Distorted Copulas: Constructions and Tail Dependence

Abstract

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C _ψ: [0, 1]² → [0, 1], C _ψ(x, y) = ψ{C[ψ⁻¹(x), ψ⁻¹(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails.

Keywords:

• Copula
• Distorted probability
• Isomorphic transformation
• Tail dependence
• TP₂ property

Mathematics Subject Classification:

• 60E05
• 62H20

Acknowledgments

We would like to express our gratitude to an anonymous reviewer for her/his comments that help us in providing a better presentation of this manuscript.

Part of this work was conducted when the first two authors were participating at the Special Semester on Stochastic with Emphasis on Finance, September 3–December 5, 2008, organized by RICAM (Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences), Linz, Austria.

The second author also thanks the Department of Knowledge-Based Mathematical Systems (Linz, Austria) for the kind hospitality.