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2012 Bayes minimax estimators of a location vector for densities in the Berger class
Dominique Fourdrinier, Fatiha Mezoued, William E. Strawderman
Electron. J. Statist. 6(none): 783-809 (2012). DOI: 10.1214/12-EJS694


We consider Bayesian estimation of the location parameter θ of a random vector X having a unimodal spherically symmetric density f(xθ2) when the prior density π(θ2) 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(xθ2) and M is the marginal with respect to F(xθ2)=1/2xθ2f(t) dt under the condition inf t0F(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 aπ(θ2)2/π(θ2)+2 c2Δπ(θ2)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 bg(θ2) where b also depends on f and where g(θ2) is a superharmonic upper bound of π(θ2)2/π(θ2). Examples illustrate the theory.


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Dominique Fourdrinier. Fatiha Mezoued. William E. Strawderman. "Bayes minimax estimators of a location vector for densities in the Berger class." Electron. J. Statist. 6 783 - 809, 2012.


Published: 2012
First available in Project Euclid: 9 May 2012

zbMATH: 1281.62067
MathSciNet: MR2988429
Digital Object Identifier: 10.1214/12-EJS694

Primary: 46E35, 62C10, 62C20

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society


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