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Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary

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  • Kuriki, Satoshi
  • Takemura, Akimichi

Abstract

We give James-Stein type estimators of a multivariate normal mean vector by shrinkage towards a closed convex set K with a smooth or piecewise smooth boundary. The rate of shrinkage is determined by the curvature of the boundary of K at the projection point onto K. By considering a sequence of polytopes Kj converging to K, we show that a particular estimator we propose is the limit of a sequence of shrinkage estimators towards Kj given by M. E. Bock (1982). In fact our estimators reduce to the James-Stein estimator and to the Bock estimator when K is a point and a convex polyhedron, respectively. Therefore they can be considered as natural extensions of these estimators. Furthermore we apply the same method to the problem of improving the restricted mle by shrinkage towards the origin in the multivariate normal mean model where the mean vector is restricted to a closed convex cone with a smooth or piecewise smooth boundary. We demonstrate our estimators in two settings, one shrinking to a ball and the other shrinking to the cone of nonnegative definite matrices.

Suggested Citation

  • Kuriki, Satoshi & Takemura, Akimichi, 2000. "Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 79-111, October.
    • Handle: RePEc:eee:jmvana:v:75:y:2000:i:1:p:79-111
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    References listed on IDEAS

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    1. Bock, M. E., 1985. "Minimax estimators that shift towards a hypersphere for location vectors of spherically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 17(2), pages 127-147, October.
    2. Dykstra, Robert R., 1988. "Trails South: The Wagon-Road Economy in the Dodge City-Panhandle Region. By C. Robert Haywood. Norman: University of Oklahoma Press, 1986. xv + 312 pp. Maps, illustrations, notes, bibliography, and in," Business History Review, Cambridge University Press, vol. 62(1), pages 157-158, April.
    3. Judge, G.G. & Yancey, T.A. & Bock, M.E. & Bohrer, R., 1984. "The non-optimality of the inequality restricted estimator under squared error loss," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 165-177.
    4. Jere R. Behrman & Ryan Schneider, 1993. "An International Perspective on Pakistani Human Capital Investments in the Last Quarter Century," The Pakistan Development Review, Pakistan Institute of Development Economics, vol. 32(1), pages 1-68.
    5. Satoshi Kuriki & Akimichi Takemura, 2000. "Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X 2 Distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(1), pages 1-14, March.
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    Cited by:

    1. Yuzo Maruyama & William Strawderman, 2005. "Necessary conditions for dominating the James-Stein estimator," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 157-165, March.
    2. Kato, Kengo, 2009. "On the degrees of freedom in shrinkage estimation," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1338-1352, August.
    3. Dominique Fourdrinier & William Strawderman & Martin Wells, 2006. "Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 73-92, March.
    4. Tiefeng Ma & Shuangzhe Liu & S. Ahmed, 2014. "Shrinkage estimation for the mean of the inverse Gaussian population," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(6), pages 733-752, August.

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