Lorenz-generated bivariate Archimedean copulas

Andrea Fontanari,Pasquale Cirillo,Cornelis W Oosterlee

doi:10.1515/demo-2020-0011

Andrea Fontanari, Pasquale Cirillo + Show 1 more

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https://doi.org/10.1515/demo-2020-0011

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Abstract

Abstract A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these “Lorenz copulas”, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An “alchemy” of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.

Highlights

The paper introduces and studies Lorenz copulas, a novel class of non-strict bivariate Archimedean copulas [35], whose generators are de ned in terms of Lorenz curves [28]
Many of the properties of these “Lorenz copulas", from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve
Even if we are not able to prove it, we conjecture that all Lorenz curves connected to lognormal-like random variables [25] can be suitable generators for non-strict Archimedean copulas characterized by asymptotic upper tail dependence

Summary

Introduction

The paper introduces and studies Lorenz copulas, a novel class of non-strict bivariate Archimedean copulas [35], whose generators are de ned in terms of Lorenz curves [28]. Many of the properties of these “Lorenz copulas", from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve.

Results

Conclusion