A new family of smooth copulas with arbitrarily irregular densities

TL;DR

This paper introduces a broad family of copulas with irregular densities derived from univariate densities, demonstrating that copula regularity cannot be assumed and revealing complex behaviors in their density functions.

Contribution

The authors construct copulas with highly irregular densities from arbitrary univariate densities, challenging assumptions about copula regularity and demonstrating diverse pathological behaviors.

Findings

Copula densities can be highly irregular, even unbounded in every neighborhood.

Constructed copulas inherit pathological behaviors from underlying univariate densities.

All constructed copulas maintain certain smoothness properties despite irregular densities.

Abstract

Copulas are known to satisfy a number of regularity properties, and one might therefore believe that their densities, when they exist, admit a certain degree of regularity themselves. We show that this is not true in general by constructing a broad family of copulas which admit densities that can hardly be considered regular. The copula densities are constructed from arbitrary univariate densities supported on the unit interval, and we show by example that the copula densities can inherit pathological behaviour from the underlying univariate densities. In particular, we construct a nontrivial univariate density which is unbounded in every open set of the unit interval, and show that it induces a copula density which is finite everywhere but unbounded in every neighborhood of the unit hypercube. Nevertheless, all of our copulas are shown to enjoy attractive smoothness properties.