# Spatial Risk Measure for Max-Stable and Max-Mixture Processes

## TL;DR

This paper introduces a spatial risk measure based on the variance of damage functions for max-stable, inverse max-stable, and max-mixture processes, providing a unified framework for different dependence structures in spatial extremes.

## Contribution

It generalizes the spatial risk measure to multiple models including max-stable, inverse max-stable, and max-mixture processes, and evaluates it through simulation.

## Key findings

- Risk measure effectively captures spatial dependence in extreme processes.
- Simulation results demonstrate the applicability across different models.
- Provides a tool for assessing spatial risk in environmental extremes.

## Abstract

In this paper, we consider isotropic and stationary max-stable, inverse max-stable and max-mixture processes $X=(X(s))\_{s\in\bR^2}$ and the damage function $\cD\_X^{\nu}= |X|^\nu$ with $0<\nu<1/2$. We study the quantitative behavior of a risk measure which is the variance of the average of $\cD\_X^{\nu}$ over a region $\mathcal{A}\subset \bR^2$.} This kind of risk measure has already been introduced and studied for \vero{some} max-stable processes in \cite{koch2015spatial}. %\textcolor{red}{In this study, we generalised this risk measure to be applicable for several models: asymptotic dependence represented by max-stable, asymptotic independence represented by inverse max-stable and mixing between of them.} We evaluated the proposed risk measure by a simulation study.