Electronic Journal of Statistics

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

Dominique Fourdrinier, Fatiha Mezoued, and William E. Strawderman

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Abstract

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.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 783-809.

Dates
First available in Project Euclid: 9 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1336568106

Digital Object Identifier
doi:10.1214/12-EJS694

Mathematical Reviews number (MathSciNet)
MR2988429

Zentralblatt MATH identifier
1281.62067

Subjects
Primary: 62C10: Bayesian problems; characterization of Bayes procedures 62C20: Minimax procedures 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
Bayes estimators minimax estimators spherically symmetric distributions location parameter quadratic loss superharmonic priors Sobolev spaces

Citation

Fourdrinier, Dominique; Mezoued, Fatiha; Strawderman, William E. Bayes minimax estimators of a location vector for densities in the Berger class. Electron. J. Statist. 6 (2012), 783--809. doi:10.1214/12-EJS694. https://projecteuclid.org/euclid.ejs/1336568106


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References

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