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

CHEN Yanguang

Department of Urban and Economic Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871; E-mail: chenyg@pku.edu.cn

Abstract 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