Bayesian inference for multivariate extreme value distributions

TL;DR

This paper develops a Bayesian inference methodology for general max-stable distributions in multivariate and spatial extremes, utilizing full likelihoods and demonstrating improved efficiency over existing methods.

Contribution

It extends Bayesian inference to general max-stable models using full likelihoods and establishes conditions for asymptotic normality of the posterior median.

Findings

Posterior median is more efficient than composite likelihood estimators.

Method enables Bayesian model comparison for multivariate extremes.

Simulation confirms improved estimator performance.

Abstract

Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. (2016, Ann. Appl. Stat., to appear) use a Bayesian approach to fit a Brown--Resnick process to extreme temperatures. In this paper, we extend this idea to a methodology that is applicable to general max-stable distributions and that uses full likelihoods. We further provide simple conditions for the asymptotic normality of the median of the posterior distribution and verify them for the commonly used models in multivariate and spatial extreme value statistics. A simulation study shows that this point estimator is considerably more efficient than the composite likelihood estimator in a frequentist framework. From a Bayesian perspective, our approach opens the way for new techniques such as Bayesian model comparison in multivariate and spatial extremes.