## Eaton’s Markov chain, its conjugate partner and $$\mathcal P$$-admissibility.(English)Zbl 0945.62012

Summary: Suppose that $$X$$ is a random variable with density $$f(x |\theta)$$ and that $$\pi(\theta |x)$$ is a proper posterior corresponding to an improper prior $$\nu(\theta)$$. The prior is called $${\mathcal P}$$-admissible if the generalized Bayes estimator of every bounded function of $$\theta$$ is almost-$$\nu$$-admissible under squared error loss. M. L. Eaton [ibid. 20, No. 3, 1147-1179 (1992; Zbl 0767.62002)] showed that recurrence of the Markov chain with transition density $R(\eta |\theta)=\int \pi(\eta |x)f(x |\theta)dx$ is a sufficient condition for $${\mathcal P}$$-admissibility of $$\nu(\theta)$$. We show that Eaton’s Markov chain is recurrent if and only if its conjugate partner, with transition density $$\widetilde R(y |x)=\int f(y |\theta)\pi(\theta |x)d\theta$$, is recurrent. This provides a new method of establishing $${\mathcal P}$$-admissibility. Often, one of these two Markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience.
For example, when $$X$$ is Poisson $$(\theta)$$ and an improper gamma prior is placed on $$\theta$$, the Markov chain defined by $$\widetilde R(y |x)$$ is equivalent to a branching process with immigration. We use this type of argument to establish $${\mathcal P}$$-admissibility of some priors when $$f$$ is a negative binomial mass function and when $$f$$ is a gamma density with known shape.

### MSC:

 62C15 Admissibility in statistical decision theory 60J27 Continuous-time Markov processes on discrete state spaces 60J05 Discrete-time Markov processes on general state spaces

Zbl 0767.62002
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