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 , X β , σ 2 Σ } can be written as Gy , where G 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 G in the BLUE of parametric functions under M . These properties include the uniqueness of G , the maximal and minimal possible ranks of G and Cov ( Gy ) , as well as identifying conditions for various equalities for G . 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.