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Equalities between OLSE, BLUE and BLUP in the linear model

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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.

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Acknowledgments

Sincere thanks go to two anonymous referees whose comments greatly improved the manuscript. The work of Y. Liu was supported by NSFC under grant 11271259.

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Authors and Affiliations

  1. Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

    Stephen J. Haslett

  2. Department of Forest Sciences, University of Helsinki, P.O. Box 27, 00014 , Helsinki, Finland

    Jarkko Isotalo

  3. Department of Applied Mathematics, Shanghai Finance University, Shanghai, 201209, China

    Yonghui Liu

  4. School of Information Sciences, University of Tampere, 33014 , Tampere, Finland

    Simo Puntanen

Authors
  1. Stephen J. Haslett
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  2. Jarkko Isotalo
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  3. Yonghui Liu
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  4. Simo Puntanen
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Correspondence to Simo Puntanen.

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Haslett, S.J., Isotalo, J., Liu, Y. et al. Equalities between OLSE, BLUE and BLUP in the linear model. Stat Papers 55, 543–561 (2014). https://doi.org/10.1007/s00362-013-0500-7

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  • DOI: https://doi.org/10.1007/s00362-013-0500-7

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