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Minimaxity in predictive density estimation with parametric constraints

Author

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  • Kubokawa, Tatsuya
  • Marchand, Éric
  • Strawderman, William E.
  • Turcotte, Jean-Philippe

Abstract

This paper is concerned with estimation of a predictive density with parametric constraints under Kullback–Leibler loss. When an invariance structure is embedded in the problem, general and unified 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 uniform 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,∞). Similar findings 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

  • Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E. & Turcotte, Jean-Philippe, 2013. "Minimaxity in predictive density estimation with parametric constraints," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 382-397.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:382-397
    DOI: 10.1016/j.jmva.2013.01.001
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    References listed on IDEAS

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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;The Institute of Statistical Mathematics, vol. 57(1), pages 129-143, March.
    2. 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.
    3. Hartigan, J. A., 2004. "Uniform priors on convex sets improve risk," Statistics & Probability Letters, Elsevier, vol. 67(4), pages 285-288, May.
    4. Tatsuya Kubokawa, 2004. "Minimaxity in Estimation of Restricted Parameters," CIRJE F-Series CIRJE-F-270, CIRJE, Faculty of Economics, University of Tokyo.
    5. 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.
    6. 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.
    7. 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.
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    Cited by:

    1. Fourdrinier, Dominique & Marchand, Éric & Strawderman, William E., 2019. "On efficient prediction and predictive density estimation for normal and spherically symmetric models," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 18-25.
    2. Matsuda, Takeru & Strawderman, William E., 2016. "Pitman closeness properties of point estimators and predictive densities with parametric constraints," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 101-106.
    3. Tatsuya Kubokawa & Éric Marchand & William E. Strawderman, 2014. "On Predictive Density Estimation for Location Families under Integrated L 2 and L 1 Losses," CIRJE F-Series CIRJE-F-935, CIRJE, Faculty of Economics, University of Tokyo.
    4. Yasuyuki Hamura & Tatsuya Kubokawa, 2022. "Bayesian predictive density estimation with parametric constraints for the exponential distribution with unknown location," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(4), pages 515-536, May.
    5. Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E., 2015. "On predictive density estimation for location families under integrated squared error loss," Journal of Multivariate Analysis, Elsevier, vol. 142(C), pages 57-74.
    6. Hisayuki Tsukuma & Tatsuya Kubokawa, 2015. "Minimaxity in estimation of restricted and non-restricted scale parameter matrices," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 261-285, April.

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