Testing for a {\delta}-neighborhood of a generalized Pareto copula

TL;DR

This paper introduces a chi-square goodness-of-fit test to determine if a copula is near a generalized Pareto copula, aiding in extreme value analysis for multivariate distributions and processes.

Contribution

It proposes a novel statistical test for assessing proximity to a generalized Pareto copula in any dimension, including a graphical tool for threshold selection.

Findings

Test effectively identifies neighborhoods of GPCs in multivariate data.

Graphical tool improves decision-making on hypothesis rejection.

Applicable to stochastic processes for extreme value modeling.

Abstract

A multivariate distribution function F is in the max-domain of attraction of an extreme value distribution if and only if this is true for the copula corresponding to F and its univariate margins. Aulbach et al. (2012a) have shown that a copula satisfies the extreme value condition if and only if the copula is tail equivalent to a generalized Pareto copula (GPC). In this paper we propose a chi-square goodness-of-fit test in arbitrary dimension for testing whether a copula is in a certain neighborhood of a GPC. The test can be applied to stochastic processes as well to check whether the corresponding copula process is close to a generalized Pareto process. Since the p-value of the proposed test is highly sensitive to a proper selection of a certain threshold, we also present a graphical tool that makes the decision, whether or not to reject the hypothesis, more comfortable.