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Equalities between OLSE, BLUE and BLUP in the linear model. (English) Zbl 1334.62110
Summary: We consider equalities between the ordinary least squares estimator (OLSE), the best linear unbiased estimator (BLUE) and the best linear unbiased predictor (BLUP) in the general linear model \(\{\mathbf y,\mathbf X\boldsymbol\beta,\mathbf V\}\) extended with the new unobservable future value \(\mathbf y_\ast\) of the response whose expectation is \(\mathbf X_\ast\boldsymbol\beta\). 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 BLUP and provide new results giving upper bounds for the Euclidean norm of the difference between the \(\mathrm{BLUP}(\mathbf y_\ast)\) and \(\mathrm{BLUE}(\mathbf X_\ast\boldsymbol\beta)\) and between the \(\mathrm{BLUP}(\mathbf y_\ast)\) and \(\mathrm{OLSE}(\mathbf X_\ast\mathbf\beta)\). A remark is made on the application to small area estimation.

62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
62H20 Measures of association (correlation, canonical correlation, etc.)
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