Multivariate threshold exceedances and extremal graphical models in the presence of multiple extreme directions

Abstract

Consider a random vector representing risk factors, and suppose that we are interested in extreme scenarios. The threshold exceedances method is widely used in extreme value theory. It uses the fact that, asymptotically, exceedances over a high threshold can be modeled using a multivariate generalized Pareto distribution. In the literature, statistical practice of this method has been discussed only when all risks are large simultaneously. This condition is not realistic, for example, when some of the risks are nearly independent. To address this limitation, we first develop a parametric model that accommodates cases where only some risk factors are extreme, while others remain moderate. We introduce the concept of extreme directions to describe these cases, and prove that our construction encompasses the full range of possible max-stable dependence structures. Second, we propose an estimation procedure for this model, leading to an algorithm capable of identifying extreme directions. Third, we extend the framework of extremal graphical models by incorporating both multiple extreme directions and extremal independence, offering a more flexible and realistic approach to modeling extreme events. Finally, we apply our developed methods to two real-world datasets: first, to discharge measurements at stations along the Danube River, and second, to financial portfolio losses from stocks listed on the NYSE, AMEX, and NASDAQ.(SC - Sciences) -- UCL, 202