Consistent and Robust Bayes Procedures for Location Based on Partial Information

Abstract

We consider Bayes procedures for a location parameter $\theta$ that are robust with respect to the shape of the distribution $F$ of the data. The case where $F$ is fixed (nonrandom) and the case where $F$ has a Dirichlet distribution are both treated. The procedures are based on the posterior distributions of the location parameter given the partial information contained in a robust estimate of location. We show consistency and asymptotic normality of the procedures and give instances where the Bayes procedure based on the full sample diverges while the Bayes procedures based on partial information converges and is asymptotically normal. Finally, we show that robust confidence procedures can be given a Bayesian interpretation.