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Equivalence of predictors under real and over-parameterized linear models. (English) Zbl 1369.62118
Summary: Assume that a real linear model \(\mathbf{y}=\mathbf{X\beta}+\mathbf{\varepsilon}\) is over-parameterized as \(\mathbf{y}=\mathbf{X\beta}+\mathbf{Z\gamma}+\mathbf{\varepsilon}\) by adding some new regressors \(\mathbf{Z\gamma}\). In such a case, results of statistical inferences of the unknown parameters \(\mathbf{\beta}\) and \(\mathbf{\varepsilon}\) under the two models are not necessarily the same. This paper aims at characterizing relationships between the best linear unbiased predictors (BLUPs) of the joint vector \(\mathbf{\phi}=\mathbf{K\beta}+\mathbf{J\varepsilon}\) of the unknown parameters in the two models. In particular, we derive necessary and sufficient conditions for the BLUPs of \(\mathbf{\phi}\) to be equivalent under the real model and its over-parameterized counterpart.

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
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