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Extremes

, Volume 21, Issue 1, pp 115–145 | Cite as

Multivariate peaks over thresholds models

  • Holger Rootzén
  • Johan Segers
  • Jennifer L. Wadsworth
Open Access
Article
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Abstract

Multivariate peaks over thresholds modelling based on generalized Pareto distributions has up to now only been used in few and mostly two-dimensional situations. This paper contributes theoretical understanding, models which can respect physical constraints, inference tools, and simulation methods to support routine use, with an aim at higher dimensions. We derive a general point process model for extreme episodes in data, and show how conditioning the distribution of extreme episodes on threshold exceedance gives four basic representations of the family of generalized Pareto distributions. The first representation is constructed on the real scale of the observations. The second one starts with a model on a standard exponential scale which is then transformed to the real scale. The third and fourth representations are reformulations of a spectral representation proposed in Ferreira and de Haan (Bernoulli 20(4), 1717–1737, 2014). Numerically tractable forms of densities and censored densities are found and give tools for flexible parametric likelihood inference. New simulation algorithms, explicit formulas for probabilities and conditional probabilities, and conditions which make the conditional distribution of weighted component sums generalized Pareto are derived.

Keywords

Extreme values Multivariate generalized Pareto distribution Peaks over threshold likelihoods Simulation of extremes 

AMS 2000 Subject Classifications

62G32 60G70 62E10 
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Notes

Acknowledgments

Anna Kiriliouk has throughout participated in the discussions leading to this paper. We thank her and two referees for many helpful comments. Research supported by the Knut and Alice Wallenberg foundation. Johan Segers was funded by contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the “Communauté française de Belgique” and by IAP research network Grant P7/06 of the Belgian government.

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Chalmers and Gothenburg UniversityGöteborgSweden
  2. 2.Université catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Lancaster UniversityLancasterUK

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