Bayes minimax estimators of a location vector for densities in the Berger class

Dominique Fourdrinier, Fatiha Mezoued, William E. Strawderman

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5 Scopus citations


We consider Bayesian estimation of the location parameter θ of a random vector X having a unimodal spherically symmetric density f({double pipe}x-θ{double pipe}2) when the prior density π({double pipe}θ{double pipe}2) is spherically symmetric and superharmonic. We study minimaxity of the generalized Bayes estimator δπ(X)=X+∇M(X)/m(X) under quadratic loss, where m is the marginal associated to f({double pipe}x-θ{double pipe}2) and M is the marginal with respect to F({double pipe}x-θ{double pipe}2)=1/2∫{double pipe}x-θ{double pipe}2∞f(t) dt under the condition inf t≥0F(t)/f(t)=c>0 (see Berger [1]). We adopt a common approach to the cases where F(t)/f(t) is nonincreasing or nondecreasing and, although details differ in the two settings, this paper complements the article by Fourdrinier and Strawderman [7] who dealt with only the case where F(t)/f(t) is nondecreasing. When F(t)/f(t) is nonincreasing, we show that the Bayes estimator is minimax provided a {double pipe}∇π({double pipe}θ{double pipe}2){double pipe}2/π({double pipe}θ{double pipe}2)+2 c2 Δπ({double pipe}θ{double pipe}2)≤0 where a is a constant depending on the sampling density. When F(t)/f(t) is nondecreasing, the first term of that inequality is replaced by b g({double pipe}θ{double pipe}2) where b also depends on f and where g({double pipe}θ{double pipe}2) is a superharmonic upper bound of {double pipe}∇π({double pipe}θ{double pipe}2){double pipe}2/π({double pipe}θ{double pipe}2). Examples illustrate the theory.

Original languageEnglish (US)
Pages (from-to)783-809
Number of pages27
JournalElectronic Journal of Statistics
StatePublished - 2012

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Bayes estimators
  • Location parameter
  • Minimax estimators
  • Quadratic loss
  • Sobolev spaces
  • Spherically symmetric distributions
  • Superharmonic priors


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