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

In this paper we consider the linear sufficiency of Fy for Xβ, for Zu and for Xβ+ Zu, when dealing with the linear mixed model y= Xβ+ Zu+ e. In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the BLUE of Xβ under the original model can be obtained as AFy for some matrix 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 BLUE s and/or BLUP s under one mixed model continue to be BLUE s and/or 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.