Bayesian methods for outliers in uniform and Pareto samples

Abstract

We begin by reviewing the current literature on outliers and look at what has been done both classically and from a Bayesian viewpoint. We then extend these Bayesian ideas to model outliers in uniform and Pareto samples. We consider the problem of deciding if there are any outliers in a sample from a uniform distribution. For a sample from a one parameter uniform distribution we show that the largest observation in the sample has the smallest conditional predictive ordinate. Hence we derive the Bayes factor for testing whether it is an outlier when the amount of contamination is known and unknown using two di erent outlier models. Then we investigate this problem when we have multiple outliers, assuming that our outliers are generated by the same probability distribution or by di erent probability distributions. Similarly for two parameter uniform samples we show that the most extreme observation in the sample has the smallest conditional predictive ordinate. Hence we derive the Bayes factors for testing whether extreme observations are outliers using the stricter outlier model that we had for the one parameter case. We consider the problem of deciding if there are any outliers in a sample from a Pareto distribution. For a sample from a univariate Pareto distribution we show that the largest observation in the sample has the smallest conditional predictive ordinate and derive the Bayes factor for testing whether Abstract 4 it is an outlier when the amount of contamination is known and unknown. Then we investigate this problem when we have multiple outliers, assuming that our outliers are generated by the same probability distribution or by di erent probability distributions. Finally we extend these ideas to the multivariate case both when the marginal samples are independent of one another and when there are correlations/partial correlations.