A dependence measure for multivariate and spatial extreme values: properties and inference

Schlather, Martin ; Tawn, Jonathan A.

Document Type: Article
Year of publication: 2003
The title of a journal, publication series: Biometrika : A Journal for the Statistical Study of Biological Problems
Volume: 90
Issue number: 1
Page range: 139-156
Place of publication: Oxford
Publishing house: Oxford Univ. Press
ISSN: 0006-3444
Publication language: English
Institution: School of Business Informatics and Mathematics > Mathematische Statistik (Schlather 2012-)
Subject: 310 Statistics
Keywords (English): Dependence measure, Extreme value, Max‐stable process, Multivariate extreme‐valu
Abstract: We present properties of a dependence measure that arises in the study of extreme values in multivariate and spatial problems. For multivariate problems the dependence measure characterises dependence at the bivariate level, for all pairs and all higher orders up to and including the dimension of the variable. Necessary and sufficient conditions are given for subsets of dependence measures to be self‐consistent, that is to guarantee the existence of a distribution with such a subset of values for the dependence measure. For pairwise dependence, these conditions are given in terms of positive semidefinite matrices and non‐differentiable, positive definite functions. We construct new nonparametric estimators for the dependence measure which, unlike all naive nonparametric estimators, impose these self‐consistency properties. As the new estimators provide an improvement on the naive methods, both in terms of the inferential and interpretability properties, their use in exploratory extreme value analyses should aid the identification of appropriate dependence models. The methods are illustrated through an analysis of simulated multivariate data, which shows that a lack of self‐consistency is frequently a problem with the existing estimators, and by a spatial analysis of daily rainfall extremes in south‐west England, which finds a smooth decay in extremal dependence with distance.

Dieser Datensatz wurde nicht während einer Tätigkeit an der Universität Mannheim veröffentlicht, dies ist eine Externe Publikation.

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