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A Bayes minimax result for spherically symmetric unimodal distributions

Author

Listed:
  • Dominique Fourdrinier

    (Normandie Université, Université de Rouen)

  • Fatiha Mezoued

    (École Nationale Supérieure de Statistique et d’Économie Appliquée (ENSSEA))

  • William E. Strawderman

    (Rutgers University)

Abstract

We consider Bayesian estimation of the location parameter $$\theta $$ θ of a random vector X having a unimodal spherically symmetric density $$f(\Vert x - \theta \Vert ^2)$$ f ( ‖ x - θ ‖ 2 ) for a spherically symmetric prior density $$\pi (\Vert \theta \Vert ^2)$$ π ( ‖ θ ‖ 2 ) . In particular, we consider minimaxity of the Bayes estimator $$\delta _\pi (X)$$ δ π ( X ) under quadratic loss. When the distribution belongs to the Berger class, we show that minimaxity of $$\delta _\pi (X)$$ δ π ( X ) is linked to the superharmonicity of a power of a marginal associated to a primitive of f. This leads to proper Bayes minimax estimators for certain densities $$f(\Vert x - \theta \Vert ^2)$$ f ( ‖ x - θ ‖ 2 ) .

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:3:d:10.1007_s10463-016-0553-1
    DOI: 10.1007/s10463-016-0553-1
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    References listed on IDEAS

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    1. 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.
    2. 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.
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