A Study of the Equivalence of the BLUEs between a Partitioned Singular Linear Model and Its Reduced Singular Linear Models
BaoXueZHANG, BaiSenLIU, ChangYuLU · Acta Scientiarum Naturalium Universitatis Sunyatseni · 2004
Consider the partitioned linear regression model A = (y, X1β1 + X2β2, σ^2V) and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) =σ^2V, where σ^2 is an unknown positive scalar, V is an n × n known symmetric nonnegative definite matrix, X = (X1 : X2) is an n× (p+q) known design matrix with rank(X) = r ≤ (p+q),and β = (β'1 : β'2)' with β1 and β2 being p × 1 and q × 1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M2X1β1 under the model A and its best linear unbiased estimators under the reduced linear models of A are given,where M2 = I - X2X2^+. Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M2X1β1 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.