Abstract

Copulas are useful tools for modeling dependence, with the Archimedean copula being representative. Archimedean copulas are defined by a single function called a generator, and due to their simplicity and flexibility, they are widely used in research despite of the lack in statistical interpretability. In this presentation, we explain how the Frank copula, one of the representative Archimedean copulas, is derived from a framework of the minimum information copula that seeks under given conditions the copula closest to independence in terms of Kullback-Leibler divergence, or maximizing the Shannon entropy in other words. Specifically, we state that Frank copula is obtained as the minimum information copula under fixed Kendall’s τ, or MICK in short, supported by both theoretical perspectives and numerical confirmation. Theoretically, the local dependence function plays a key role; the local dependence function of the Frank copula is known to be proportional to its copula density, and the variation of the Shannon entropy is closely related to it. Furthermore, we present a numerical algorithm that calculates the checkerboard approximation of MICK in a greedy manner, which repeatedly adjusts the probability mass locally to maintain constant local dependence in every region. As a result, the obtained copula is observed to be identical to a discretized Frank copula. Our result implies the equivalence between the Frank copula and MICK, enabling a new interpretation that using the Frank copula is based solely on extracting information from Kendall’s τ when determining joint distributions.