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December, 1984 Admissibility, Difference Equations and Recurrence in Estimating a Poisson Mean
Iain Johnstone
Ann. Statist. 12(4): 1173-1198 (December, 1984). DOI: 10.1214/aos/1176346786


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.


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Iain Johnstone. "Admissibility, Difference Equations and Recurrence in Estimating a Poisson Mean." Ann. Statist. 12 (4) 1173 - 1198, December, 1984.


Published: December, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0557.62006
MathSciNet: MR760682
Digital Object Identifier: 10.1214/aos/1176346786

Primary: 62C15
Secondary: 60J10 , 62F10

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

Rights: Copyright © 1984 Institute of Mathematical Statistics


Vol.12 • No. 4 • December, 1984
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