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

TL;DR

This paper introduces a new class of goodness-of-fit tests for max-stable processes used in modeling spatial extremes, comparing nonparametric and parametric estimators of the dependence structure.

Contribution

It proposes a novel testing framework based on extremal coefficients and pairwise pseudo-likelihood, with bootstrap methods for p-value approximation, applicable in high-dimensional spatial data.

Findings

Tests perform well in simulations with dimension 10.

Method effectively distinguishes between different max-stable models.

Application to rainfall data demonstrates practical utility.

Abstract

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 {\em 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 {\em one- or two-level parametric bootstrap}. Conditions for the asymptotic validity of these resampling procedures are given based on the work of Genest and R\'emillard. 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.