A Study of the Equivalence of the BLUEs between a Partitioned Singular Linear Model and Its Reduced Singular Linear Models

Abstract

Consider the partitioned linear regression model ${\user1{\mathcal{A}}} = {\left( {y,X_{1} \beta _{1} + X_{2} \beta _{2} ,\sigma ^{2} V} \right)}$ and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ² V, where σ² 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 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 M ₂ X ₁β₁under the model ${\user1{\mathcal{A}}}$ and its best linear unbiased estimators under the reduced linear models of ${\user1{\mathcal{A}}}$ 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 ₁β₁ under the model ${\user1{\mathcal{A}}}$ and those under its reduced linear models are established. Lastly, we also study the connections between the model ${\user1{\mathcal{A}}}$ and its linear transformation model.