Bayes linear estimator in the general linear model

The Bayes linear estimator is derived by minimizing the Bayes risk with respect to the squared loss function. Non-unbiased estimators such as ordinary ridge, typical shrinkage, fractional rank, and restricted least squares estimators, as well as classical linear unbiased estimators such as ordinary least squares and generalized least squares estimators, are either Bayes linear estimators or their limit points. In this paper, we discuss the statistical properties and optimality of Bayes linear estimators. First, we explore properties of Bayes linear estimators such as linear sufficiency and linear completeness. Second, we derive necessary and sufficient conditions under which two Bayes linear estimators coincide. In particular, several examples, including Rao's mixed-effects model and the general linear model with a spatial error process, demonstrate that our results can lead to a more efficient estimation procedure. Finally, we establish equivalence conditions for the equality of residual sums of squares when Bayes linear estimators are considered.