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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

Citations:

Zbl 0767.62002
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