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