Consistency of Bayesian inference for multivariate max-stable distributions

The Annals of Statistics · 2022 · Volume 50 · Issue 3

Abstract

Predicting extreme events is important in many applications in risk analysis. Extreme-value theory suggests modelling extremes by max-stable distributions. The Bayesian approach provides a natural framework for statistical prediction. Although various Bayesian inferential procedures have been proposed in the literature of univariate extremes and some for multivariate extremes, the study of their asymptotic properties has been left largely untouched. In this paper we focus on a semiparametric Bayesian method for estimating max-stable distributions in arbitrary dimension. We establish consistency of the pertaining posterior distributions for fairly general, well-specified max-stable models, whose margins can be short-, light- or heavy-tailed. We then extend our consistency results to the case where data are samples of block maxima whose distribution is only approximately a max-stable one, which represents the most realistic inferential setting.

Keywords

60G70, 62C10, 62G20, 62G32, Angular measure, Bernstein polynomials, Extreme-value copula, multivariate max-stable distribution, nonparametric estimation, Pickands dependence function, posterior consistency

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