A matrix handling of predictions of new observations under a general random-effects model

Assume that a general linear random-effects model $\by = \bX\bbe + \bve$ is given, and new observations in the future follow the linear model $\by_{\!f} = \bX_{\!f}\bbe + \bve_{\!f}$. This paper shows how to establish all possible best linear unbiased predictors (BLUPs) under the general linear random-effects model with original and new observations from the original observation vector $\by$ under a most general assumption on the covariance matrix among the random vectors $\bbe$, $\bve$ and $\bve_{\!f}$. It utilizes a standard method of solving optimization problem in the L\"owner partial ordering on a constrained quadratic matrix-valued function, and obtains analytical expressions of the BLUPs, including those for $\by_{\!f}$, $\bX_{\!f}\bbe$ and $\bve_{\!f}$. In particular, some fundamental equalities for the BLUPs are established under the linear random-effects model.