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A weak functional framework for applications in statistics

Adel Blouza, Dominique Fourdrinier, and Patrice Lepelletier

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Abstract

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

Source
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

Dates
First available in Project Euclid: 14 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1331731614

Digital Object Identifier
doi:10.1214/11-IMSCOLL807

Mathematical Reviews number (MathSciNet)
MR3202505

Zentralblatt MATH identifier
1326.62015

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

Keywords
loss estimation spherically symmetric distributions Green formula Sobolev spaces

Rights
Copyright © 2012, Institute of Mathematical Statistics

Citation

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. https://projecteuclid.org/euclid.imsc/1331731614


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References

  • [1] Adams, R. A. (1978). Sobolev spaces, Academic Press, Inc.
  • [2] Dautray, R. and Lions, J.-L. (1987). Analyse mathématique et calcul numérique pour les sciences et les techniques, volume 2, Masson, Paris, New York.
  • [3] Fourdrinier, D. and Lepelletier, P. (2008). Estimating a general function of a quadratic function. The Annals of the Institute of Statistical Mathematics 60 85–119.
  • [4] Fourdrinier, D. and Strawderman, W. E. (2003). On Bayes and unbiased estimators of loss. The Annals of the Institute of Statistical Mathematics 55 4, 803–816.
  • [5] Fourdrinier, D. and Strawderman, W. E. (2008). Generalized Bayes minimax estimators of location vector for spherically symmetric distributions. Journal of Multivariate Analysis, 99, 4, 735–750,.
  • [6] Fourdrinier, D. and Wells, M. T. (1995) Estimation of a loss function for spherically symmetric distributions in the general linear model. The Annals of Statistics, 23 2, 571–592.
  • [7] Johnstone, I. (1988) On admissibility of some unbiased estimates of loss, In Statistical Decision Theory and Related Topics IV, S. S. Gupta and J. O. Berger, Editors, 1 361–380, Springer-Verlag, New York.
  • [8] Lepelletier, P. (2004) Sur les régions de confiance : amélioration, estimation d’un degré de confiance conditionnel, PhD thesis, UMR CNRS 6085, Université de Rouen, France.
  • [9] Lu, K. and Berger, J. O. (1989). Estimated confidence procedures for multivariate normal means. Journal of Statistical Planning and Inference 23 1–19.
  • [10] Stein, C. (1981). Estimation of the mean of multivariate normal distribution. The Annals of Statistics 9 6 1135–1151.
  • [11] Treves, F. (1967). Topological vector spaces, distributions and kernels, Academic Press, New York, London.