Empirical likelihood-based inferences for the Lorenz curve

In this paper, we discuss empirical likelihood-based inferences for the Lorenz curve. The profile empirical likelihood ratio statistics for the Lorenz ordinate are defined under the simple random sampling and the stratified random sampling designs. It is shown that the limiting distributions of the profile empirical likelihood ratio statistics are scaled Chi-square distributions with one degree of freedom. We also derive the limiting processes of the associated empirical likelihood-based Lorenz processes. Hybrid bootstrap and empirical likelihood intervals for the Lorenz ordinate are proposed based on the newly developed empirical likelihood theory. Extensive simulation studies are conducted to compare the relative performances of various confidence intervals for Lorenz ordinates in terms of coverage probability and average interval length. The finite sample performances of the empirical likelihood-based confidence bands are also illustrated in simulation studies. Finally, a real example is used to illustrate the application of the recommended intervals.     

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.     

一种收入分配洛伦兹曲线拟合的线性模型研究

洛伦兹曲线是用来描述社会收入分配状况的一种曲线,能够精确有效地拟合洛伦兹曲线是进行收入分配定量分析的基础.基于洛伦兹曲线的经济学规律和数学特点,提出了一种新的指数函数多项式拟合洛伦兹曲线的线性模型,并讨论研究模型的矩阵广义逆和线性最小二乘求解方法.通过与5个典型的非线性拟合模型的实例对比分析,验证了方法具有收敛稳定、精度高、对噪声不敏感的优点.     

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.     

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.     

Numerical and physical modeling of the dynamics of the Lorenz system

This paper describes a modification of the method of power series for the construction of approximate solutions of the Lorenz system. The results of a computer-aided simulation are presented. Also, physical modeling of the dynamics of the Lorenz system for processes occurring in an electric circuit is considered.     

广义Lorenz 曲线的非参数统计推断

本文讨论了广义Lorenz 曲线的经验似然统计推断. 在简单随机抽样、分层随机抽样和整群随机抽样下, 本文分别定义了广义Lorenz 坐标的pro le 经验似然比统计量, 得出这些经验似然比的极限分布为带系数的自由度为1 的χ² 分布. 对于整个Lorenz 曲线, 基于经验似然方法类似地得出相应的极限过程. 根据所得的经验似然理论, 本文给出了bootstrap 经验似然置信区间构造方法, 并通过数据模拟, 对新给出的广义Lorenz 坐标的bootstrap 经验似然置信区间与渐近正态置信区间以及bootstrap 置信区间等进行了对比研究. 对整个Lorenz 曲线, 基于经验似然方法对其置信域也进行了模拟研究. 最后我们将所推荐的置信区间应用到实例中.     

Shape-restricted inference for Lorenz curves using duality theory

We propose a new methodology for estimating curves under a convexity restriction based on Fenchel duality and wavelet approximations. In contrast to approaches where a possibly non-convex estimator is convexified at a second stage, our procedure allows us to construct directly an estimator with a convex shape. The method is applied to the estimation of the Lorenz curve. Applications to estimation of average value at risk, as well as multivariate generalisations to Lorenz surfaces are mentioned. We show asymptotic efficiency which demonstrates that the convexity is achieved at no extra cost.     

Construction of concentration measures for General Lorenz curves using Riemann-Stieltjes integrals

Lorenz curves were invented to model situations of inequality in real life and applied in econometrics (distribution of wealth or poverty), biometrics (distribution of species richness), and informetrics (distribution of literature over their producers). Different types of Lorenz curves are hereby found in the literature, and in each case a theory of good concentration measures is presented. The present paper unifies these approaches by presenting one general model of concentration measure that applies to all these cases. Riemann-Stieltjes integrals are hereby needed where the integrand is a convex function and the integrator a function that generalizes the inverse of the derivative of the Lorenz function, in case this function is not everywhere differentiable.Calling this general measure C we prove that, if we have two Lorenz functions f, g such that f < g, then C(f) < C(g). This general proof contains the many partial results that are proved before in the literature in the respective special cases.     

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.     

两同心球间旋转流动类Lorenz方程组的静态分歧

对同心球间旋转流动的N av ier-S tokes方程谱展开后进行三模态截断,讨论了所得到的类Lorenz型方程组的分歧问题.给出了静态奇异点的条件,并计算出解分支.首先,简要介绍了Lorenz方程组以及用Lorenz截断法讨论非线性问题的意义,其次,推导同心球间旋转流动N av ier-S tokes方程的流函数-涡度形式,最后,讨论同心球间旋转流动的类Lorenz型方程组的分歧问题.     

两同心球间旋转流动类Lorenz方程组的分歧分析

0引言两个同心旋转球之间的流动又称为球Couette流动.作为一个简单的模型,研究它能够为揭示流动失稳转捩至湍流这一重大理论课题的规律提供线索;同时,由于球Couette流动更象全球大气流动,研究它也能成为研究大气物理提供一个粗略的模型,为这一方面     

The dependence of the height of a Lorenz curve of a Zipf function on the size of the system

The Lorenz curve of a Zipf function describes, graphically, the relation between the fraction of the items and the fraction of the sources producing these items. Hence it generalizes the so-called 80/20-rule to general fractions.In this paper we examine the relation of such Lorenz curves with the size of the system (expressed by the total number of sources). We prove that the height of such a Lorenz curve is an increasing function of the total number of sources.In other words, we show that the share of items as a function of the corresponding share of sources increases with increasing size of the system. This conclusion is opposite (but not in contradiction) to a conclusion of Aksnes and Sivertsen (studied in an earlier paper of Egghe) but where “share of sources” is replaced by “number of sources”.     

A universal unfolding of the Lorenz system

A universal unfolding of the Lorenz system is derived and studied in this paper. Both rigorous theoretical analysis and numerical simulations show that the Lorenz system, the Chen system, and the Lü system belong to the same universal unfolding. Therefore, they all have similar dynamical behaviors in the sense that if the Lorenz system has limit cycles produced from a Hopf bifurcation for a certain set of parameter values, then the other two systems also have limit cycles from the same set of parameter values; and if the Lorenz, Chen, and Lü systems are chaotic for some parameter values (for example, some typical parameter values), respectively, then the homotopic system for the Lorenz system and the Chen system, and the homotopic system for these three systems, are all chaotic within the entire domain of these homotopic parameters.     

A complete bifurcation diagram of nonlocal bifurcations of singular points in the Lorenz system

For the system of Lorenz equations in the parameter space we construct a complete bifurcation diagram of all homoclinic and heteroclinic separatrix contours of singular points that exist in the system. These constructs include the existence surface of a homoclinic butterfly, the existence half-surface of homoclinic loops of saddle-focus separatrices, and the existence curve of a heteroclinic separatrix contour joining a saddle-node with two saddle-foci.     

The Inequality Between Two Distributions: Applications to the Analysis of Diversity Data

A new measure called diversity difference is proposed for the inequality of a pair of distributions. The diversity difference measure satisfies eight properties of a measure of inequality. This measure is simple to calculate and provides easily interpreted results.Existing inequality measures examine the distribution of a single variable whose data are arranged in a monotonic order. The new measure can employ multiple variables and does not require each to be monotonic but can be used if the data happen to be monotonic. The pair of distributions is useful for organizational diversity data because one of the distributions represents the actual proportions of employees in any class or set of classes and the other distribution is the benchmark or anchoring distribution. Data from the measure can be displayed in diversity difference trees for quick interpretation.The diversity difference measure can be arranged to define a Lorenz curve. An example with three classes (gender, race, and age) is employed to provide examples of the measure, the resulting Lorenz curve, and the disparity ratio.     

Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems and two linearly coupled Lorenz systems. Some sufficient conditions for synchronization are attained through rigorous mathematical theory. Compared with the results in the reference [Chaos, Solitons & Fractals 2002;14:529], the sufficient condition for the synchronization of two linearly coupled Lorenz systems is simpler and less conservative. Numerical simulations are provided for illustration and verification.     

Classifying 3-trip Lorenz knots

The topological types of closed periodic solutions of the Lorenz equations are in one-to-one correspondence with aperiodic positive words on two generators. The number of syllables in such a word is called the trip number of the corresponding knot. Classifications for knots with trip numbers 1 and 2 are known. This paper gives a complete classification for 3-trip Lorenz knots.     

Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of Lorenz equations

The computational uncertainty principle (CUP) is applied to explain the experimental formulae of the critical time of decoupling for Lorenz equations (LEs). We apply the multiple precision (MP) library in obtaining the long-time solution of LEs, and based on the classic Taylor scheme, we developed a high-performance parallel Taylor solver to do the computation. The new solver is several hundreds times faster than the reported solvers developed in MATHEMATICA software, and it has the ability to yield longer solutions of LEs, up to t ∼ 10⁴ LTU (Lorenz time unit). Further, we notice that the two computation processes with different precisions or orders will produce the reliable correct reference solutions before they have a significant difference. According to this property we propose an approach for maintaining the correct numerical solution. The new solver and the solution validation approach are used to identify and correct an erroneous solution reported in a previous study.