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Quadratic prediction and quadratic sufficiency in finite populations

Author

Listed:
  • Liu, Xu-Qing
  • Wang, Dong-Dong
  • Rong, Jian-Ying

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.

Suggested Citation

  • Liu, Xu-Qing & Wang, Dong-Dong & Rong, Jian-Ying, 2009. "Quadratic prediction and quadratic sufficiency in finite populations," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1979-1988, October.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:9:p:1979-1988
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    References listed on IDEAS

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    1. Drygas, Hilmar, 1985. "Linear sufficiency and some applications in multilinear estimation," Journal of Multivariate Analysis, Elsevier, vol. 16(1), pages 71-84, February.
    2. Liu, Xu-Qing & Rong, Jian-Ying & Liu, Xiu-Ying, 2008. "Best linear unbiased prediction for linear combinations in general mixed linear models," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1503-1517, September.
    3. Mueller, Jochen, 1987. "Sufficiency and completeness in the linear model," Journal of Multivariate Analysis, Elsevier, vol. 21(2), pages 312-323, April.
    4. Markiewicz, Augustyn, 1998. "Comparison of linear restricted models with respect to the validity of admissible and linearly sufficient estimators," Statistics & Probability Letters, Elsevier, vol. 38(4), pages 347-354, July.
    5. Liu, Xu-Qing & Wang, Dong-Dong & Rong, Jian-Ying, 2009. "Quadratic prediction problems in multivariate linear models," Journal of Multivariate Analysis, Elsevier, vol. 100(2), pages 291-300, February.
    6. Heiligers, Berthold & Markiewicz, Augustyn, 1996. "Linear sufficiency and admissibility in restricted linear models," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 105-111, October.
    7. Liu, Xu-Qing & Rong, Jian-Ying, 2007. "Quadratic prediction problems in finite populations," Statistics & Probability Letters, Elsevier, vol. 77(5), pages 483-489, March.
    8. Liu, Xu-qing & Rong, Jian-ying, 2007. "Nonnegative quadratic estimation and quadratic sufficiency in general linear models," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1180-1194, July.
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