A test for Archimedeanity in bivariate copula models

TL;DR

This paper introduces a new statistical test to determine if a bivariate copula is Archimedean, based on properties of associativity and the Fréchet-upper bound, with proven convergence and bootstrap-based critical values.

Contribution

The paper presents a novel test for Archimedeanity in bivariate copulas, combining two characterization measures and establishing its theoretical properties and finite sample performance.

Findings

Test statistic based on associativity and upper bound properties.

Weak convergence of the test statistic is proven.

Simulation study demonstrates finite sample effectiveness.

Abstract

We propose a new test for the hypothesis that a bivariate copula is an Archimedean copula. The test statistic is based on a combination of two measures resulting from the characterization of Archimedean copulas by the property of associativity and by a strict upper bound on the diagonal by the Fr\'echet-upper bound. We prove weak convergence of this statistic and show that the critical values of the corresponding test can be determined by the multiplier bootstrap method. The test is shown to be consistent against all departures from Archimedeanity if the copula satisfies weak smoothness assumptions. A simulation study is presented which illustrates the finite sample properties of the new test.