On properties of BLUEs under general linear regression models

The best linear unbiased estimator (BLUE) of parametric functions of the regression coefficients under a general linear model M={y,Xβ,σ2Σ}M={y,Xβ,σ2Σ} can be written as GyGy, where GG is the solution of a consistent linear matrix equation composed by the given matrices in the model and their generalized inverses. In the past several years, a useful tool—the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use this algebraic method to give a comprehensive investigation to various algebraic and statistical properties of the projection matrix GG in the BLUE of parametric functions under MM. These properties include the uniqueness of GG, the maximal and minimal possible ranks of GG and Cov(Gy)Cov(Gy), as well as identifying conditions for various equalities for GG. In addition, necessary and sufficient conditions were established for equalities of projection matrices in the BLUEs of parametric functions under the original model and its transformed models.