Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being

Abstract

The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries.

Lorenz map, inequality ordering and curves based on multidimensional rearrangements

“…Gajdos and Weymark [2005] discuss multivariate versions of the Gini index based exclusively on functionals of individually reordered marginals, as opposed to the multivariate reorderings proposed here. Koshevoy and Mosler [1997], Banerjee [2016], Grothe et al [2021], Mosler [2013], Sarabia and Jorda [2020] also propose multivariate Gini indices based on multivariate Lorenz curve proposals. Finally, optimal transport rearrangements are used to define multivariate comonotonicity and quantiles in Ekeland et al [2012], and Puccetti and Scarsini [2010].…”