A hierarchy of lorenz curves based on the generalized tukey's lambda distribution

A hierarchy of Lorenz curves based on the generalized Tukey's Lambda distribution is proposed. Representations of the corresponding distribution and density function are also provided, together with popular inequality measures. Estimation methods are suggested. Finally, a comparison with other parametric families of Lorenz curves is established.     

洛伦兹曲线的半参数估计

洛伦兹曲线与基尼系数是研究社会收入分配差异的重要工具.社会收入分配是一个复杂的过程,用尽可能精确的曲线给出洛伦兹曲线的估计进而给出基尼系数的估计,历来是统计学者和经济学者的工作目标.基于将参数方法与非参数方法相结合的思想给出洛伦兹曲线的半参数估计,进而导出基尼系数的估计,并据此进行了实证分析.     

Generating Ordered Families of Lorenz Curves by Strongly Unimodal Distributions

From any strongly unimodal density on the real line, it is possible to generate a one-parameter family of Lorenz curves. The resulting families of Lorenz curves are Lorenz ordered with respect to the indexing parameter. Symmetry of the unimodal density results in the generation of symmetric Lorenz curves. A related characterization of the normal distribution is presented.     

A new generalized Weibull distribution in income economic inequality curves

Researchers have been developing various extensions and modified forms of the Weibull distribution to enhance its capability for modeling and fitting different data sets. In this note, we investigate the potential usefulness of the new modification to the standard Weibull distribution called odd Weibull distribution in income economic inequality studies. Some mathematical and statistical properties of this model are proposed. We obtain explicit expressions for the first incomplete moment, quantile function, Lorenz and Zenga curves and related inequality indices. In addition to the well-known stochastic order based on Lorenz curve, the stochastic order based on Zenga curve is considered. Since the new generalized Weibull distribution seems to be suitable to model wealth, financial, actuarial and especially income distributions, these findings are fundamental in the understanding of how parameter values are related to inequality. Also, the estimation of parameters by maximum likelihood and moment methods is discussed. Finally, this distribution has been fitted to United States and Austrian income data sets and has been found to fit remarkably well in compare with the other widely used income models.     

Relationship between the weighted distributions and some inequality measures

In this paper, we study the relationships between the weighted distributions and the parent distributions in the context of Lorenz curve, Lorenz ordering and inequality measures. These relationships depend on the nature of the weight functions and give rise to interesting connections. The properties of weighted distributions for general weight functions are also investigated. It is shown how to derive and to determine characterizations related to Lorenz curve and other inequality measures for the cases weight functions are increasing or decreasing. Some of the results are applied for special cases of the weighted distributions. We represent the reliability measures of weighted distributions by the inequality measures to obtain some results. Length-biased and equilibrium distributions have been discussed as weighted distributions in the reliability context by concentration curves. We also review and extend the problem of stochastic orderings and aging classes under weighting. Finally, the relationships between the weighted distribution and transformations are discussed.     

The Lamé class of Lorenz curves

In this paper, the class of Lamé Lorenz curves is studied. This family has the advantage of modeling inequality with a single parameter. The family has a double motivation: it can be obtained from an economic model and from simple transformations of classical Lorenz curves. The underlying cumulative distribution functions have a simple closed form, and correspond to the Singh–Maddala and Dagum distributions, which are well known in the economic literature. The Lorenz order is studied and several inequality and polarization measures are obtained, including Gini, Donaldson–Weymark–Kakwani, Pietra, and Wolfson indices. Some extensions of the Lamé family are obtained. Fitting and estimation methods under two different data configurations are proposed. Empirical applications with real data are given. Finally, some relationships with other curves are included.     

Estimating Lorenz Curves Using a Dirichlet Distribution

The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population. When a particular functional form of the Lorenz curve is specified, it is typically estimated by linear or nonlinear least squares estimation techniques that have good properties when the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor normally distributed. This article proposes and applies a new methodology that recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the income proportions are distributed as a Dirichlet distribution. Five Lorenz curve specifications are used to demonstrate the technique. Maximum likelihood estimates under the Dirichlet distribution assumption provide better fitting Lorenz curves than nonlinear least squares and another estimation technique that has appeared in the literature.     

Lorenz ranking of income distributions

Based on the stochastic comparison of the Lorenz curves of income distributions, five partial orderings of income distributions are obtained. Three of these orderings are the well known star shaped, stochastic and the Lorenz orderings. The other two are new and are studied in some detail. The weakest ordering which is called the Lorenz area ordering is of special importance since it enables us to compare interesting Lorenz curves. This latter ordering leads to a class of income inequality measures which are identical with the linear inequality measures considered by Mehran (1976). A discussion of these measures is presented together with an application to part of Kunzet's (1963) data.     

Statistical inference for the difference of two Lorenz curves

Testing the Lorenz dominance is of importance in economic and social sciences. In this article, we propose new tools to do inferences for the difference of two Lorenz curves. The asymptotic normality of the proposed smoothed nonparametric estimator is proved. We also propose a smoothed jackknife empirical likelihood (JEL) method which avoids to estimate the complicate asymptotic variance. It is proved that the proposed JEL ratio statistics converge to the standard chi-square distribution. Simulation studies and real data analysis are also conducted, and show encouraging finite-sample performance.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

An extended Lindley distribution

In this paper we introduce an extension of the Lindley distribution which offers a more flexible model for lifetime data. Several statistical properties of the distribution are explored, such as the density, (reversed) failure rate, (reversed) mean residual lifetime, moments, order statistics, Bonferroni and Lorenz curves. Estimation using the maximum likelihood and inference of a random sample from the distribution are investigated. A real data application illustrates the performance of the distribution.     

The beta generalized Rayleigh distribution with applications to lifetime data

For the first time, we propose a new distribution so-called the beta generalized Rayleigh distribution that contains as special sub-models some well-known distributions. Expansions for the cumulative distribution and density functions are derived. We obtain explicit expressions for the moments, moment generating function, mean deviations, Bonferroni and Lorenz curves and densities of the order statistics and their moments. We estimate the parameters by maximum likelihood and provide the observed information matrix. The usefulness of the new distribution is illustrated through two real data sets that show that it is quite flexible in analyzing positive data instead of the generalized Rayleigh and Rayleigh distributions.     

Zenga Distribution and Inequality Ordering

The aim of this article is to establish an ordering related to the inequality for the recently introduced Zenga distribution. In addition to the well-known order based on the Lorenz curve, the order based on I(p) curve is considered. Since the Zenga distribution seems to be suitable to model wealth, financial, actuarial, and, especially, income distributions, these findings are fundamental in the understanding of how parameter values are related to inequality. This investigation shows that for the Zenga distribution, two of the three parameters are inequality indicators.     

The McDonald extended distribution: properties and applications

We study a five-parameter lifetime distribution called the McDonald extended exponential model to generalize the exponential, generalized exponential, Kumaraswamy exponential and beta exponential distributions, among others. We obtain explicit expressions for the moments and incomplete moments, quantile and generating functions, mean deviations, Bonferroni and Lorenz curves and Gini concentration index. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The applicability of the new model is illustrated by means of a real data set.     

A concentration approach to sensitivity studies in statistical estimation problems

It is shown that the concept of concentration is of potential interest in the sensitivity study of some parameters and related estimators. Basic ideas are introduced for a real parameter θ>0 together with graphical representations using Lorenz curves of concentration. Examples based on the mean, standard deviation and variance are provided for some classical distributions. This concentration approach is also discussed in relation with influence functions. Special emphasis is given to the average concentration of an estimator which provides a sensitivity measure allowing one to compare several estimators of the same parameter. Properties of this measure are investigated through simulation studies and its practical interest is illustrated by examples based on the trimmed mean and the Winsorized variance.     

General properties for the beta extended half-normal model

We formulate and study a four-parameter lifetime model called the beta extended half-normal distribution. This model includes as sub-models the exponential, extended half-normal and half-normal distributions. We derive expansions for the new density function which do not depend on complicated functions. We obtain explicit expressions for the moments and incomplete moments, generating function, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. In addition, the model parameters are estimated by maximum likelihood. We provide the observed information matrix. The new model is modified to cope with possible long-term survivors in the data. The usefulness of the new distribution is shown by means of two real data sets.     

A new class of generalized Lindley distributions with applications

A new four-parameter class of generalized Lindley (GL) distribution called the beta-generalized Lindley (BGL) distribution is proposed. This class of distributions contains the beta-Lindley, GL and Lindley distributions as special cases. Expansion of the density of the BGL distribution is obtained. The properties of these distributions, including hazard function, reverse hazard function, monotonicity property, shapes, moments, reliability, mean deviations, Bonferroni and Lorenz curves are derived. Measures of uncertainty such as Renyi entropy and s-entropy as well as Fisher information are presented. Method of maximum likelihood is used to estimate the parameters of the BGL and related distributions. Finally, real data examples are discussed to illustrate the applicability of this class of models.     

A new compound class of log-logistic Weibull–Poisson distribution: model,properties and applications

A new class of distributions called the log-logistic Weibull–Poisson distribution is introduced and its properties are explored. This new distribution represents a more flexible model for lifetime data. Some statistical properties of the proposed distribution including the expansion of the density function, quantile function, hazard and reverse hazard functions, moments, conditional moments, moment generating function, skewness and kurtosis are presented. Mean deviations, Bonferroni and Lorenz curves, Rényi entropy and distribution of the order statistics are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators and width of the confidence interval for each parameter and finally applications of the model to real data sets are presented to illustrate the usefulness of the proposed distribution.     

The Kumaraswamy Burr XII distribution: theory and practice

For the first time, a five-parameter distribution, called the Kumaraswamy Burr XII (KwBXII) distribution, is defined and studied. The new distribution contains as special models some well-known distributions discussed in lifetime literature, such as the logistic, Weibull and Burr XII distributions, among several others. We obtain the complete moments, incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and reliability of the KwBXII distribution. We provide two representations for the moments of the order statistics. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. For different parameter settings and sample sizes, various simulation studies are performed and compared to the performance of the KwBXII distribution. Three applications to real data sets demonstrate the usefulness of the proposed distribution and that it may attract wider applications in lifetime data analysis.