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Asymptotic admissibility of priors and elliptic differential equations

John A. Hartigan

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Abstract

We evaluate priors by the second order asymptotic behaviour of the corresponding estimators. Under certain regularity conditions, the risk differences between efficient estimators of parameters taking values in a domain D, an open connected subset of Rd, are asymptotically expressed as elliptic differential forms depending on the asymptotic covariance matrix V. Each efficient estimator has the same asymptotic risk as a “local Bayes” estimate corresponding to a prior density p. The asymptotic decision theory of the estimators identifies the smooth prior densities as admissible or inadmissible, according to the existence of solutions to certain elliptic differential equations. The prior p is admissible if the quantity pV is sufficiently small near the boundary of D. We exhibit the unique admissible invariant prior for V=I, D=Rd{0}. A detailed example is given for a normal mixture model.

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), 117-130

Dates
First available in Project Euclid: 14 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1331731616

Digital Object Identifier
doi:10.1214/11-IMSCOLL809

Mathematical Reviews number (MathSciNet)
MR3202507

Zentralblatt MATH identifier
1326.62020

Subjects
Primary: 62C20: Minimax procedures
Secondary: 62F10: Point estimation 62P30: Applications in engineering and industry

Keywords
Birge ratio consensus value jackknife estimator matrix weighted means meta-analysis normal mean shrinkage estimator

Rights
Copyright © 2012, Institute of Mathematical Statistics

Citation

Hartigan, John A. Asymptotic admissibility of priors and elliptic differential equations. Contemporary Developments in Bayesian Analysis and Statistical Decision Theory: A Festschrift for William E. Strawderman, 117--130, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2012. doi:10.1214/11-IMSCOLL809. https://projecteuclid.org/euclid.imsc/1331731616


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