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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.

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