Statistics and Its Interface

Volume 8 (2015)

Number 1

Special Issue on Extreme Theory and Application (Part II)

Guest Editors: Yazhen Wang and Zhengjun Zhang

Functional regular variations, Pareto processes and peaks over threshold

Pages: 9 – 17

DOI: http://dx.doi.org/10.4310/SII.2015.v8.n1.a2

Authors

Clément Dombry (Laboratoire de Mathématiques de Besançon, Université de Franche Comté, Besançon, France)

Mathieu Ribatet (Département de Mathématiques, Université Montpellier 2, Montpellier, France)

Abstract

History: The latest developments of extreme value theory focus on the functional framework and much effort has been put in the theory of max-stable processes and functional regular variations. Paralleling the univariate extreme value theory, this work focuses on the exceedances of a stochastic process above a high threshold and their connections with generalized Pareto processes. More precisely we define an exceedance through a homogeneous cost functional $\ell$ and show that the limiting (rescaled) distribution is a $\ell$-Pareto process whose spectral measure can be characterized. Three equivalent characterizations of the $\ell$-Pareto process are given using either a constructive approach, either a homogeneity property or a peak over threshold stability property. We also provide non parametric estimators of the spectral measure and give some examples.

Keywords

extreme value theory, functional regular variations, generalized Pareto process, peaks over threshold

2010 Mathematics Subject Classification

60G70

Full Text (PDF format)