On quadratic prediction problems in finite populations

For a finite population and its linear model, Liu and Rong proposed a notion of optimal invariant quadratic unbiased prediction (OIQUP) and offered two methods for studying this notion, in which the first is incomplete. In this note, we mainly aim at fulfilling the first approach used by Liu and Rong by considering a transformed matrix equation set through permutation matrix techniques. Solvability of the matrix equation set, optimality of the resulting predictor, and equivalence of the representations of OIQUP, derived in this note and by Liu and Rong, are investigated in detail. In addition, an application to predicting population variance is conducted based on a simulated population.