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Stein Phenomenon in Estimation of Means Restricted to a Polyhedral Convex Cone

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Listed:
  • Hisayuki Tsukuma

    (Graduate School of Economics, University of Tokyo)

  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)

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 gain.

Suggested Citation

  • Hisayuki Tsukuma & Tatsuya Kubokawa, 2005. "Stein Phenomenon in Estimation of Means Restricted to a Polyhedral Convex Cone," CIRJE F-Series CIRJE-F-342, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2005cf342
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    References listed on IDEAS

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    1. Griffith, Rachel & Redding, Stephen J. & Simpson, Helen, 2002. "Productivity Convergence and Foreign Ownership at the Establishment Level," CEPR Discussion Papers 3765, C.E.P.R. Discussion Papers.
    2. Kyoji Fukao & Hyeog Ug Kwon, 2005. "Why Did Japan's TFP Growth Slow Down in the Lost Decade? An Empirical Analysis Based on Firm-Level Data of Manufacturing Firms," Discussion papers 05004, Research Institute of Economy, Trade and Industry (RIETI).
    3. Dollar, David & Wolff, Edward N., 1988. "Convergence Of Industry Labor Productivity Among Advanced Economies: 1963-1982," Working Papers 88-19, C.V. Starr Center for Applied Economics, New York University.
    4. Kyoji Fukao & Hyeog Ug Kwon, 2006. "Why Did Japan'S Tfp Growth Slow Down In The Lost Decade? An Empirical Analysis Based On Firm-Level Data Of Manufacturing Firms," The Japanese Economic Review, Japanese Economic Association, vol. 57(2), pages 195-228.
    5. Davidson, Russell & MacKinnon, James G., 1993. "Estimation and Inference in Econometrics," OUP Catalogue, Oxford University Press, number 9780195060119.
    6. Dollar, David & Wolff, Edward N, 1988. "Convergence of Industry Labor Productivity among Advanced Economies, 1963-1982," The Review of Economics and Statistics, MIT Press, pages 549-558.
    7. Timothy Dunne & Mark J. Roberts & Larry Samuelson, 1989. "The Growth and Failure of U. S. Manufacturing Plants," The Quarterly Journal of Economics, Oxford University Press, pages 671-698.
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