A note on upper-patched generators for Archimedean copulas

Abstract

Abstract. The class of multivariate Archimedean copulas is defined by using a real-valued function called the generator of the copula. This generator satisfies some properties, including d-monotonicity. We propose here a new basic transformation of this generator, preserving these properties, thus ensuring the validity of the transformed generator and inducing a proper valid copula. This transformation acts only on a specific portion of the generator, it allows both the non-reduction of the likelihood on a given dataset, and the choice of the upper tail dependence coefficient of the transformed copula. Numerical illustrations show the utility of this construction, which can improve the fit of a given copula both on its central part and its tail.Mathematics Subject Classification. 62H20, 62E20, 60E05, 62H05.

Lorenz-generated bivariate Archimedean copulas

“…While is not di cult to verify that a non-strict Archimedean copula C(u, v) is not able to model independence directly i.e. C(u, v) ≠ uv ≠ Π (u, v), it can be extremely useful when dealing with phenomena that exhibit upper tail dependence, or when one is interested in the dependence structure of random quantities that do not take on low quantiles at the same time [5,11,26]. In economics, for instance, a situation in which a non-strict copula could be a viable tool for data modeling is given by the presence of minimum production cost (including minimum wages), or the existence of some sort of technological frontier [43].…”