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Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

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  • Tsukuma, Hisayuki
  • Kubokawa, Tatsuya

Abstract

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:1:p:141-164
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    References listed on IDEAS

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    1. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    2. Hartigan, J. A., 2004. "Uniform priors on convex sets improve risk," Statistics & Probability Letters, Elsevier, vol. 67(4), pages 285-288, May.
    3. Tatsuya Kubokawa, 2004. "Minimaxity in Estimation of Restricted Parameters," CIRJE F-Series CIRJE-F-270, CIRJE, Faculty of Economics, University of Tokyo.
    4. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    5. Berry, J. Calvin, 1990. "Minimax estimation of a bounded normal mean vector," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 130-139, October.
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    Cited by:

    1. Kubokawa, Tatsuya & Strawderman, William E., 2011. "A unified approach to non-minimaxity of sets of linear combinations of restricted location estimators," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1429-1444, November.
    2. Chang, Yuan-Tsung & Matsuda, Takeru & Strawderman, William E., 2019. "A note on improving on a vector of coordinate-wise estimators of non-negative means via shrinkage," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 143-150.
    3. Hisayuki Tsukuma, 2012. "Simultaneous estimation of restricted location parameters based on permutation and sign-change," Statistical Papers, Springer, vol. 53(4), pages 915-934, November.
    4. Lakshmi Kanta Patra & Suchandan Kayal & Somesh Kumar, 2021. "Minimax estimation of the common variance and precision of two normal populations with ordered restricted means," Statistical Papers, Springer, vol. 62(1), pages 209-233, February.
    5. Tatsuya Kubokawa, 2010. "Minimax Estimation of Linear Combinations of Restricted Location Parameters," CIRJE F-Series CIRJE-F-723, CIRJE, Faculty of Economics, University of Tokyo.
    6. Tatsuya Kubokawa & William E. Strawderman, 2011. "A Unified Approach to Non-minimaxity of Sets of Linear Combinations of Restricted Location Estimators," CIRJE F-Series CIRJE-F-786, CIRJE, Faculty of Economics, University of Tokyo.
    7. 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.
    8. Tatsuya Kubokawa & William E. Strawderman, 2010. "Non-minimaxity of Linear Combinations of Restricted Location Estimators and Related Problems," CIRJE F-Series CIRJE-F-749, CIRJE, Faculty of Economics, University of Tokyo.
    9. 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.

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