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

S. J. Haslett (), X. Q. Liu (), A. Markiewicz () and S. Puntanen ()
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S. J. Haslett: The Australian National University
X. Q. Liu: Nanjing University of Aeronautics and Astronautics
A. Markiewicz: Poznań University of Life Sciences
S. Puntanen: University of Tampere

Statistical Papers, 2020, vol. 61, issue 1, No 20, 385-401

Abstract: Abstract In this paper we consider the linear sufficiency of $$\mathbf {F}\mathbf {y}$$Fy for $$\mathbf {X}\varvec{\beta }$$Xβ, for $$\mathbf {Z}\mathbf {u}$$Zu and for $$\mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}$$Xβ+Zu, when dealing with the linear mixed model $$\mathbf {y}= \mathbf {X}\varvec{\beta }+ \mathbf {Z}\mathbf {u}+ \mathbf {e}$$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 $${{\mathrm{BLUE}}}$$BLUE of $$\mathbf {X}\varvec{\beta }$$Xβ under the original model can be obtained as $$\mathbf {A}\mathbf {F}\mathbf {y}$$AFy 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 $${{\mathrm{BLUE}}}$$BLUEs and/or $${{\mathrm{BLUP}}}$$BLUPs under one mixed model continue to be $${{\mathrm{BLUE}}}$$BLUEs and/or $${{\mathrm{BLUP}}}$$BLUPs 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.

Keywords: Best linear unbiased estimator; Best linear unbiased predictor; Linear mixed model; Linear model; Misspecified model; 62J05; 62J10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s00362-017-0943-3

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