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Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

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  • Maruyama, Yazo
  • Takemura, Akimichi
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    Abstract

    We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular, we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 99 (2008)
    Issue (Month): 1 (January)
    Pages: 50-73

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    Handle: RePEc:eee:jmvana:v:99:y:2008:i:1:p:50-73

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    Keywords: Admissibility Spherically symmetric distribution Minimaxity Regularly varying function Harmonic prior;

    References

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    1. Brandwein, Ann Cohen, 1979. "Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 9(4), pages 579-588, December.
    2. Bock, M. E., 1985. "Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 17(2), pages 127-147, October.
    3. Maruyama, Yuzo, 2004. "Stein's idea and minimax admissible estimation of a multivariate normal mean," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 88(2), pages 320-334, February.
    4. Maruyama, Yuzo, 1998. "A Unified and Broadened Class of Admissible Minimax Estimators of a Multivariate Normal Mean," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 64(2), pages 196-205, February.
    5. Maruyama, Yuzo, 2003. "Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 84(2), pages 274-283, February.
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    Cited by:
    1. Maruyama, Yuzo, 2009. "An admissibility proof using an adaptive sequence of smoother proper priors approaching the target improper prior," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 100(8), pages 1845-1853, September.

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