Equalities between OLSE, BLUE and BLUP in the linear model

Stephen Haslett (), Jarkko Isotalo (), Yonghui Liu () and Simo Puntanen ()

Statistical Papers, 2014, vol. 55, issue 2, 543-561

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

Keywords: Best linear unbiased estimator; Best linear unbiased predictor; Gauss–Markov model; Generalized inverse; Linear model; Ordinary least squares; 15A42; 62J05; 62H12; 62H20 (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s00362-013-0500-7

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