Bayesian Inference and Prediction based on the Peaks Over Threshold Method

Abstract

In this work we focus on the Peaks Over Threshold (POT) method, which is arguably the most popular approach in the univariate extreme values literature for analysing extreme events. In this setting, we investigate a Bayesian inferential procedure with rigorous theoretical guarantees that allows to extrapolate extreme events in the very far of the tail of the data distribution with a simple uncertainty quantification. An important purpose in risk analysis is the prediction of future events that are more severe than those yet seen. Leveraging on the proposed Bayesian approach we derive a posterior predictive distribution that can be used for forecasting the size and occurrence of extreme events. We show that such a posterior predictive distribution is an accurate estimator of the true predictive distribution of extreme events.