June 2022 Consistency of Bayesian inference for multivariate max-stable distributions
Simone A. Padoan, Stefano Rizzelli
Author Affiliations +
Ann. Statist. 50(3): 1490-1518 (June 2022). DOI: 10.1214/21-AOS2160

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.

Funding Statement

Simone Padoan is supported by the Bocconi Institute for Data Science and Analytics (BIDSA), Italy.

Acknowledgements

The authors are grateful to Michael Falk, Bas Kleijn and Anthony Davison for their valuable support and help. The authors are also grateful to two anonymous referees for their constructive suggestions, which have undoubtedly improved the presentation of this work. Stefano Rizzelli has carried out part of the work on this paper while being a postdoctoral fellow at the Chair of Statistics STAT of the École Polythechnique Fédŕale de Lausanne.

Citation

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Simone A. Padoan. Stefano Rizzelli. "Consistency of Bayesian inference for multivariate max-stable distributions." Ann. Statist. 50 (3) 1490 - 1518, June 2022. https://doi.org/10.1214/21-AOS2160

Information

Received: 1 September 2020; Revised: 1 December 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441129
zbMATH: 07547939
Digital Object Identifier: 10.1214/21-AOS2160

Subjects:
Primary: 62G20 , 62G32
Secondary: 60G70 , 62C10

Keywords: Angular measure , Bernstein polynomials , Extreme-value copula , multivariate max-stable distribution , nonparametric estimation , Pickands dependence function , posterior consistency

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • June 2022
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