The generalized Pareto process; with a view towards application and simulation

TL;DR

This paper extends the peaks-over-threshold method to the space of continuous functions through the generalized Pareto process, enabling better simulation of extreme events like storms based on observed data.

Contribution

It introduces a novel functional framework for the generalized Pareto distribution, removing reliance on max-stable processes and broadening application scope.

Findings

Established the peaks-over-threshold approach in function space.

Developed a method to simulate wind fields from extreme storm data.

Provided a theoretical foundation for functional extreme value analysis.

Abstract

In extreme value statistics, the peaks-over-threshold method is widely used. The method is based on the generalized Pareto distribution characterizing probabilities of exceedances over high thresholds in $\mathbb {R}^d$. We present a generalization of this concept in the space of continuous functions. We call this the generalized Pareto process. Differently from earlier papers, our definition is not based on a distribution function but on functional properties, and does not need a reference to a related max-stable process. As an application, we use the theory to simulate wind fields connected to disastrous storms on the basis of observed extreme but not disastrous storms. We also establish the peaks-over-threshold approach in function space.