A Bayes minimax result for spherically symmetric unimodal distributions

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 ) .