On tail behavior in Bayesian location inference

Abstract

The asymptotic behavior in the right tail of the hazard rate function is considered to compare probability distributions. Using this tail ordering, the position of the posterior distribution with respect to the prior and the likelihood distributions is analyzed for a Bayesian location problem, and it is proved that, under rather general conditions, the posterior distribution is equivalent to the lightest-tailed distribution, except when both the likelihood and the prior are very heavy-tailed distributions. The relationship between the posterior distributions based on random samples of sizes n and 1, respectively, is also studied, as well as its dependence on the relative position of the prior distribution and the model for observations in the hazard rate scale.