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

Abstract

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. (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 BLUEs and/or BLUPs under one mixed model continue to be BLUEs and/or BLUPs under the other mixed model. In particular, we describe the mutual relations of the conditions. These problems were approached differently by Rong & Liu (2010) and we will show how their results are related to those obtained by our approach.

Equalities between the BLUEs and BLUPs under the partitioned linear fixed model and the corresponding mixed model

“…This follows from the requirement [3,4]. For the general reviews of the BLUPproperties, see, e.g., Tian [23,24].…”

“…) is one solution to the equation A(X : VM) = (X : 0); recall that µ = Xβ is always estimable in F . The matrix G is unique for the choice of generalized inverses marked as " − " but to obtain uniqueness for G (which somewhat simplifies our considerations) we have to choose the Moore-Penrose inverse W + in the end of the expression (4).…”

“…It appears that the concept of the linear model with new future observations is a powerful tool for these considerations. For further references we may mention [15], [22], [25], and [4].…”