Quadratic prediction problems in multivariate linear models

Linear and quadratic prediction problems in finite populations have become of great interest to many authors recently. In the present paper, we mainly aim to extend the problem of quadratic prediction from a general linear model, of form y=Xβ+e,e∼N(0,σ2V), to a multivariate linear model, denoted by Y=XB+E,V ec(E)∼N(0,Σ⊗V) with Y=(yij)n×q=(y1,…,yq). Firstly, the optimal invariant quadratic unbiased (OIQU) predictor and the optimal invariant quadratic (potentially) biased (OIQB) predictor of Y′HY for any particular symmetric nonnegative definite matrix H satisfying HX=0 are derived. Secondly, we consider predicting a′Y′HYb and tr(Y′HY). The corresponding restricted OIQU predictor and restricted OIQB predictor for them are given. In addition, we also offer four concluding remarks. One concerns the generalization of predicting a′Y′HYb and tr(Y′HY), and the others are concerned with three possible extensions from multivariate linear models to growth curve models, to restricted multivariate linear models, and to matrix elliptical linear models.