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,X11*X22,2V) and its four reduced linear models, where y is an n 1 observable random vector with E(y) = X and dispersion matrix Var(y) = 2 V, where 2 is an unknown positive scalar, V is an n n known symmetric nonnegative definite matrix, X = (X 1 : X 2) is an n(p*q) known design matrix with rank(X) = r (p*q), and = ( 1: 2 ) with 1 and 2 being p1 and q1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M 2 X 11under the model A and its best linear unbiased estimators under the reduced linear models of A are given, where M 2 = I -X 2 X 2 * . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M 2 X 11 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.