Eaton's Markov chain, its conjugate partner and P-admissibility

• 成果类型: Article
• 署名作者: Hobert, JP; Robert, CP
• 署名单位: State University System of Florida; University of Florida; Institut Polytechnique de Paris; ENSAE Paris
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1999
• 页码: 361-373
• 关键词: discrete exponential-families; estimators; recurrence; distributions; Poisson; transience; schemes

摘要

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 P-admissible if the generalized Bayes estimator of every bounded function of theta is almost-nu-admissible under squared error loss. Eaten showed that recurrence of the Markov chain with transition density R(eta\theta) = integral pi(eta\x)f(x\theta) dx is a sufficient condition for P-admissibility of nu(theta). We show that Eaten's Markov chain is recurrent if and only if its conjugate partner, with transition density (R) over tilde(y\x) = integral f(y\theta) pi(theta\x)d theta, is recurrent. This provides a new method of establishing 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 (R) over tilde(y\x) is equivalent to a branching process with immigration. We use this type of argument to establish P-admissibility of some priors when f is a negative binomial mass function and when f is a gamma density with known shape.