The general mixed linear model can be written as y = Xβ + Zu + e. In this paper, we mainly deal with two problems. Firstly, the problem of predicting a general linear combination of fixed effects and realized values of random effects in a general mixed linear model is considered and an explicit representation of the best linear unbiased predictor (BLUP) is derived. In addition, we apply the resulting conclusion to several special models and offer an alternative to characterization of BLUP. Secondly, we recall the notion of linear sufficiency and consider it as regards the BLUP problem and characterize it in several different ways. Further, we study the concepts of linear sufficiency, linear minimal sufficiency and linear completeness, and give relations among them. Finally, four concluding remarks are given.

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Mixed linear model, True model, Misspecified model, Dispersion, Random regression coefficient model, Compound symmetric, 62H12, 62F10, 62M20, 15A09,

a b s t r a c tLinear 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, σ 2 V), to a multivariate linear model, denoted by. . , y q ). 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.

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