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Minimaxity in Predictive Density Estimation with Parametric Constraints

Author

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  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)

  • Éric Marchand

    (Département de mathématiques, Université de Sherbrooke,)

  • William E. Strawderman

    (Department of Statistics and Biostatistics, Rutgers University,)

  • Jean-Philippe Turcotte

    (Département de mathématiques, Université de Sherbrooke,)

Abstract

This paper is concerned with estimation of a predictive density with parametric constraints under Kullback-Leibler loss. When an invariance structure is embed- ded in the problem, general and uni ed conditions for the minimaxity of the best equivariant predictive density estimator are derived. These conditions are applied to check minimaxity in various restricted parameter spaces in location and/or scale families. Further, it is shown that the generalized Bayes estimator against the uni- form prior over the restricted space is minimax and dominates the best equivariant estimator in a location family when the parameter is restricted to an interval of the form [a0;1). Similar ndings are obtained for scale parameter families. Finally, the presentation is accompanied by various observations and illustrations, such as normal, exponential location, and gamma model examples.

Suggested Citation

  • Tatsuya Kubokawa & Éric Marchand & William E. Strawderman & Jean-Philippe Turcotte, 2012. "Minimaxity in Predictive Density Estimation with Parametric Constraints," CIRJE F-Series CIRJE-F-843, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2012cf843
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2012/2012cf843.pdf
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    References listed on IDEAS

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    1. José Manuel Corcuera & Federica Giummolè, 1999. "A Generalized Bayes Rule for Prediction," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(2), pages 265-279, June.
    2. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2008. "Stein's phenomenon in estimation of means restricted to a polyhedral convex cone," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 141-164, January.
    3. Éric Marchand & William Strawderman, 2005. "Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 129-143, March.
    4. Hartigan, J. A., 2004. "Uniform priors on convex sets improve risk," Statistics & Probability Letters, Elsevier, vol. 67(4), pages 285-288, May.
    5. Tatsuya Kubokawa, 1994. "Double shrinkage estimation of ratio of scale parameters," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(1), pages 95-116, March.
    6. Kengo Kato, 2009. "Improved prediction for a multivariate normal distribution with unknown mean and variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 531-542, September.
    7. Tatsuya Kubokawa, 2004. "Minimaxity in Estimation of Restricted Parameters," CIRJE F-Series CIRJE-F-270, CIRJE, Faculty of Economics, University of Tokyo.
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    Cited by:

    1. Hisayuki Tsukuma & Tatsuya Kubokawa, 2012. "Minimaxity in Estimation of Restricted and Non-restricted Scale Parameter Matrices," CIRJE F-Series CIRJE-F-858, CIRJE, Faculty of Economics, University of Tokyo.

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