## The Annals of Statistics

### Admissibility, Difference Equations and Recurrence in Estimating a Poisson Mean

Iain Johnstone

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346786

Digital Object Identifier
doi:10.1214/aos/1176346786

Mathematical Reviews number (MathSciNet)
MR760682

Zentralblatt MATH identifier
0557.62006

JSTOR