Further properties of linear prediction sufficiency and the BLUPs in the linear model with new observations
Augustyn Markiewicz, Simo Puntanen · Afrika Statistika · 2013
A linear statistic $\\mathrm{Fy}$ is called linearly sufficient for the estimable parametric function of $\\mathrm X_{*} \\beta$ under the linear model $\\mathscr M = \\{ \\mathrm y, \\mathrm X \\beta, \\mathrm V \\}$ if there exists a matrix $\\mathrm A$ such that $\\mathrm {AFy}$ is the best linear unbiased estimator, BLUE, for $\\mathrm X_{*} \\beta$. 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 $\\mathrm{Fy}$ with respect to $\\mathrm{y}_{*}$, $\\mathrm X_{*} \\beta$, and $\\varepsilon_{*}$, when the random vector $\\mathrm{y}_{*}$ comes from $\\mathrm{y}_{*} = \\mathrm X_{*} \\beta + \\varepsilon_{*}$, and the prediction is based on the linear model $\\mathscr M$. 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 $\\varepsilon_{*}$. We also apply our results into the linear mixed model.