Nonparametric Bayesian Inference on Bivariate Extremes

TL;DR

This paper introduces a nonparametric Bayesian method for modeling bivariate extremes, using an infinite-dimensional spectral measure, enabling flexible inference and prediction of rare events.

Contribution

It develops a novel Bayesian nonparametric model for spectral measures in bivariate extremes, with a trans-dimensional MCMC algorithm for inference.

Findings

The method accurately predicts extreme outcomes in simulations.

Applied to Danish fire insurance data, it effectively estimates rare event probabilities.

The approach is flexible and dense within the class of spectral measures.

Abstract

The tail of a bivariate distribution function in the domain of attraction of a bivariate extreme-value distribution may be approximated by the one of its extreme-value attractor. The extreme-value attractor has margins that belong to a three-parameter family and a dependence structure which is characterised by a spectral measure, that is a probability measure on the unit interval with mean equal to one half. As an alternative to parametric modelling of the spectral measure, we propose an infinite-dimensional model which is at the same time manageable and still dense within the class of spectral measures. Inference is done in a Bayesian framework, using the censored-likelihood approach. In particular, we construct a prior distribution on the class of spectral measures and develop a trans-dimensional Markov chain Monte Carlo algorithm for numerical computations. The method provides a bivariate predictive density which can be used for predicting the extreme outcomes of the bivariate distribution. In a practical perspective, this is useful for computing rare event probabilities and extreme conditional quantiles. The methodology is validated by simulations and applied to a data-set of Danish fire insurance claims.