Lorenz-Generated Bivariate Archimedean Copulas

Abstract

An alternative generating mechanism for non-strict bivariate Archimedean copulas via the curve of a positive random variable is proposed. 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 curve. For instance, by selecting random variables with lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An alchemy of curves that can be used as general framework to build multiparametric families of copulas is also discussed.