Title: Extremes with Random Covariates
Abstract: Classical extreme value statistics assumes that observations are independent and identically distributed. The observations are drawn from a distribution whose tail is approximated by a Generalized Pareto Distribution (GPD). In applications where the tail distributions may vary according to covariates, the parameters of the GPD are often modelled by parametric functions of the covariates. The goal of this article is to provide a framework for inference on the heterogeneity present among extreme values. To this end, we expand the current regular variation framework to include heterogeneity in the tails, which we call the proportional tail model. We provide a framework for inference by showing consistency and joint asymptotic normality for the parameters of the proportional tail model.
Title: Flexible Tail Modelling of Stochastic Processes via Domain-Scaling
Abstract: Extreme events in environmental sciences, such as severe wind storms or heavy precipitation, are typically modelled using stochastic processes in space and/or time. In many applications, threshold exceedances of such processes appear to be more localized the more extreme they are. Many classical statistical models in spatio-temporal extremes are based on regular variation and cannot model this effect. To address this limitation, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate the localization of extremes. This framework is inspired by the convergence of domain-scaled maxima of Gaussian processes to a max-stable process. The resulting approach is able to cover a broad range of tail behaviours, from Gaussian to Pareto-type processes.
This is joint work with Kirstin Strokorb (University of Bath) and Raphaël de Fondeville (Federal Statistical Office).