2021/03/04 by Baíllo, Amparo, Cárcamo, Javier, Mora-Corral, Carlos
#Applications (stat.AP) #Econometrics (econ.EM) #FOS: Computer and information sciences #FOS: Economics and business
paper · doi:10.48550/arxiv.2103.03286
We find the set of extremal points of Lorenz curves with fixed Gini index and compute the maximal L1-distance between Lorenz curves with given values of their Gini coefficients. As an application we introduce a bidimensional index that simultaneously measures relative inequality and dissimilarity between two populations. This proposal employs the Gini indices of the variables and an L1-distance between their Lorenz curves. The index takes values in a right-angled triangle, two of whose sides characterize perfect relative inequality-expressed by the Lorenz ordering between the underlying distributions. Further, the hypotenuse represents maximal distance between the two distributions. As a consequence, we construct a chart to, graphically, either see the evolution of (relative) inequality and distance between two income distributions over time or to compare the distribution of income of a specific population between a fixed time point and a range of years. We prove the mathematical results behind the above claims and provide a full description of the asymptotic properties of the plug-in estimator of this index. Finally, we apply the proposed bidimensional index to several real EU-SILC income datasets to illustrate its performance in practice.