Estimating the limiting shape of bivariate scaled sample clouds: with additional benefits of self-consistent inference for existing extremal dependence properties

TL;DR

This paper introduces the first statistical method to estimate the limiting shape of scaled sample clouds in bivariate extremes, enabling unified and self-consistent inference of extremal dependence properties with practical applications like sea wave height analysis.

Contribution

It develops the first inference procedure for the limit set of scaled sample clouds, linking it to extremal dependence frameworks and providing self-consistent estimators.

Findings

Estimator performs well across various distributions.

Extremal dependence estimators show significant improvements.

Application to sea wave heights confirms weakening dependence with distance.

Abstract

The key to successful statistical analysis of bivariate extreme events lies in flexible modelling of the tail dependence relationship between the two variables. In the extreme value theory literature, various techniques are available to model separate aspects of tail dependence, based on different asymptotic limits. Results from Balkema and Nolde (2010) and Nolde (2014) highlight the importance of studying the limiting shape of an appropriately-scaled sample cloud when characterising the whole joint tail. We now develop the first statistical inference for this limit set, which has considerable practical importance for a unified inference framework across different aspects of the joint tail. Moreover, Nolde and Wadsworth (2022) link this limit set to various existing extremal dependence frameworks. Hence, a by-product of our new limit set inference is the first set of self-consistent estimators for several extremal dependence measures, avoiding the current possibility of contradictory conclusions. In simulations, our limit set estimator is successful across a range of distributions, and the corresponding extremal dependence estimators provide a major joint improvement and small marginal improvements over existing techniques. We consider an application to sea wave heights, where our estimates successfully capture the expected weakening extremal dependence as the distance between locations increases.