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 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 and its best linear unbiased estimators under the reduced linear models of 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 and those under its reduced linear models are established. Lastly, we also study the connections between the model and its linear transformation model. *This work is supported by the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 10226024), Postdoctoral Foundation of China and Lab. of Math. for Nonlinear Sciences at Fudan University. This research is supported in part by The International Organizing Committee and The Local Organizing Committee at the University of Tampere for this Workshop **The work is supported in part by an NSF grant of China. Results in this paper were presented by the first author at The Eighth International Workshop on Matrices and Statistics: Tampere, Finland, August 1999