Locally Optimal Tests of Independence for Archimedean Copula Families

Abstract

A multivariate distribution can be decoupled into its marginal distributions and a copula function, a distribution function with uniform marginals. Copulas are well suited for modelling the dependence between multivariate random variables independent of their marginal distributions. Applications range from survival analysis over extreme value theory to econometrics. In recent years, copulas have attracted increased attention in financial statistics, in particular regarding modelling issues for high-dimensional problems like value-at-risk or portfolio credit risk. The well studied subclass of Archimedean copulas can be expressed as a function of a one-dimensional generating function ϕ. This class has become popular due to its richness in various distributional attributes providing flexibility in modelling. Here, we present locally optimal tests of independence for Archimedean copula families that are parameterized by a dependence parameter ϑ, where ϑ = 0 denotes independence of the marginal distributions. Under the general assumption of L 2 -differentiability at ϑ = 0 we calculate tangents of the underlying parametric families. For selected examples the optimal tests are calculated and connections to well-known correlation functionals are presented.