A general prediction analysis to linear random-effects models with restrictions and new observations

This paper presents a unified approach to the problem of best linear unbiased prediction (BLUP) of a joint vector of all unknown parameters in a general linear random-effects model (LRM) with restrictions and new observations via some state-of-the-art tools in matrix mathematics. We first establish the fundamental matrix equation and the exact algebraic expression for calculating the Best Linear Unbiased Predictor (BLUP) of a joint vector of all unknown parameters in the LRM by directly solving a constrained quadratic matrix-valued function optimization problem. We then present a variety of special cases and uniform decompositions of the BLUP, as well as many algebraic and statistic properties of the BLUP. In particular, we show how to use various matrix rank/inertia formulas and matrix tricks in establishing and simplifying various complicated matrix expressions related to the covariances matrices of BLUPs, and to derive necessary and sufficient conditions for equalities and equalities of covariance matrices of BLUPs to hold. The whole work in this paper provides a precise algebraic study to LRMs, and can be utilized as standard tools in statistical analysis and inference of various types of LRM.