A new derivation of BLUPs under random-effects model

This paper considers predictions of vectors of parameters under a general linear model $$\mathbf{y}= \mathbf{X}{\pmb {\beta }}+ {\pmb {\varepsilon }}$$ y = X β + ε with the random coefficients $${\pmb {\beta }}$$ β satisfying $${\pmb {\beta }}=\mathbf{A}{\pmb {\alpha }}+ {\pmb {\gamma }}$$ β = A α + γ . It utilizes a standard method of solving constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and obtains the best linear unbiased predictor (BLUP) of given vector $$\mathbf{F}{\pmb {\alpha }}+ \mathbf{G}\varvec{\gamma } + \mathbf{H}{\pmb {\varepsilon }}$$ F α + G γ + H ε of the unknown parameters in the model. Some special cases of the BLUPs are also presented. In particular, a general decomposition equality $$\mathbf{y}= \mathrm{BLUE}(\mathbf{X}\mathbf{A}{\pmb {\alpha }}) + \mathrm{BLUP}(\mathbf{X}{\pmb {\gamma }}) + \mathrm{BLUP}({\pmb {\varepsilon }})$$ y = BLUE ( X A α ) + BLUP ( X γ ) + BLUP ( ε ) is proved under the random-effects model. A further problem on BLUPs of new observations under the random-effects model is also addressed. Copyright Springer-Verlag Berlin Heidelberg 2015