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.     

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.     

Two-part fractional regression model with conditional FDH responses: an application to Brazilian agriculture

We extend the notion of a two-part fractional regression model with conditional free disposal hull efficiency responses to accommodate two-stage regression analysis. The two-part regression model includes the binomial model with a nonlinear specification for the expected response in (0,1] and is a more general formulation in the context of fractional regressions. We use nonlinear least squares to assess the effect of covariates in the conditional efficiency response. The approach is applied to Brazilian agricultural county data, as reported in the Brazilian agricultural census of 2006. The efficiency measure is output oriented and assumes variable returns to scale. Output is rural gross income and inputs are land expenses, labor expenses and expenses on other technological inputs. The covariates affecting production are credit, technical assistance, a rural development index, income concentration, measured by the Gini index, and regional dummies. Overall Brazilian rural production performance responds positively to all covariates.

     

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.

     

基于凹凸组合的一类洛伦兹曲线模型

为了统计和分析一个国家和地区的收入分配情况,经济学界往往通过入户调查获得家庭收入与消费等数据,采用洛伦兹曲线模型来进行数据拟合.洛伦兹曲线模型拟合效果的好坏,直接影响着收入分配的描述.本文构建了一类凹凸组合的洛伦兹曲线模型,并针对19个国家的收入分配数据进行了实证分析.结果显示该模型具有较好的拟合效果,其基尼系数能较好地描述收入分配现状,对反映和监测居民之间的贫富差距具有重要意义.     

Social welfare and the gini coefficient revisited

The question of conflict between the rankings of income distributions with the same mean by the Gini coefficient and by individualistic social welfare functions is re-examined. The negative result of Newbery (1970) is extended. However, positive results are obtainable which reverse Newbery's conclusion by admitting into the individual's utility index a measure of his position in the income distribution, or of his deprivation with respect to others' incomes.     

Quantile estimations via modified Cholesky decomposition for longitudinal single-index models

Quantile regression is a powerful complement to the usual mean regression and becomes increasingly popular due to its desirable properties. In longitudinal studies, it is necessary to consider the intra-subject correlation among repeated measures over time to improve the estimation efficiency. In this paper, we focus on longitudinal single-index models. Firstly, we apply the modified Cholesky decomposition to parameterize the intra-subject covariance matrix and develop a regression approach to estimate the parameters of the covariance matrix. Secondly, we propose efficient quantile estimating equations for the index coefficients and the link function based on the estimated covariance matrix. Since the proposed estimating equations include a discrete indicator function, we propose smoothed estimating equations for fast and accurate computation of the index coefficients, as well as their asymptotic covariances. Thirdly, we establish the asymptotic properties of the proposed estimators. Finally, simulation studies and a real data analysis have illustrated the efficiency of the proposed approach.

     

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.     

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).     

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.     

A GENERATIVE MODEL OF INCOME DISTRIBUTION: FORMALIZATION WITH ITERATED INVESTMENT GAME

The purpose of this paper is to formalize a simple model that theoretically connects individuals' rational choice at the micro level to income distribution, which is subject to the Gibrat's law empirically, as social structure at macro level. We use an iterated investment game as a baseline model in which a player has a binary choice between investing and not investing. Given parameters which prescribe the payoff structure of the game are the prize density γ and the rate of return R. Method of analysis is a simulation with computation. We investigate changes in the Gini coefficient and skewness of the total profit distribution, as the parameters varied as follows: 0 ≤ γ ≤ 1, R = 0.5, 1, 2, 3, and n (the number of times that the game is repeated) = 5, 10. As a result of analysis, we derive the implication that the Gini coefficient increases up to critical point, where 0 ≤ γ ≤ 1/(R + 1), then decreases as prize density increases, where 1/(R + 1) < γ ≤ 1. Furthermore, we show that our model, with cumulative effect, generates a lognormal distribution under the condition that 1/(R + 1) < γ ≤ 1.     

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.     

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.     

都市圈农村收入差异及影响因素分析——以北京市怀柔区为例

对都市圈农村居民收入差异进行研究将有助于正确的认识和指导都市农村经济社会发展的实践,准确地判断和选择合适的都市农村经济发展模式,提高农村居民的收入水平,促进地区的生产发展.综合运用直方图、箱线图、基尼系数对怀柔区农村收入差异及其变化情况进行了分析.结合怀柔区资源禀赋特点对造成农村收入差异的地理和交通区位、就业结构、旅游开发、农村投资等重要形成因素进行了理论分析,在此基础上,筛选出了五环距离、景区级别、是否民俗村等关键解释变量,并依据上述变量进行了回归分析.     

On the asymptotic equilibrium of a population system with migration

In the analysis of economic and social issues of a country (or any larger or smaller socio-economic unit) the demographic dynamics of the considered population often play a crucial role. Very current emergencies in this respect are e.g. ageing, longevity risk, state-run healthcare etc. Over the last decade migration between EU countries also became an important issue, and in recent years the uncontrolled migration from non-EU countries is also a major concern. Therefore, the better theoretical understanding of the evolutionary mechanism of age-classified populations interacting via migration, is a timely modelling-methodological task. This paper is a preliminary demographic methodological contribution to a further research in support of socio-economic modelling and decision making concerning migration issues.It is known that in the framework of the classical age-specific Leslie model, under simple demographic conditions, a closed population in the long term tends to an equilibrium age distribution. As the main theoretical result of the paper, a similar convergence is proved for a system of several populations with migration between them, and this long-term behaviour (convergence theorem) is extended to systems of sex-structured populations. Based on the latter model, medium term projections are also analysed concerning the effect of migration among countries on the development of the old-age dependency ratio (the proportion of pensioner age classes to active ones), which is an aggregate scalar indicator of ageing, a major concern in most industrialized countries. Illustrative simulation analysis is carried out with data from three European countries.     

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.     

Analysis of the third European survey on working conditions with composite indicators

A composite indicator Working conditions for comparing European countries is constructed from data of the Third European Survey on Working Conditions. The main findings are as follows: (a) European countries differ with respect to working conditions statistically more significantly than with respect to earnings; it implies a quite accurate discrimination threshold in ranking countries with respect to working conditions, (b) working conditions and earnings positively correlate over the whole of Europe but correlate little within single countries; it indicates at the prevailing role of national determinants over professional or social specificities as contributing to the average working conditions, and (c) earnings play no essential role in subjective estimations, including job satisfaction, which mainly depends on working conditions; consequently, more attention should be paid to improving the latter.     

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.

     

Extended Gini-Type Measures of Risk and Variability

ABSTRACT

The aim of this paper is to introduce a risk measure, Extended Gini Shortfall (EGS), that extends the Gini-type measures of risk and variability by taking risk aversion into consideration. Our risk measure is coherent and catches variability, an important concept for risk management. The analysis is made under the Choquet integral representations framework. We expose results for analytic computation under well-known distribution functions. Furthermore, we provide a practical application.