Extremal dependence of random scale constructions

TL;DR

This paper investigates the extremal dependence properties of bivariate vectors formed by a random scale construction, analyzing how tail heaviness and dependence influence asymptotic independence or dependence, with implications for risk modeling.

Contribution

It provides a comprehensive analysis of extremal dependence in random scale models, unifying and extending existing results, and introduces new models capturing both dependence regimes.

Findings

Heavier tails in R increase extremal dependence.

Lighter tails in R preserve the dependence structure of (W1,W2).

Both asymptotic independence and dependence are possible in heavy-tailed cases.

Abstract

A bivariate random vector can exhibit either asymptotic independence or dependence between the largest values of its components. When used as a statistical model for risk assessment in fields such as finance, insurance or meteorology, it is crucial to understand which of the two asymptotic regimes occurs. Motivated by their ubiquity and flexibility, we consider the extremal dependence properties of vectors with a random scale construction $(X_1,X_2)=R(W_1,W_2)$, with non-degenerate $R>0$ independent of $(W_1,W_2)$. Focusing on the presence and strength of asymptotic tail dependence, as expressed through commonly-used summary parameters, broad factors that affect the results are: the heaviness of the tails of $R$ and $(W_1,W_2)$, the shape of the support of $(W_1,W_2)$, and dependence between $(W_1,W_2)$. When $R$ is distinctly lighter tailed than $(W_1,W_2)$, the extremal dependence of $(X_1,X_2)$ is typically the same as that of $(W_1,W_2)$, whereas similar or heavier tails for $R$ compared to $(W_1,W_2)$ typically result in increased extremal dependence. Similar tail heavinesses represent the most interesting and technical cases, and we find both asymptotic independence and dependence of $(X_1,X_2)$ possible in such cases when $(W_1,W_2)$ exhibit asymptotic independence. The bivariate case often directly extends to higher-dimensional vectors and spatial processes, where the dependence is mainly analyzed in terms of summaries of bivariate sub-vectors. The results unify and extend many existing examples, and we use them to propose new models that encompass both dependence classes.