Equalities between OLSE, BLUE and BLUP in the linear model
We consider equalities between the ordinary least squares estimator (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathrm {OLSE} $$</EquationSource> </InlineEquation>), the best linear unbiased estimator (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathrm {BLUE} $$</EquationSource> </InlineEquation>) and the best linear unbiased predictor (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\mathrm {BLUP} $$</EquationSource> </InlineEquation>) in the general linear model <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$$</EquationSource> </InlineEquation> extended with the new unobservable future value <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$ \mathbf y _{*}$$</EquationSource> </InlineEquation> of the response whose expectation is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$ \mathbf X _{*}\varvec{\beta }$$</EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\mathrm {BLUP} $$</EquationSource> </InlineEquation> and provide new results giving upper bounds for the Euclidean norm of the difference between the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\mathrm {BLUP} ( \mathbf y _{*})$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$$</EquationSource> </InlineEquation> and between the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$\mathrm {BLUP} ( \mathbf y _{*})$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$$</EquationSource> </InlineEquation>. A remark is made on the application to small area estimation. Copyright Springer-Verlag Berlin Heidelberg 2014
