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Multivariate peaks over thresholds models

Journal article
Authors Holger Rootzén
J. Segers
J. L. Wadsworth
Published in Extremes
Volume 21
Issue 1
Pages 115-145
ISSN 1386-1999
Publication year 2018
Published at Department of Mathematical Sciences
Pages 115-145
Language en
Keywords Extreme values, Multivariate generalized Pareto distribution, Peaks over threshold likelihoods, tail dependence, m-estimator, extremes, distributions, exceedances, statistics
Subject categories Probability Theory and Statistics


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.

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