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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{A}}} = {\left( {y,X_{1} \beta _{1} + X_{2} \beta _{2} ,\sigma ^{2} V} \right)} $$\end{document} and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ2V, where σ2 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 M2X1β1under the model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{A}}} $$\end{document} and its best linear unbiased estimators under the reduced linear models of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{A}}} $$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{A}}} $$\end{document} and those under its reduced linear models are established. Lastly, we also study the connections between the model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\user1{\mathcal{A}}} $$\end{document} and its linear transformation model.