The Annals of Statistics

Admissibility, Difference Equations and Recurrence in Estimating a Poisson Mean

Iain Johnstone

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Consider estimation of a Poisson mean $\lambda$ based on a single observation $x$, using estimator $d(x)$ and loss function $(d(x) - \lambda)^2/\lambda$. The goal is to decide (in)admissibility of $d(x)$. To every generalized Bayes estimator there corresponds a unique reversible birth and death process $\{X_t\}$ on $\mathbb{Z}_+$. Under side conditions $d(x)$ is admissible if and only if it is generalized Bayes and $\{X_t\}$ is recurrent. Explicit equivalent conditions exist in terms of difference equations and minimization problems. The theory is a discrete, univariate counterpart to Brown's (1971) diffusion characterization of admissibility in estimation of a multivariate normal mean. A companion paper discusses simultaneous estimation of several Poisson means.

Article information

Ann. Statist., Volume 12, Number 4 (1984), 1173-1198.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62C15: Admissibility
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 62F10: Point estimation

Admissibility birth and death process Poisson mean difference equations Dirichlet problem recurrence


Johnstone, Iain. Admissibility, Difference Equations and Recurrence in Estimating a Poisson Mean. Ann. Statist. 12 (1984), no. 4, 1173--1198. doi:10.1214/aos/1176346786.

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