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Minimaxity in Predictive Density Estimation with Parametric Constraints

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  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)

  • �ric Marchand

    (D�partement de math�matiques, Universit� de Sherbrooke,)

  • William E. Strawderman

    (Department of Statistics and Biostatistics, Rutgers University,)

  • Jean-Philippe Turcotte

    (D�partement de math�matiques, Universit� de Sherbrooke,)

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    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.

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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2012/2012cf843.pdf
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    Bibliographic Info

    Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-843.

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    Length: 27 pages
    Date of creation: Feb 2012
    Date of revision:
    Handle: RePEc:tky:fseres:2012cf843

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    1. 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, vol. 57(1), pages 129-143, March.
    2. Tatsuya Kubokawa, 2004. "Minimaxity in Estimation of Restricted Parameters," CIRJE F-Series CIRJE-F-270, CIRJE, Faculty of Economics, University of Tokyo.
    3. Kengo Kato, 2009. "Improved prediction for a multivariate normal distribution with unknown mean and variance," Annals of the Institute of Statistical Mathematics, Springer, vol. 61(3), pages 531-542, September.
    4. José Manuel Corcuera, 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.
    5. Hartigan, J. A., 2004. "Uniform priors on convex sets improve risk," Statistics & Probability Letters, Elsevier, vol. 67(4), pages 285-288, May.
    6. Tatsuya Kubokawa, 1994. "Double shrinkage estimation of ratio of scale parameters," Annals of the Institute of Statistical Mathematics, Springer, vol. 46(1), pages 95-116, March.
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