基尼系数与洛伦兹曲线的等价分类

Equivalence Partition of Gini Coefficient and Lorentz Curve

基尼系数是测度社会收入分配不均等程度的统计指标。计算基尼系数的方法有很多种,学者们针对不同的数据提出了不同的计算公式,但这些公式在相同条件下是等价的。文章给出了各种基尼系数的数学关系,并分别讨论了连续型和离散型收入分布的基尼系数公式。将基尼系数公式与洛伦兹曲线分别划分为几个等价类,其中一些公式是首次提出的。在离散样本的条件下,给出了两类分组基尼系数的比较结果,并且给出一个反例说明了洛伦兹曲线不一定是下凸的。

Gini coefficient is a statistical index to measure the inequality degree of social income distribution. There are many methods to calculate gini coefficient, and scholars have put forward different calculation formulas for different data, but these formulas are equivalent under the same conditions. This paper presents the mathematical relations of various Gini coefficients and discusses the Gini coefficient formulas for continuous and discrete income distributions, respectively, and then divides the Gini coefficient formula and Lorentz curve into several equivalent classes, some of which are proposed for the first time. Finally, under the discrete samples, the paper presents the comparison results of two kinds of grouped Gini coefficients, alsoproviding counterexample to demonstrate that the Lorentz curve is not necessarily downward convex.