Stein's phenomenon in estimation of means restricted to a polyhedral convex cone
AbstractThis 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.
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Bibliographic InfoArticle provided by Elsevier in its journal Journal of Multivariate Analysis.
Volume (Year): 99 (2008)
Issue (Month): 1 (January)
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Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description
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- 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.
- 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.
- 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.
- 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.
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