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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.

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