Admissibility, difference equations and recurrence in estimating a Poisson mean. (English) Zbl 0557.62006

Johnstone, Iain

Ann. Stat. 12, 1173-1198 (1984).

Let x be distributed according to a Poisson distribution with mean \(\eta\in (0,\infty)\). For estimating \(\eta\), let the loss function be \(L(d,\eta)=(d-\eta)^ 2/\eta,\) where \(d=d(x)\) denotes an estimator of \(\eta\). If d(x) is admissible and its first nonzero value occurs at \(x=r+1\geq 0\) then there is a finite measure P(d\(\eta)\) on [0,\(\infty)\) such that \(d(x)=d_ p(x)=p_ x/p_{x-1}\), \(x\geq r+1\), where \(p_ x=\int \eta^{x-r}P(d\eta).\)
Let Q(d\(\eta)\) be a finite measure on [0,\(\infty)\) and let \(q_ x=\int \eta^{x-r}Q(d\eta)\), \(u^ 2_ x=q_ x/p_ x\), \(Du_ x=u_ x- u_{x-1}\) and \(a_ x=(p^ 2_ x/p_{x-1})x!\). Consider the side conditions (i) \(d_ p(x)-x\leq M(1+\sqrt{x})\) and (ii) \(d_ p(x+1)-d_ p(x)\in 0(1)\). It is shown that \(d_ p\) is admissible if (i) and (ii) hold and \(\min_{S}\sum^{\infty}_{r+1}(Du_ s)^ 2a_ s=0\), where \(S=\{u_ x:Z\to R|\) \(u_ r=1\), \(u_{\infty}=\lim_{x\to \infty}u_ x=0\}\) and Z denotes the set of positive integers.

Reviewer: K.Alam

MSC:

62C15	Admissibility in statistical decision theory
62F10	Point estimation
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

difference equations; recurrence; Dirichlet problem; generalized; Bayes estimator; reversible birth and death process; minimization; problems; Poisson distribution; loss function; finite measure