Modeling extremal dependence in multivariate and spatial problems: a practical perspective

TL;DR

This work addresses how to model and infer extremal dependence in high-dimensional multivariate and spatial data using extreme value theory. It combines parametric, semi-parametric, and nonparametric tools, with a focus on angular measures $H$ and Pickands functions $A$ to capture dependence in the tails, implemented in the ExtremalDep package. Key contributions include flexible Bayesian and frequentist inference for extremal dependence, nonparametric approximations via Bernstein polynomials, and practical procedures for joint tail probabilities, return levels, and extreme quantile regions, demonstrated across diverse environmental and financial applications. The work emphasizes accessible, decision-oriented outputs (joint/conditional tail probabilities, extreme regions, simulations) to support risk assessment and planning in real-world settings.

Abstract

From environmental sciences to finance, there are growing needs for assessing the risk of more extreme events than those observed. Extrapolating extreme events beyond the range of the data is not obvious and requires advanced tools based on extreme value theory. Furthermore, the complexity of risk assessments often requires the inclusion of multiple variables. Extreme value theory provides very important tools for the analysis of multivariate or spatial extreme events, but these are not easily accessible to professionals without appropriate expertise. This article provides a minimal background on multivariate and spatial extremes and gives simple yet thorough instructions on how to analyse high-dimensional extremes using the R package ExtremalDep. After briefly introducing the statistical methodologies, we focus on road testing the package's toolbox through several real-world applications.

Figures (2)

• Figure 1: From left to right, the first two panels display angular densities corresponding to strong and weak dependence between two maxima, respectively. In the strong dependence case, the probability mass concentrates around $1/2$, whereas in the weak dependence case it concentrates near the vertices of $(0,1)$. The third and fourth panels show the corresponding Pickands dependence functions, which are close to the boundary cases $\max(w,1-w)$ and $1$, respectively.
• Figure 2: Estimated angular densities of the models from Table \ref{['tab:prop_prior']}.