On the Closed Form of Optimal Invariant Quadratic Predictors in Finite Populations

For a finite population and its linear model Y  =  X β +  e , the problem of deriving optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor (OIQUP) and optimal invariant quadratic (potentially) biased predictor (OIQBP) for the population quadratic quantities, Y ′ HY , is of interest and has been previously considered by Liu and Rong (2007 Liu , X. , Rong , J. ( 2007 ). Quadratic prediction problems in finite populations . Statist. Probab. Lett. 77 : 483 – 489 .[Crossref], [Web of Science ®] , [Google Scholar]). In this note, we mainly aim at motivating the problems of OIQUP and OIQBP by showing that the unique closed form of OIQUP and OIQBP is just the one given by Liu and Rong through permutation matrix techniques.