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Admissible estimation, Dirichlet principles and recurrence of birth-death chains on $${\mathbb{Z}}^ p_+$$. (English) Zbl 0592.62009
The problem considered is that of estimating unknown means $$\lambda_ i$$ of each of p independent Poisson observations $$X_ i$$, with loss $\sum^{p}_{i=1}\lambda_ i^{-1}(d_ i(x)-\lambda_ i)^ 2$ incured from an estimator $$d(x)=(d_ i(x);\quad 1\leq i\leq p),$$ where $$x=(x_ 1,...,x_ p).$$
The paper provides probabilistic descriptions and explicit criteria for admissibility of d(x). It has been known for some time that the ML estimator $$d(x)=x$$ is inadmissible for $$p\geq 2$$, a result analogous to the famous discovery of Stein in the multivariate normal case. The theory developed in this paper is parallel to the one obtained earlier for the normal case [Admissibility from recurrence via Poincaré inequalities in the Gaussian case. Manuscript. (1983)].
Noting that the search for admissible rules may, roughly, be confined to the class of generalized Bayes procedures, the author associates a reversible Markov chain $$\{X_ t\}$$ on $$Z^+_ p$$ with each generalized Bayes procedure. The main results are that the question of admissibility of such a procedure can be described as a variational problem in probabilistic potential theory associated with $$\{X_ t\}$$ and that admissibility is in fact equivalent to recurrence of $$\{X_ t\}$$. The variational problem is closely related to the Dirichlet principle for reversible chains, studied recently by D. Griffeath and T. M. Liggett [Ann. Probab. 10, 881-895 (1982; Zbl 0498.60090)] and T. Lyons [ibid. 11, 393-402 (1983; Zbl 0509.60067)].
Reviewer: B.-H.Lindqvist

MSC:
 62C15 Admissibility in statistical decision theory 62F10 Point estimation 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J45 Probabilistic potential theory
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