A new analysis of optimal estimation and prediction under linear mixed models

A linear statistical model including both fixed but unknown parameters and random unknown parameters is called a linear mixed model. The aim of this paper is to provide a unified study on a series of fundamental and important optimal estimation and prediction problems in the contexts of linear mixed models and their transformed models. We shall establish a mathematical procedure for solving some optimal estimation and prediction problems on a given linear mixed model and its transformed models using some precise analytical tools in matrix theory. The coverage includes constructing a general vector composed of all unknown parameters in the context of a linear mixed model and its transformed models, defining the best linear unbiased predictors of the vector, deriving the analytical expressions of the best linear unbiased predictors, and discussing a variety of theoretical performances and properties of best linear unbiased predictors. As extensions, we discuss the derivation of best linear unbiased predictors of future observations under a linear mixed model.