LAUSR

In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$ F y for $$\mathbf {X}\varvec{\beta }$$ X β , for $$\mathbf {Z}\mathbf {u}$$ Z u and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$ X β + Z u , when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$ y = X β + Z u + e . In particular, we explore the relations between these sufficiency properties. The usual definition of linear sufficiency means, for example, that the $${{\mathrm{BLUE}}}$$ BLUE of $$\mathbf {X}\varvec{\beta }$$ X β under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$ A F y for some matrix $$\mathbf {A}$$ 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 $${{\mathrm{BLUE}}}$$ BLUE s and/or $${{\mathrm{BLUP}}}$$ BLUP s under one mixed model continue to be $${{\mathrm{BLUE}}}$$ BLUE s and/or $${{\mathrm{BLUP}}}$$ 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.