Admissible estimation, Dirichlet principles and recurrence of birth-death chains on ℤ p +

Abstract

We establish a connection between admissible simultaneous estimation and recurrence of reversible Markov chains on ℤ + . Specifically, to each generalized Bayes estimator of the mean of a vector of p independent Poisson variables for a weighted quadratic loss, we associate a variational problem and a reversible birth and death chain on ℤ + . The variational problem is closely related to the Dirichlet principle for reversible chains studied recently by Griffeath, Liggett and Lyons. Under side conditions, admissibility of the estimator is equivalent to zero infimal energy in the variational problem and to recurrence of the Markov chain. This yields analytic and probabilistic criteria for inadmissibility which are applied to establish a broad class of results and previous conjectures.