We find the set of extremal points of Lorenz curves with fixed Gini index and compute the maximal $L^1$-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 $L^1$-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.
翻译:我们用固定基尼指数来找到洛伦茨曲线的极限点,并计算洛伦茨曲线与基尼系数的给定值之间的最大值 $L$1$-距离。作为应用,我们引入了一个双维指数,同时测量两个人口之间的相对不平等和差异。这个提议采用变量的基尼指数和洛伦茨曲线之间的一个$L$1美元-距离。这个指数在右角三角中取值,其中两面以洛伦茨在基底分布间排序中表示的完美相对不平等为特征。此外,低温值代表两个分布之间的最大距离。因此,我们用图形绘制一个图表,要么看(相对的)不平等和两个人口之间时间分配之间的距离的演变,要么比较特定人口在一个固定时间点和若干年的范围之间的收入分配。我们证明了上述索赔背后的数学结果,并完整地描述了该指数的顶部估量值值值的精确度特性。此外,我们用图示了该指数的若干实际收入指数。我们用图示了该指数的双维指数。