Necessary and Sufficient Conditions for the Least Squares Estimator to Be the Best Linear Unbiased Estimator in the General Gauss- Markov Model

Abstract:

      In this paper, we discuss the decomposition of the space μ(X : V) and the invariance with respect to the choice of a generalized inverse of matrix X in the general Gauss-Markov model. In Theorem 1, we give necessary and sufficient conditions for the least squares estimator P_xy = BLUE(Xβ) under the general Gauss-Markov model M = {y,Xβ,σ²V}. In Theorem 2, we prove that P_xy= BLUE(Xβ) under model M and invariant with respect to the choice of a generalized inverse of matrix X are equivalent.