A sub-asymptotic model for bivariate threshold exceedances

TL;DR

This paper introduces a flexible sub-asymptotic model for bivariate threshold exceedances that captures various tail dependence behaviors and uses neural Bayes for inference, demonstrated on rainfall data.

Contribution

It proposes a novel parametric class for bivariate extremes that includes the multivariate GP as a limit and allows for natural dependence evolution.

Findings

Model effectively captures diverse tail dependence behaviors.

Neural Bayes estimation provides accurate inference.

Application to rainfall data demonstrates model flexibility.

Abstract

Extreme value theory offers a statistical framework for quantifying the risk of rare events, with the generalized Pareto (GP) distribution providing the canonical limit model for univariate threshold exceedances. In many applications, however, extremes are intrinsically multivariate, requiring models that capture both marginal tail behaviours and joint extremal dependencies. Under asymptotic dependence, the multivariate GP distribution represents a suitable modelling family, but when asymptotic independence arises, sub-asymptotic models are needed. In this work, we propose and study a flexible sub-asymptotic parametric class to model bivariate threshold exceedances. Our new model accommodates a broad range of tail dependence behaviours and contains the standardised multivariate GP distribution as a limiting case while retaining margins that converge to univariate GP tails. Our formulation allows extremal dependence to evolve naturally with the marginal parameters on the original data scale, facilitating direct computation and interpretation of failure probabilities. Model inference is done via a likelihood-free neural Bayes estimation approach, with tailored prior specifications. An extensive simulation study and an application to Belgian rainfall extremes illustrate the estimation framework and the flexibility of the model.