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Statistical Inference for Max-Stable Processes by Conditioning on Extreme Events

Part of: Distribution theory Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  22 February 2016

Sebastian Engelke ,
Alexander Malinowski ,
Marco Oesting  and
Martin Schlather
Sebastian Engelke*
Affiliation:
Université de Lausanne and Georg-August-Universität Göttingen
Alexander Malinowski*
Affiliation:
Universität Mannheim and Georg-August-Universität Göttingen
Marco Oesting*
Affiliation:
Universität Mannheim
Martin Schlather*
Affiliation:
Universität Mannheim
*
∗ Postal address: Université de Lausanne, UNIL-Dorigny, Bâtiment Extranef, 1015 Lausanne, Switzerland. Email address: sebastian.engelke@unil.ch
∗∗ Postal address: Institut für Mathematik, Universität Mannheim, A5, 6, 68131 Mannheim, Germany.
∗∗∗∗ Postal address: INRA, UMR 518 Math. Info. Appli., Rue Claude Bernard, 75005 Paris, France. Email address: marco.oesting@agroparistech.fr
∗∗∗∗∗ Email address: schlather@math.uni-mannheim.de
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Abstract

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In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.

Keywords

Extreme value statisticsincremental representationmax-stable processmixed moving maximapeaks-over-threshold

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference 62E20: Asymptotic distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 478 - 495
Copyright
© Applied Probability Trust 

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