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Exact simulation of max-stable processes.

Authors
  • Dombry, Clément1
  • Engelke, Sebastian2
  • Oesting, Marco3
  • 1 Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 Route de Gray, 25030 Besançon cedex, France. , (France)
  • 2 École Polytechnique Fédérale de Lausanne, EPFL-FSB-MATHAA-STAT, Station 8, 1015 Lausanne, Switzerland , [email protected] , (Switzerland)
  • 3 Universität Siegen, Department Mathematik, Walter-Flex-Straße 3, 57068 Siegen, Germany , [email protected] , (Germany)
Type
Published Article
Journal
Biometrika
Publication Date
Jun 01, 2016
Volume
103
Issue
2
Pages
303–317
Identifiers
PMID: 27279659
Source
Medline
Keywords
License
Unknown

Abstract

Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.

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