On Outlier Rejection Phenomena in Bayes Inference

Abstract

SUMMARY Inference is considered for a location parameter given a random sample. Outliers are not explicitly modelled, but rejection of extreme observations occurs naturally in any Bayesian analysis of data from distributions with suitably thick tails. For other distributions outlier rejection behaviour can never occur. These phenomena motivate new definitions of outlier-proneness and outlier-resistance. The definitions and methodology are Bayesian but the conclusions also have meaning for nonBayesians because they are proved for arbitrary prior distributions. Thus, for example, the t distribution is said to be outlier-prone because it is shown that any admissible inference procedure applied to a t sample will effectively ignore extreme outlying observations regardless of prior information. On the other hand, the normal distribution, for example, is said to be outlier-resistant because it never allows outlier rejection, regardless of prior information.