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Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

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  • Fourdrinier, Dominique
  • Strawderman, William E.

Abstract

Let X~f([short parallel]x-[theta][short parallel]2) and let [delta][pi](X) be the generalized Bayes estimator of [theta] with respect to a spherically symmetric prior, [pi]([short parallel][theta][short parallel]2), for loss [short parallel][delta]-[theta][short parallel]2. We show that if [pi](t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator [delta]0(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e-[alpha]t[beta] and e-[alpha]t+[beta][phi](t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .

Suggested Citation

  • Fourdrinier, Dominique & Strawderman, William E., 2008. "Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 735-750, April.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:4:p:735-750
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    References listed on IDEAS

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    1. Cellier, D. & Fourdrinier, D. & Strawderman, W. E., 1995. "Shrinkage Positive Rule Estimators for Spherically Symmetrical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 53(2), pages 194-209, May.
    2. Maruyama, Yuzo, 2003. "Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 274-283, February.
    3. Strawderman, William E., 1974. "Minimax estimation of location parameters for certain spherically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 4(3), pages 255-264, September.
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    1. Fourdrinier, Dominique & Strawderman, William E., 2016. "Stokes’ theorem, Stein’s identity and completeness," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 224-231.
    2. Dominique Fourdrinier & Fatiha Mezoued & William E. Strawderman, 2017. "A Bayes minimax result for spherically symmetric unimodal distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 543-570, June.
    3. Dominique Fourdrinier & Tatsuya Kubokawa & William E. Strawderman, 2023. "Shrinkage Estimation of a Location Parameter for a Multivariate Skew Elliptic Distribution," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 808-828, February.
    4. Tsukuma, Hisayuki, 2010. "Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1483-1492, July.

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