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Quadratic prediction and quadratic sufficiency in finite populations
Authors:Xu-Qing Liu  Dong-Dong Wang  Jian-Ying Rong  
Institution:aDepartment of Computing Science, Huaiyin Institute of Technology, Huai’an 223003, PR China;bDepartment of Foundation Courses, Huai’an College of Information Technology, Huai’an 223003, PR China
Abstract:The problem of quadratic prediction for population quadratic quantities in finite populations has been considered in the literature. In this paper, we mainly aim at extending the ordinary quadratic prediction problems to a general case, and derive the representations of the two essentially unique optimal predictors: one is an optimal invariant quadratic unbiased predictor, and the other is an optimal invariant quadratic (potentially) biased predictor. Further, we show that the two predictors are nonnegative and reasonable by considering an extreme situation, and apply resulting conclusions to a special model with a compound symmetric variance matrix. In addition, we propose a notion of quadratic sufficiency with regard to the optimal prediction problems by employing materials derived in the first part, and investigate corresponding characterizations in detail.
Keywords:Finite population  Linear model  Nonnegativity  OIQBP  OIQUP  Optimal invariant quadratic (potentially) biased predictor  Optimal invariant quadratic unbiased predictor  Quadratic prediction  Quadratic sufficiency
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