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A matrix handling of predictions of new observations under a general random-effects model. (English) Zbl 1329.62321
Summary: Linear regression models that include random effects are commonly used to analyze longitudinal and correlated data. Assume that a general linear random-effects model $$y=X\beta+\varepsilon$$ with $$\beta=A\alpha+\gamma$$ is given, and new observations in the future follow the linear model $$y_f=X_f\beta+\varepsilon_f$$. This paper shows how to establish a group of matrix equations and analytical formulas for calculating the best linear unbiased predictor (BLUP) of the vector $$\phi=F\alpha+G\gamma+H\varepsilon+H_f\varepsilon_f$$ of all unknown parameters in the two models under a general assumption on the covariance matrix among the random vectors $$\gamma$$, $$\varepsilon$$ and $$\varepsilon_f$$ via solving a constrained quadratic matrix-valued function optimization problem. Many consequences on the BLUPs of $$\phi$$ and their covariance matrices, as well as additive decomposition equalities of the BLUPs with respect to its components are established under various assumptions.

##### MSC:
 62J05 Linear regression; mixed models 15A09 Theory of matrix inversion and generalized inverses 62H12 Estimation in multivariate analysis
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