On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation

TL;DR

This paper introduces the concept of weak conditional convergence for bivariate copulas, showing its equivalence to pointwise convergence in certain classes and exploring implications for nonparametric estimation.

Contribution

It establishes the equivalence of weak conditional and pointwise convergence for Archimedean and Extreme Value copulas, and demonstrates that any copula can be approximated by checkerboard copulas.

Findings

Weak conditional convergence is equivalent to pointwise convergence for specific copula classes.

Any copula can be approximated by a sequence of checkerboard copulas.

Implications for dependence measures and nonparametric estimation are discussed.

Abstract

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance - in fact, given samples of a copula $C$ the corresponding empirical copulas do not converge weakly conditional to $C$ with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula $C$ is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas.