Extremal properties of the Theil and Gini measures of inequality

Quality & Quantity - Two popular inequality measures used in the study of income and wealth distributions are the Gini (G) and Theil (T) indices. Several bounds on these inequality measures are...     

A micro-macro linkage in the measurement of inequality: Another look at the Gini coefficient

The aim of this paper is to help readers understand the significance of the Gini coefficient. We have two major formulae for the Gini coefficient. One of the formulae bases on the idea of aggregation of the microscopic differences between individuals' incomes or wealth. The idea that underlies the other formula is macroscopic presentation of distribution or concentration of income or wealth. We will show an unabridged proof of the equivalence between these formulae to examine how the two conceptions of the measurement of inequality are linked to each other.     

Edgeworth expansions and normalizing transforms for inequality measures

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second-order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second-order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n⁻¹)$O\left(n^{-1}\right)$. This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n^−3/2)$O\left(n^{-3/2}\right)$. We illustrate our procedure with an application to regional inequality measurement in Côte d’Ivoire.     

一种改进的Gini系数

Gini系数是以Lorenz曲线为基础而构造的一个用以衡量收入或财富分配不均等程度的测度指数。在收入分配理论中，Gini系数是一个重要的度量收入分配不平等程度的指标，但由于Gini系数与Lorenz曲线之间不是“一对一”的对应关系，即不同的Lorenz曲线具有相同的Gini系数，从而导致该指标的应用存在着一些缺陷。本文分析了Gini系数不能区分Lorenz曲线的原因，其次通过构造一种加权的Gini系数，对原Gini系数进行了修正，从而解决了原Gini系数无法区分不同的Lorenz曲线的问题。     

Asymptotic and bootstrap inference for inequality and poverty measures

A random sample drawn from a population would appear to offer an ideal opportunity to use the bootstrap in order to perform accurate inference, since the observations of the sample are IID. In this paper, Monte Carlo results suggest that bootstrapping a commonly used index of inequality leads to inference that is not accurate even in very large samples, although inference with poverty indices is satisfactory. We find that the major cause is the extreme sensitivity of many inequality indices to the exact nature of the upper tail of the income distribution. This leads us to study two non-standard bootstraps, the m out of n bootstrap, which is valid in some situations where the standard bootstrap fails, and a bootstrap in which the upper tail is modelled parametrically. Monte Carlo results suggest that accurate inference can be achieved with this last method in moderately large samples.     

The estimation of a family of measures of economic inequality

This paper shows that a broad class of measures of inequality (including those of Herfindahl and Theil) can be readily (and accurately) estimated from grouped data. The methods are similar to those developed earlier by the author for estimating the Gini index. The results are illustrated by estimating Thiel's index from tax data and show that the standard textbook grouping method of assuming that all incomes in any class are at the mid-point can lead to serious error in the resulting estimate of inequality.     

基尼系数和中等收入群体比重的关联性分析

基尼系数是测量和判断收入分配状况最常用的综合指标之一,而中等收入群体比重则是判断社会稳定与否的重要标志。本文通过重新定义低、中、高三类收入群体以及采用多项式函数形式拟合洛伦兹曲线,建立基尼系数和中等收入群体比重之间的数量对应关系。实证分析表明:基尼系数和中等收入比重之间存在负向相关关系。2001年中国的基尼系数约为0．44,相应的中等收入群体比重大致在44%左右。     

关于基尼系数若干问题的再研究--与部分学者商榷

基尼系数是测量收入不平等的一个重要指标,近年来有很多学者对此进行了深入研究,本文主要对有关学者的研究内容进行商榷与补充。首先,实际计算基尼系数应首选离散公式,并且根据具体的抽样方法选择无偏或有效的估计公式;其次,本文论证了对数正态分布下基尼系数G与标准差d的关系式：G=2Ф（σ/√2）-1;再次,针对基尼系数存在的不足,改进基尼系数成为一些学者的研究重点,本文发现一些改进并没有解决基尼系数与洛伦茨曲线的非一一对应问题,并且可操作性较差,通过介绍S基尼系数和E基尼系数,本文说明了基尼系数改进的总体思路;最后,针对学者提出的“部分分布决定性定理（PD定理）”,本文论证了其推理过程中存在的问题,说明该定理是错误的。     

关于基尼系数分解分析的讨论

本文通过几何分解的方法阐述了基尼系数在不同群体中进行分解的过程,并在此基础上对群体间基尼系数的应用和群体内基尼系数的进一步分解进行了讨论。说明由于交叉项的存在,基尼系数只能对不同收入群体进行有限的分解,对群体内基尼系数的进一步分解是极为勉强的。只有当各群体收入重叠程度较低时,方可对交叉项进行忽略,此时对群体内基尼系数的进一步分解才具有较清晰的经济含义。     

论基尼系数的局限及调整

本文从理论上对基尼系数在反映贫富差距程度时存在的局限性进行了分析，并从低收入阶层的相对处境理应更加受到重视的角度出发，构建能够兼顾反映居民整体收入差异和低收入阶层相对处境的指标——调整基尼系数。结合对有关资料进行的实证分析，本文的结论是，用调整基尼系数反映贫富差距比用基尼系数更为合理。     

基尼系数与样本信息含量

基尼系数是度量收入不平等的主要指标,采用何种方法估计总体基尼系数取决于样本的信息含量。对于“分组数据”样本,需要使用参数法对总体收入分布或洛伦兹曲线进行假定来估计总体基尼系数。本文提出通过计算样本基尼系数的上下界来检验总体收入分布或洛伦兹曲线假定的正确性,并运用该方法对程永宏(2006)关于农村家庭人均纯收入分布的假定进行了检验。     

The controversial link between entrepreneurial activity and inequality

International Entrepreneurship and Management Journal - This study illustrates how entrepreneurship may contribute to enhancing economic growth and prosperity by reducing inequality and exclusion....     

基尼系数与收入分配

一、基尼系数与贫富差距 自古以来,有没有一种社会形态是贫富完全平均没有任何差异的呢?有,那就是原始社会早期,那时候物质财富极度贫乏,什么东西都是平均分配的。而从原始社会后期到现在,贫富差距就不同程度地存在着,“朱门酒肉臭,路有冻死骨”,绝不仅仅是中国古代贫富差距的写照。在现代发达国家,“富裕中的贫困”仍是普遍现象。在美国,一个普通医生年薪约20万美元,一个普通警官年薪为5万美元,而一个普通农场工人一年只能挣2万美元。即使同是医生,收入差距也相当大,横向和纵向不平等无处不在。     

质疑《一种改进的Gini系数》

《一种改进的Gini系数》登载于《数量经济技术经济研究》2005年第5期。鉴于不同的Lorenz曲线可能对应于同一个Gini系数,作者提出了一种改进方法,即通过平移、旋转、反射等线性变换并增加矩量,使得不同的Lorenz曲线能够对应不同的Gini系数。当然这时得到的已经是修正的Gini系数了。     

Reliable inference for the Gini index

Although attention has been given to obtaining reliable standard errors for the plug-in estimator of the Gini index, all standard errors suggested until now are either complicated or quite unreliable. An approximation is derived for the estimator by which it is expressed as a sum of IID random variables. This approximation allows us to develop a reliable standard error that is simple to compute. A simple but effective bias correction is also derived. The quality of inference based on the approximation is checked in a number of simulation experiments, and is found to be very good unless the tail of the underlying distribution is heavy. Bootstrap methods are presented which alleviate this problem except in cases in which the variance is very large or fails to exist. Similar methods can be used to find reliable standard errors of other indices which are not simply linear functionals of the distribution function, such as Sen’s poverty index and its modification known as the Sen–Shorrocks–Thon index.