A Bayes minimax result for spherically symmetric unimodal distributions

Dominique Fourdrinier (), Fatiha Mezoued () and William E. Strawderman ()
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Dominique Fourdrinier: Normandie Université, Université de Rouen
Fatiha Mezoued: École Nationale Supérieure de Statistique et d’Économie Appliquée (ENSSEA)
William E. Strawderman: Rutgers University

Annals of the Institute of Statistical Mathematics, 2017, vol. 69, issue 3, No 3, 543-570

Abstract: 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 ) .

Keywords: Bayes estimators; minimax estimators; Spherically symmetric distributions; Location parameter; Unimodal densities; Quadratic loss; Superharmonic priors (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10463-016-0553-1

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