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Equalities between OLSE, BLUE and BLUP in the linear model

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  • Stephen Haslett
  • Jarkko Isotalo
  • Yonghui Liu
  • Simo Puntanen

Abstract

We consider equalities between the ordinary least squares estimator ( $$\mathrm {OLSE} $$ ), the best linear unbiased estimator ( $$\mathrm {BLUE} $$ ) and the best linear unbiased predictor ( $$\mathrm {BLUP} $$ ) in the general linear model $$\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$$ extended with the new unobservable future value $$ \mathbf y _{*}$$ of the response whose expectation is $$ \mathbf X _{*}\varvec{\beta }$$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $$\mathrm {BLUP} $$ and provide new results giving upper bounds for the Euclidean norm of the difference between the $$\mathrm {BLUP} ( \mathbf y _{*})$$ and $$\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$$ and between the $$\mathrm {BLUP} ( \mathbf y _{*})$$ and $$\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$$ . A remark is made on the application to small area estimation. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:stpapr:v:55:y:2014:i:2:p:543-561
    DOI: 10.1007/s00362-013-0500-7
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    Cited by:

    1. Nesrin Güler & Melek Eriş Büyükkaya & Melike Yiğit, 2022. "Comparison of Covariance Matrices of Predictors in Seemingly Unrelated Regression Models," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(3), pages 801-809, September.
    2. Changli Lu & Yuqin Sun & Yongge Tian, 2018. "Two competing linear random-effects models and their connections," Statistical Papers, Springer, vol. 59(3), pages 1101-1115, September.
    3. Yongge Tian & Wenxing Guo, 2016. "On comparison of dispersion matrices of estimators under a constrained linear model," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(4), pages 623-649, November.
    4. Tian, Yongge & Zhang, Xuan, 2016. "On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 105-112.
    5. 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.
    6. Bo Jiang & Yuqin Sun, 2019. "On the equality of estimators under a general partitioned linear model with parameter restrictions," Statistical Papers, Springer, vol. 60(1), pages 273-292, February.

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