Spring Seminar Series - February 24, 2003
University
of Minnesota
School of Statistics
College of
Liberal Arts
Evaluating Improper Priors for a Geometric
Success Probability
James P. Hobert
Department of Statistics
University of Florida
Monday, February 24, 2003
4:00 PM, B29
Ford Hall
Minneapolis, East Bank Campus
Social at 3:30 PM, 300
Ford Hall
Abstract
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 posterior is called strongly admissible if the formal Bayes estimator
of every bounded function of $\theta$ is admissible under squared error loss.
Strong admissibility of $\pi$ can be viewed as an endorsement of the improper
prior $\nu$. 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 $\pi$. Hobert and Robert (1999, Annals of
Statistics) showed that $W$ is recurrent if and only if a related Markov
chain, $V$, is recurrent. In the case where $X$ is a geometric random variable
with success probability $\theta$, a fairly thorough analysis of the Markov
chain $V$ is possible. This leads to a simple, checkable sufficient condition
for the strong admissibility of $\pi$. (This is joint work with Christian
Robert and Jason Schweinsberg.)