Title: 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.
Title: Estimating probabilities of multivariate failure sets based on pairwise tail dependence coefficients
Abstract: Estimating the probability of extreme events involving multiple risk factors is a critical challenge in fields such as finance and climate science. We propose a semi-parametric approach to estimate the probability that a multivariate random vector falls into an extreme failure set, based on the information in the tail pairwise dependence matrix (TPDM) only. The TPDM provides a partial summary of tail dependence for all pairs of components of the random vector. We propose an efficient algorithm to obtain approximate completely positive decompositions of the TPDM, enabling the construction of a max-linear model whose TPDM approximates that of the original random vector. We also provide conditions under which the approximation turns out to be exact. Based on the decompositions, we can construct max-linear random vectors to estimate failure probabilities, exploiting its computational simplicity. The algorithm allows to obtain multiple decompositions efficiently. Finally, we apply our framework to estimate probabilities of extreme events for real-world datasets, including industry portfolio returns and maximal wind speeds, demonstrating its practical utility for risk assessment.