Some properties of linear sufficiency and the BLUPs in the linear mixed model

Abstract

In this paper we consider the linear sufficiency of $\mathbf {F}\mathbf {y}$ for $\mathbf {X}\varvec{\beta }$, for $\mathbf {Z}\mathbf {u}$ and for $\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$, when dealing with the linear mixed model $\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the ${{\mathrm{BLUE}}}$ of $\mathbf {X}\varvec{\beta }$ under the original model can be obtained as $\mathbf {A}\mathbf {F}\mathbf {y}$ for some matrix $\mathbf {A}$. Liu et al. (J Multivar Anal 99:1503–1517, 2008) introduced a slightly different definition for the linear sufficiency and we study its relation to the standard definition. We also consider the conditions under which ${{\mathrm{BLUE}}}$s and/or ${{\mathrm{BLUP}}}$s under one mixed model continue to be ${{\mathrm{BLUE}}}$s and/or ${{\mathrm{BLUP}}}$s under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong and Liu (Stat Pap 51:445–453, 2010) and we will show how their results are related to those obtained by our approach.