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A weak functional framework for applications in statistics

Adel Blouza, Dominique Fourdrinier, and Patrice Lepelletier

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For the problem of estimating a general loss of the form c(xθ2), Stein’s identity is particularly relevant in deriving unbiased estimators of loss when x is used as an estimate of θ and is distributed as $\mathcal{N}_{p}(\theta,I)$, and when c is the identity function. In [3], Fourdrinier and Lepelletier show that extensions to other distributions (actually, to spherically symmetric distributions) and to general functions c are conceivable, but through another approach involving a Green’s formula. Somewhat surprisingly, the statistical context induces an unusual weak functional framework. The main goal of this paper is to present such an analytic context.

Chapter information

Dominique Fourdrinier, Éric Marchand and Andrew L. Rukhin, eds., Contemporary Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2012), 90-103

First available in Project Euclid: 14 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C05: General considerations 62C12: Empirical decision procedures; empirical Bayes procedures 26B20: Integral formulas (Stokes, Gauss, Green, etc.) 46F10: Operations with distributions

loss estimation spherically symmetric distributions Green formula Sobolev spaces

Copyright © 2012, Institute of Mathematical Statistics


Blouza, Adel; Fourdrinier, Dominique; Lepelletier, Patrice. A weak functional framework for applications in statistics. Contemporary Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman, 90--103, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2012. doi:10.1214/11-IMSCOLL807.

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