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Asymptotic admissibility of priors and elliptic differential equations

John A. Hartigan

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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

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

First available in Project Euclid: 14 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

Copyright © 2012, Institute of Mathematical Statistics


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.

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