivalence of the BLUEs between a partitioned singular linear model and its reduced singular linear models

Consider the partitioned linear regression model A = (y,X(1)beta(1) + X(2)beta(2),sigma(2)V) and its four reduced linear models, where y is an n x 1 observable random vector with E(y) = Xbeta and dispersion matrix Var(y) = sigma(2)V, where sigma(2) is an unknown positive scalar, V is an n x n known symmetric nonnegative definite matrix, X = (X-1 : X-2) is an nx(p+q) known design matrix with rank(X) = r less than or equal to (p+q), and beta = (beta' (1): beta'(2) )' with beta(1) and beta(2) being px1 and qx1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M(2)X(1)beta(1)under the model A and its best linear unbiased estimators under the reduced linear models of A are given, where M-2 = I -X2X2+ . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M(2)X(1)beta(1) under the model A and those under its reduced linear models are established. Lastly, we also study the connections between the model A and its linear transformation model.