A goodness-of-fit test for bivariate extreme-value copulas

TL;DR

This paper introduces a statistical test for verifying if a bivariate extreme-value copula's dependence function fits a specified parametric family, using a Cramér-von Mises statistic and bootstrap methods.

Contribution

It proposes a novel goodness-of-fit test for the Pickands dependence function within the class of extreme-value copulas, including validation and power assessment.

Findings

Test effectively distinguishes true from false models in simulations.

Bootstrap procedure accurately estimates the test statistic's distribution.

Extension to left-tail decreasing dependence structures broadens applicability.

Abstract

It is often reasonable to assume that the dependence structure of a bivariate continuous distribution belongs to the class of extreme-value copulas. The latter are characterized by their Pickands dependence function. In this paper, a procedure is proposed for testing whether this function belongs to a given parametric family. The test is based on a Cram\'{e}r--von Mises statistic measuring the distance between an estimate of the parametric Pickands dependence function and either one of two nonparametric estimators thereof studied by Genest and Segers [Ann. Statist. 37 (2009) 2990--3022]. As the limiting distribution of the test statistic depends on unknown parameters, it must be estimated via a parametric bootstrap procedure, the validity of which is established. Monte Carlo simulations are used to assess the power of the test and an extension to dependence structures that are left-tail decreasing in both variables is considered.