Further properties of linear prediction suciency and the BLUPs in the linear model with new observations

A linear statistic Fy is called linearly sufficient for the estimable parametric function of X*β under the linear model M = {y,Xβ,V} if there exists a matrix A such that AFy is the best linear unbiased estimator, BLUE, for X*β. The concept of linear sufficiency with respect to a predictable random vector is defined in the corresponding way but considering best linear unbiased predictor, BLUP, instead of BLUE. In this paper, we consider the linear sufficiency of Fy with respect to y*, X*β, and ∈*, when the random vector y* comes from y* = Xβ + ∈*, and the prediction is based on the linear model M. Our main results concern the mutual relations of these sufficiencies. In addition, we give an extensive review of some interesting properties of the covariance matrices of the BLUPs of ∈*. We also apply our results into the linear mixed model.Keywords: Best linear unbiased estimator; Best linear unbiased predictor; Linear sufficiency; Linear mixed model; Transformed linear mode