Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions with residual vector

We consider estimation of the mean vector, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\theta $$</EquationSource> </InlineEquation>, of a spherically symmetric distribution with known scale parameter under quadratic loss and when a residual vector is available. We show minimaxity of generalized Bayes estimators corresponding to superharmonic priors with a non decreasing Laplacian of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\pi (\Vert \theta \Vert ^{2})$$</EquationSource> </InlineEquation>, under certain conditions on the generating function <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$f(\cdot )$$</EquationSource> </InlineEquation> of the sampling distributions. The class of sampling distributions includes certain variance mixtures of normals. Copyright Springer-Verlag Berlin Heidelberg 2014