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

Author

Listed:
  • S. J. Haslett

    (The Australian National University)

  • X. Q. Liu

    (Nanjing University of Aeronautics and Astronautics
    Huaiyin Institute of Technology)

  • A. Markiewicz

    (Poznań University of Life Sciences)

  • S. Puntanen

    (University of Tampere)

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.

Suggested Citation

  • S. J. Haslett & X. Q. Liu & A. Markiewicz & S. Puntanen, 2020. "Some properties of linear sufficiency and the BLUPs in the linear mixed model," Statistical Papers, Springer, vol. 61(1), pages 385-401, February.
    • Handle: RePEc:spr:stpapr:v:61:y:2020:i:1:d:10.1007_s00362-017-0943-3
    DOI: 10.1007/s00362-017-0943-3
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    References listed on IDEAS

    as
    1. Stephen Haslett & Simo Puntanen, 2011. "On the equality of the BLUPs under two linear mixed models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 381-395, November.
    2. Liu, Xin & Wang, Qing-Wen, 2013. "Equality of the BLUPs under the mixed linear model when random components and errors are correlated," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 297-309.
    3. Radosław Kala & Simo Puntanen & Yongge Tian, 2017. "Some notes on linear sufficiency," Statistical Papers, Springer, vol. 58(1), pages 1-17, March.
    4. Rao, C. Radhakrishna, 1973. "Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix," Journal of Multivariate Analysis, Elsevier, vol. 3(3), pages 276-292, September.
    5. Radosław Kala & Paweł Pordzik, 2009. "Estimation in singular partitioned, reduced or transformed linear models," Statistical Papers, Springer, vol. 50(3), pages 633-638, June.
    6. Stephen Haslett & Jarkko Isotalo & Yonghui Liu & Simo Puntanen, 2014. "Equalities between OLSE, BLUE and BLUP in the linear model," Statistical Papers, Springer, vol. 55(2), pages 543-561, May.
    7. Jian-Ying Rong & Xu-Qing Liu, 2010. "On misspecification of the dispersion matrix in mixed linear models," Statistical Papers, Springer, vol. 51(2), pages 445-453, June.
    8. Liu, Xu-Qing & Rong, Jian-Ying & Liu, Xiu-Ying, 2008. "Best linear unbiased prediction for linear combinations in general mixed linear models," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1503-1517, September.
    9. S. Haslett & S. Puntanen & B. Arendacká, 2015. "The link between the mixed and fixed linear models revisited," Statistical Papers, Springer, vol. 56(3), pages 849-861, August.
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