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

A class of goodness-of-fit tests for spatial extremes models based on max-stable processes

Pages: 45 – 62



Ivan Kojadinovic (Laboratoire de mathématiques et applications, Université de Pau et des Pays de l’Adour, Pau, France)

Hongwei Shang (HP Labs, Palo Alto, California, U.S.A.)

Jun Yan (Department of Statistics, University of Connecticut, Storrs, Conn., U.S.A.)


Parametric max-stable processes are increasingly used to model spatial extremes. Starting from the fact that the dependence structure of a max-stable process is completely characterized by an extreme-value copula, a class of goodness-of-fit tests is proposed based on the comparison between a nonparametric and a parametric estimator of the corresponding unknown multivariate Pickands dependence function. Because of the high-dimensional setting under consideration, these functional estimators are only compared at a specific set of points at which they coincide, up to a multiplicative constant, with estimators of the extremal coefficients. The nonparametric estimators of the Pickands dependence function used in this work are those recently studied by Gudendorf and Segers. The parametric estimators rely on the use of the pairwise pseudo-likelihood which extends the concept of pairwise (composite) likelihood to a rank-based context. Approximate $p$-values for the resulting margin-free tests are obtained by means of a one- or two-level parametric bootstrap. Conditions for the asymptotic validity of these resampling procedures are given based on the work of Genest and Rémillard. The finite-sample performance of the tests is investigated in dimension 10 under the Smith, Schlather and geometric Gaussian models. An application of the tests to rainfall data is finally presented.


copula, extremal coefficients, pairwise pseudo-likelihood, parametric bootstrap, Pickands dependence function, rank-based statistics

2010 Mathematics Subject Classification

Primary 62G32, 62H11. Secondary 62H15.

Published 13 February 2015