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

In this article we consider the general linear regression model A = （y,X₁β₁ + X₂β₂,σ²V） and its four reduced linear models, where V is known nonnegative definite and X = （X₁ : X₂） can be rank-deficient. The formulae for the differences between the Minimum Norm Quadratic Unbiased Estimators （MINQUEs） of σ² under the model A and its MINQUEs under reduced linear models of A are given. Further, the necessary and sufficient conditions for the equalities between the MINQUEs of σ² under A and its reduced linear models are established,