Acta Scientiarum Naturalium Universitatis Pekinensis ›› 2018, Vol. 54 ›› Issue (6): 1283-1289.DOI: 10.13209/j.0479-8023.2018.052

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An Approach Based on Euler’s Formula to Estimating Gini Coefficient of Pareto Distribution

CHEN Yanguang   

  1. Department of Urban and Economic Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871
  • Received:2017-12-26 Revised:2018-05-20 Online:2018-11-20 Published:2018-11-20
  • Contact: CHEN Yanguang, E-mail: chenyg(at)



  1. 北京大学城市与环境学院城市与经济地理系, 北京100871
  • 通讯作者: 陈彦光, E-mail: chenyg(at)
  • 基金资助:


A simple approach to estimating Gini coefficient based on Lorenz curve is proposed to solve the problem that the concentration index replacing the Gini coefficient results in deviation from imbalance measurement of spatial distributions and size distributions. In view of the scale-free distribution phenomena in the complex social and economic system, a logarithmic function of convex Lorenz curve is derived from the pure Zipf’s distribution by means of Euler’s formula for the sum of harmonic sequence. Then an approximate formula to estimate Gini coefficient can be constructed by using the parameters of the logarithmic Lorenz curve model. The formula is applied to the cities in Beijing, Tianjin and Hebei (Jing-Jin-Ji) region, and the Gini coefficients of 22 years are evaluated by the night lighting data. The results show that there is a significant difference between the Gini coefficient and the centralization index. A conclusion can be reached that the centralization index is applicable to the distributions with characteristic scales, while the proposed formula are suitable for the scale-free distributions. This work will help researchers to understand the scopes of application of the imbalance measurements and provide a reference for further developing the direct estimation methods of Gini coefficient.

Key words: Pareto distribution, Zipf’s law, Euler formula, Lorenz curve, Gini coefficient, scale-free phenomena


为了解决集中化指数代替Gini系数度量规模和空间分布不平衡引发的偏差问题, 提出一个Gini系数的近似估计思路。针对社会经济系统中的无尺度分布现象, 借助Euler公式, 基于Zipf定律, 推导一个累积分布的对数函数, 据此构造一个Gini系数的近似估计公式。将此公式应用于京津冀城镇体系, 借助夜晚灯光数据计算22年的Gini系数。结果表明Gini系数与集中化指数存在显著差异。由此得出结论: 集中化指数适用于有尺度分布现象, 所提新方法则适用于无尺度分布现象。研究结果有助于认识不平衡测度的适用范围, 并可为进一步发展Gini系数的直接估计方法提供参考。

关键词: Pareto分布, Zipf定律, Euler公式, Lorenz曲线, Gini系数, 无尺度现象

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