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Nonnegative quadratic estimation and quadratic sufficiency in general linear models

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  • Liu, Xu-qing
  • Rong, Jian-ying

Abstract

Notions of linear sufficiency and quadratic sufficiency are of interest to some authors. In this paper, the problem of nonnegative quadratic estimation for [beta]'H[beta]+h[sigma]2 is discussed in a general linear model and its transformed model. The notion of quadratic sufficiency is considered in the sense of generality, and the corresponding necessary and sufficient conditions for the transformation to be quadratically sufficient are investigated. As a direct consequence, the result on (ordinary) quadratic sufficiency is obtained. In addition, we pose a practical problem and extend a special situation to the multivariate case. Moreover, a simulated example is conducted, and applications to a model with compound symmetric covariance matrix are given. Finally, we derive a remark which indicates that our main results could be extended further to the quasi-normal case.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:6:p:1180-1194
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    References listed on IDEAS

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    1. Heiligers, Berthold & Markiewicz, Augustyn, 1996. "Linear sufficiency and admissibility in restricted linear models," Statistics & Probability Letters, Elsevier, vol. 30(2), pages 105-111, October.
    2. Gnot, Stanislaw & Grzadziel, Mariusz, 2002. "Nonnegative Minimum Biased Quadratic Estimation in Mixed Linear Models," Journal of Multivariate Analysis, Elsevier, vol. 80(2), pages 217-233, February.
    3. Drygas, Hilmar, 1985. "Linear sufficiency and some applications in multilinear estimation," Journal of Multivariate Analysis, Elsevier, vol. 16(1), pages 71-84, February.
    4. Gnot, S. & Trenkler, G. & Zmyslony, R., 1995. "Nonnegative Minimum Biased Quadratic Estimation in the Linear Regression Models," Journal of Multivariate Analysis, Elsevier, vol. 54(1), pages 113-125, July.
    5. Mueller, Jochen, 1987. "Sufficiency and completeness in the linear model," Journal of Multivariate Analysis, Elsevier, vol. 21(2), pages 312-323, April.
    6. 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.
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    Cited by:

    1. 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.
    2. 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.

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