On the Closed Form of Optimal Invariant Quadratic Predictors in Finite Populations |
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Authors: | Jian-Ying Rong Xu-Qing Liu |
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Affiliation: | 1. Department of Foundation Courses , Huai'an College of Information Technology , Huai'an, P.R. China rjy98@163.com;3. Faculty of Mathematics and Physics , Huaiyin Institute of Technology , Huai'an, P.R. China |
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Abstract: | 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. |
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Keywords: | Closed form Finite population Modified Lagrange multipliers function Optimal invariant quadratic unbiased predictor Optimal invariant quadratic (potentially) biased predictor Permutation matrix Symmetrizer matrix |
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