Linear sufficiency and completeness in the context of estimating the parametric function in the general Gauss–Markov model

In this paper we consider linear sufficiency and linear completeness in the context of estimating the estimable parametric function K ′ β under the general Gauss–Markov model { y , X β , σ 2 V } . We give new characterizations for linear sufficiency, and define and characterize linear completeness in a case of estimation of K ′ β . Also, we consider a predictive approach for obtaining the best linear unbiased estimator of K ′ β , and subsequently, we give the linear analogues of the Rao–Blackwell and Lehmann–Scheffé Theorems in the context of estimating K ′ β .