Bivariate Lorenz Curves Based on the Sarmanov–Lee Distribution

Abstract

The extension of the univariate Lorenz curve to higher dimensions is not an obvious task. In this chapter, using the definition proposed by Arnold (Pareto Distributions. International Co-operative Publishing House, Fairland (1983)), closed expressions for the bivariate Lorenz curve are given, assuming that the underlying bivariate income distribution belong to the class of bivariate distributions with given marginals described by Sarmanov (Doklady Sov. Math. 168, 596–599 (1966)) and Lee (Commun. Stat. A-Theory 25, 1207–1222 (1996)). The expression of the bivariate Lorenz curve can be easily interpreted as a convex linear combination of products of classical and concentrated Lorenz curves. A closed expression for the bivariate Gini index (Arnold, Majorization and the Lorenz durve. In: Lecture Notes in Statistics, vol. 43. Springer, New York (1987)) in terms of the classical and concentrated Gini indices of the marginal distributions is given. This index can be decomposed in two factors, corresponding to the equality within and between variables. A specific model Pareto marginal distributions is studied. Other aspects are briefly discussed.