Asymptotic behaviors of the Lorenz curve and Gini index in sampling from a length-biased distribution

In this work, we consider the nonparametric estimators of the Lorenz curve and Gini index based on a sample from the corresponding length-biased distribution. We show that this estimators are strongly consistent for the associated Lorenz curve and Gini index. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. We apply the strong Gaussian approximation technique to obtain a functional law for the iterated logarithm for the Lorenz curve. Also, we obtain an asymptotic normality for the corresponding Gini index.     

Lorenz Curve for Truncated and Censored Data

In this paper, we consider a nonparametric estimator of the Lorenz curve and Gini index when the data are subjected to random left truncation and right censorship. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. A law of the iterated logarithm for the Lorenz curve is also derived. Lastly, we obtain a central limit theorem for the corresponding Gini index.     

Gini indices and the moments of the share density function

The expected value of the share density of the income distribution can be expressed in terms of the Gini index. The variance of the share density of the income distribution is interesting because it gives a relationship between the first and the second order Gini indices. We find an expression for this variance and, as a result, we obtain some nontrivial bounds on these Gini indices. We propose new statistics on the income distribution based on the higher moments of the share density function. These new statistics are easily computable from the higher order Gini indices. Relating these moments to higher order Ginis suggests new estimates on these quantities.     

Ordering Gini indexes of multivariate elliptical risks

Gini index is a well-known tool in economics that is often used for measuring income inequality. In insurance, the index and its modifications have been used to compare the riskiness of portfolios, to order reinsurance contracts, and to summarize insurance scores (relativities). In this paper, we establish several stochastic orders between the Gini indexes of multivariate elliptical risks with the same marginals but different dependence structures. This work is motivated by the applied studies of Brazauskas et al. (2007) and Samanthi et al. (2015), who employed the Gini index to compare the riskiness of insurance portfolios. Based on extensive Monte Carlo simulations, these authors have found that the power function of the associated hypothesis test increases as portfolios become more positively correlated. The comparison of the Gini indexes (of empirically estimated risk measures) presented in this paper provides a theoretical explanation to this statistical phenomenon. Moreover, it enriches the studies of the problem of central concentration of elliptical distributions and generalizes the pd-1 order proposed by Shaked and Tong (1985).     

Apportionments with minimum Gini index of disproportionality: a Quadratic Knapsack approach

The ultimate goal of proportional apportionment methods is the minimization of disproportionality, i.e., unequal distribution of political representation among voters, or citizens. The Gini index is a well known tool for measuring inequality. In this work we propose a quotient method that minimizes the Gini index of disproportionality. Our method reduces the rounding of quotas to an instance of quadratic knapsack, a widely studied combinatorial optimization problem. Preliminary computational results, including real cases from the EU Parliament and the US House of Representatives, show that the method is effective, since the instances to solve are rather easy.     

混合总体基尼系数的下限——兼论我国城乡合在一起时基尼系数的计算

本文研究基尼系数的估计问题.国家统计局每年发布的统计年鉴包含城镇和农村个人收入的分组数据,但分点不相同并且未公布.这些情况对于估计整个社会的基尼系数带来挑战.本文对于两个总体按照一定比例混合后的新总体,针对来自原来两个总体的分组数据,给出了新总体的基尼系数的下限.并将所得的结果用于计算我国城乡合在一起时基尼系数的下限值.这些结果容易推广到更多总体混合情形,也可以应用到其它实际情况的基尼系数的估计,比如国家或地区的联合体.     

Generalized gini inequality indices

When incomes are ranked in descending order the social-evaluation function corresponding to the Gini relative inequality index can be written as a linear function withthe weights being the odd numbers in increasing order. We generalize this function by allowing the weights to be an arbitrary non-decreasing sequence of numbers. This results in a class of generalized Gini relative inequality indices and a class of generalized Gini absolute inequality indices. An axiomatic characterization of the latter class is also provided.     

Distributions and moments for estimators of Gini index in an exponential distribution

In this paper, we propose several estimators of Gini index of the two-parameter exponential distribution and obtain distributions and moments of the proposed estimators. The proposed estimators are shown to cosistency and will be compared in terms of the mean squared error (MSE) through Monte Carlo method.     

中国收入分配差距预测的GM(1,1)模型

基尼系数是现行较为通用的衡量一国收入分配差距的指标,利用灰色系统中的GM(1,1)预测模型,建立起对我国收入分配差距的预测模型,得到2005年基尼系数的预测值0.471,进而建立精度更高的新陈代谢模型,得到2006—2010年的基尼系数预测值,结果表明到2010年,基尼系数有可能临近0.5,基尼系数依然是攀升的,从而分配差距的扩大趋势依未改变,但增长幅度缓慢而平稳,这为当前及今后几年经济发展和调节收入分配差距的政策制定提供了一定的参考.     

On Thon's axiomatization of the Gini index

We give a short proof of Thon's characterization of the Gini index.     

Stochastic ordering of Gini indexes for multivariate elliptical risks

In this paper, we show that the conjecture, made by Samanthi et al. (2016), on the ordering of Gini indexes of multivariate normal risks with respect to the strength of dependence, is not true. By using the positive semi-definite ordering of covariance matrices, we can obtain the usual stochastic order of the Gini indexes for multivariate normal risks. This can be generalized to multivariate elliptical risks. We also investigate the monotonicity of the Gini indexes in the usual stochastic order when the covariance (dispersion, resp.) matrices of multivariate normal (elliptical, resp) risks increase componentwise. In addition, we derive a large deviation result for the Gini indexes of multivariate normal risks.     

Measuring income inequalities beyond the Gini coefficient

Growing interest in the analysis of interrelationships between income distribution and economic growth has recently stimulated new theoretical and empirical research. Measures such as the head-count ratio for the poverty index or the widely used Gini coefficient are aggregated indicators describing the general extent of inequality without deeper insights into income distribution among households. To derive an indicator accounting for income distribution among income groups, we propose a new approach based on an output oriented DEA model where the input value is unitized to 1 for each country and weights restrictions imposed so as to favour a higher income share in the lower quantiles. We demonstrate the merits of this approach on the quintile income breakdown data of 29 European countries. Prioritizing lower income groups’ welfare, countries such as Slovenia and Slovakia can be equally favoured by the new proposed indicator while being assessed differently by the Gini index. An intertemporal analysis reveals a slight deterioration of income distribution in the majority of 29 European countries over the period of 2007–2016 in a Rawlsian sense.

     

Distributions of maximum likelihood estimators of Lorenz curve and Gini index of exponential distribution

Summary This paper presents the maximum likelihood estimators (MLEs) of the Lorenz curve and Gini index of the exponential distribution, their exact distributions and moments. All these MLEs are shown to converge almost surely and in therth mean. Further their asymptotic distributions are obtained. Here we use only very simple arguments to derive certain results that are very useful in statistical study of ‘inequality’.     

关于洛伦兹曲线和基尼系数的统计推断

本文对于洛伦兹曲线的最一般定义,在无任何附加限制条件下,论证了基于样本数据所得到的洛仑兹曲线的通常估计量具有强相合性.在此基础上,证明了基尼系数和Schutz系数的通常估计量均具有强相合性.此外,导出了基尼系数估计量的渐进分布,并在此基础上给出了大样本情形下基尼系数的置信区间.     

Bootstrapping the Kaplan–Meier estimator on the whole line

This article is concerned with proving the consistency of Efron’s bootstrap for the Kaplan–Meier estimator on the whole support of a survival function. While previous works address the asymptotic Gaussianity of the Kaplan–Meier estimator without restricting time, we enable the construction of bootstrap-based time-simultaneous confidence bands for the whole survival function. Other practical applications include bootstrap-based confidence bands for the mean residual lifetime function or the Lorenz curve as well as confidence intervals for the Gini index. Theoretical results are complemented with a simulation study and a real data example which result in statistical recommendations.

     

基于基尼系数的区域创新投入产出熵权TOPSIS匹配性评价分析

采用匹配基尼系数对我国创新系统投入产出匹配性程度进行了测算。通过构建基于基尼系数的TOPSIS熵权评价模型,对我国大陆31个省市自治区的创新投入产出匹配性进行了评价,对各地区投入产出匹配性评价值聚类分析。研究结果表明我国创新系统投入产出呈现出不匹配的状态,形成了京津、长江三角洲和珠江三角洲三个极化区域。研究结果验证了本文构建模型的可行性和合理性,为区域创新投入产出的研究提供了新的思路。     

Multivariate Gini Indices

Two extensions of the univariate Gini index are considered:R_D, based on expected distance between two independent vectors from the same distribution with finite meanμ ^d; andR_V, related to the expected volume of the simplex formed fromd+1 independent such vectors. A new characterization ofR_Das proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. TheLorenz zonoidwas suggested as a multivariate generalization of the Lorenz curve.R_Vis, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. Whend=1, bothR_DandR_Vequal twice the area between the usual Lorenz curve and the line of zero disparity. Whend>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid:R_Dis proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, whileR_Vis the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces.     

Random effects clustering in multilevel modeling: choosing a proper partition

A novel criterion for estimating a latent partition of the observed groups based on the output of a hierarchical model is presented. It is based on a loss function combining the Gini income inequality ratio and the predictability index of Goodman and Kruskal in order to achieve maximum heterogeneity of random effects across groups and maximum homogeneity of predicted probabilities inside estimated clusters. The index is compared with alternative approaches in a simulation study and applied in a case study concerning the role of hospital level variables in deciding for a cesarean section.

     

改进的收入空间法及其在中等收入人口度量中的应用

提出了衡量社会分配公平的局部公平指数D(p)=((L'(p))/((L"(p))),给出了其确切的经济学解释.同时定义了分别对应穷人和富人的Gini指数G_1,G_2,在此基础上建立了改进的收入空间法,用于确定中等收入人口范围:[(1-1/2D(1-G_1))m,(1+1/2D(1-G_2))m],其中m为中位收入.模型对2013全国研究生数学建模竞赛E题的收入分配数据进行定量研究,得出的结论与经济学的原理"两极分化时中等收入人口降低,中问隆起时中等收入人口升高"相符合,这体现了模型的准确性与可靠性.     

Gini Correlation for Feature Screening

Acta Mathematicae Applicatae Sinica, English Series - In this paper we propose the Gini correlation screening (GCS) method to select the important variables with ultrahigh dimensional data. The new...