Quadratic prediction problems in finite populations

In this paper, we investigate quadratic prediction problems of f(Y)=Y′HYf(Y)=Y′HY with HH satisfying HX=0HX=0 in finite populations under the linear model Y=Xβ+eY=Xβ+e, e∼N(0,σ2V)e∼N(0,σ2V). We mainly aim at proposing two notions of optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor, and studying the structure of their explicit representations. In addition, we apply the main results of this paper to some special cases and finally derive a conclusion, which is not only mathematically interesting but also practically significant.