TY - JOUR

T1 - A dependence measure for multivariate and spatial extreme values: properties and inference.

AU - Tawn, Jonathan A.

AU - Schlather, Martin

N1 - RAE_import_type : Journal article RAE_uoa_type : Statistics and Operational Research

PY - 2003/3

Y1 - 2003/3

N2 - 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.

AB - 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.

KW - Dependence measure

KW - Extreme value

KW - Max-stable process

KW - Multivariate extreme-value distribution

U2 - 10.1093/biomet/90.1.139

DO - 10.1093/biomet/90.1.139

M3 - Journal article

VL - 90

SP - 139

EP - 156

JO - Biometrika

JF - Biometrika

SN - 1464-3510

IS - 1

ER -