Bayesian nonparametrics for heavy tailed distribution. Application to food risk assessment

Abstract

Based on the fact that any heavy tailed distribution can be approximated by a possibly infinite mixture of Pareto distributions, this paper proposes two Bayesian methodologies tailored to infer on distribution tails belonging to the Frèchet domain of attraction. Firstly, a Bayesian Pareto based clustering procedure is developed, where the mixing distribution is chosen to be the classical conjugate prior of the Pareto distribution. This allows the grouping of $n$ objects into a certain number of clusters according to their extremal behavior and also exhibits a new estimator for the tail index. Secondly, a nonparametric extension of the model based clustering is proposed in which the parameter of interest is the mixing distribution. Estimation of the tail probability is conducted using a Dirichlet process prior for the unknown mixing distribution. To illustrate, both methodologies are applied to simulated data sets and a real data set concerning dietary exposure to a mycotoxin called Ochratoxin A.