Fall Seminar Series - September 29, 2005
University of Minnesota
School of Statistics
College of Liberal Arts
Markov Chain Conditions for Admissibility
Jim Hobert
Department of Statistics
University of Florida
Thursday, September 29, 2005
3:30 PM, 115
Ford Hall
Minneapolis, East Bank Campus
Social at 3:00 PM, 300 Ford Hall
Abstract
Suppose that $X$ is a random vector with density $f(x|\theta)$ and
that $\pi(\theta|x)$ is a proper posterior density corresponding to an
improper prior $\nu(\theta)$. The prior is called strongly admissible
if the formal Bayes estimator of every bounded function of $\theta$ is
admissible under squared error loss. Eaton (1992, Annals of
Statistics) showed that recurrence of a certain Markov chain, $W$,
defined in terms of $f$ and $\nu$, implies the strong admissibility of
$\nu$. Hobert and Robert (1999, Annals of Statistics) showed that $W$
is recurrent if and only if a related Markov chain, $\tilde{W}$, is
recurrent. I will show that when $X$ is an exponential random
variable, a fairly thorough analysis of the Markov chain $\tilde{W}$
is possible and this leads to a simple sufficient condition for strong
admissibility. I will also explain how the relationship between $W$
and $\tilde{W}$ can be used to establish that certain perturbations of
strongly admissible priors retain strong admissibility. (This is
joint work with M. Eaton, G. Jones, D. Marchev and J.
Schweinsberg.)