Outlier Resistant Distributions: Where Does the Probability Go?

Abstract

Summary For a single observation x from a distribution with unknown location parameter θ, we investigate the relationship between the prior probability that θ lies in an interval I and the posterior probability that θ lies in I, given x, as x tends to infinity. In the case when the likelihood function is strictly unimodal, and thus outlier resistant, we give sufficient conditions to ensure that the limiting posterior probability is not greater than the prior probability, by exploiting dispersion properties of the distribution.