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Admissible estimation, Dirichlet principles and recurrence of birth-death chains on ℤ p+
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  • Published: January 1986

Admissible estimation, Dirichlet principles and recurrence of birth-death chains on ℤ p+

  • Iain Johnstone1 

Probability Theory and Related Fields volume 71, pages 231–269 (1986)Cite this article

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Summary

We establish a connection between admissible simultaneous estimation and recurrence of reversible Markov chains on ℤ p+ . Specifically, to each generalized Bayes estimator of the mean of a vector of p independent Poisson variables for a weighted quadratic loss, we associate a variational problem and a reversible birth and death chain on ℤ p+ . The variational problem is closely related to the Dirichlet principle for reversible chains studied recently by Griffeath, Liggett and Lyons. Under side conditions, admissibility of the estimator is equivalent to zero infimal energy in the variational problem and to recurrence of the Markov chain. This yields analytic and probabilistic criteria for inadmissibility which are applied to establish a broad class of results and previous conjectures.

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Author information

Authors and Affiliations

  1. Department of Statistics, Stanford University, 94305, Stanford, California, USA

    Iain Johnstone

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  1. Iain Johnstone
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Additional information

Research supported by an Australian National University Scholarship and A.D. White Fellowship at Cornell University and by NSF at Mathematical Sciences Research Institute, Berkeley and at Stanford

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Johnstone, I. Admissible estimation, Dirichlet principles and recurrence of birth-death chains on ℤ p+ . Probab. Th. Rel. Fields 71, 231–269 (1986). https://doi.org/10.1007/BF00332311

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  • Received: 21 November 1983

  • Revised: 26 February 1985

  • Issue Date: January 1986

  • DOI: https://doi.org/10.1007/BF00332311

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Keywords

  • Markov Chain
  • Stochastic Process
  • Statistical Theory
  • Variational Problem
  • Broad Class
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