Concentration curve for truncated and censored data

Concentration curve is the inverse Lorenz curve. Together, they form the basis for most measures of distributional inequality. In this paper, we consider the empirical estimator of the concentration curve when the data are subjected to random left truncation and/or right censorship. Simultaneous strong Gaussian approximations for the associated Lorenz and normed concentration processes are established under appropriate assumptions. Functional laws of the iterated logarithm for the two processes are established as easy consequences. The construction provides a solid foundation for the study of functional statistics based on the two processes.     

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.     

Structure of Stieltjes classes of moment-equivalent probability laws

A Stieltjes class is a one-parameter family of moment-equivalent distribution functions constructed by modulation of a given indeterminate distribution function F, called the center of the class. Members of a Stieltjes class are mutually absolutely continuous, and conversely, any pair of moment-equivalent and mutually absolutely continuous distribution functions can be joined by a Stieltjes class. The center of a Stieltjes class is an equally weighted mixture of its extreme members, and this places restrictions on which distributions can belong to a Stieltjes class with a given center. The lognormal law provides interesting illustrations of the general ideas. In particular, it is possible for two moment equivalent infinitely divisible distributions to be joined by a Stieltjes class, and random scaling can be used to construct new Stieltjes classes from a given Stieltjes class.     

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.     

Some Classes of Multivariate Life Distributions in Discrete Time

New classes of multivariate survival distribution functions based on monotonic behaviour of a multivariate failure rate are developed in the discrete set up. Relationship among the classes along with multivariate geometric distributions that act as boundaries of the various classes are identified.     

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.     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.     

The beta generalized Gompertz distribution

A new five-parameter continuous model called the beta generalized Gompertz distribution is introduced and studied. This distribution contains the Gompertz, generalized Gompertz, beta Gompertz, generalized exponential, beta generalized exponential, exponential and beta exponential distributions as special sub-models. Some mathematical properties of the new model are derived. We show that the density function of the new distribution can be expressed as a linear combination of Gompertz densities. We obtain explicit expressions for the moments, moment generating function, quantile function, density function of the order statistics and their moments, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The model parameters are estimated by using the maximum likelihood method of estimation and the observed information matrix is determined. Finally, an application to real data set is given to illustrate the usefulness of the proposed model.     

Meritocratic and monopoly inequality: A computer simulation of income distribution

This paper explores the properties of a model of the distribution of income in which individual income is proportional to a multiplicative function of previous income, ability, chance, a ceiling factor determined by competition among members of an income class for resources held by members of other classes, and an additive factor summarizing effects of altruism and minimal subsistence. The behavior of the model is investigated by computer simulation for combinations of values of three model parameters representing the tendency of income to grow exponentially (the Monopoly effect), the weight of the ability factor (the meritocracy effect), and the weight of the ceiling factor resulting from competitive interactions. Steady state income distributions generated by the model are characterized by measures of income inequality, exchange mobility, elite stability, and meritocracy. Results suggest that for constant Monopoly effect, the effect of the meritocracy parameter on various aggregate outcomes is nonlinear, with a range over which greater returns to ability produce lower inequality, lower exchange mobility, greater elite stability and meritocracy, for constant returns to ability, a greater Monopoly effect generally produces greater inequality, more exchange mobility, less stability of the elite, and lower meritocracy. Results also reveal a nonlinear relationship between exchange mobility and inequality, with mobility decreasing to a minimum and then increasing again as inequality increases; a nonlinear but monotonic negative relationship between elite stability and inequality, with greater inequality, associated with less stability, and a nonlinear relationship between meritocracy and inequality, with meritocracy increasing at first with inequality at low inequality levels, reaching a maximum and then decreasing as inequality increases further. These findings are interpreted in relation to major stratification trends in the course of sociocultural evolution.     

On the Gittins index in the M/G/1 queue

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground–Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground–Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic.     

A characterization of inequality measures based on bilateral inequality

Summary. This article deals with inequality measures on income distributions. These measures are assumed to fulfil common axioms like the Principle of Progressive Transfers, Normalization, and Symmetry. Strict Monotonicity will be defined and is included in the basic set of axioms. If the measures are especially based on bilateral comparisons and if two additional conditions are fulfilled the bilateral inequality measure must be proportional to the absolute distance of the two incomes involved. It is remarkable that all these conditions do not give an answer whether an inequality measure for two incomes should be a quasi-convex or a quasi-concave function. This question is controversely discussed in economic research. Our surprising answer, however, is given by use of RAO's statistic concept defining the diversity of a population.     

拟序下收入分配的不公平度量研究

收入分配的不公平度量是福利经济学研究的一个主要问题.Alain和Patrick在收入分配的不公平度量方法中引入了拟序下的绝对差异、绝对剥夺和绝对满足的概念,对它们之间的关系有过讨论,但并没有给出证明.本文在此基础之上,详细地探讨了它们三者、Lorenz准则、累进性转移、T-转移之间的关系,同时给出了完整的证明.     

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.     

The Möbius distribution on the disc

A simple new family of distributions is proposed which has support the unit disc in two dimensions. The density functions of the family are unimodal, monotonic or uniantimodal. The bivariate symmetric beta distributions, which include the uniform distribution, are special cases, but many members of the family are skew. The distributions have three parameters, one controlling orientation, one controlling degree of concentration and the third controlling skewness, or more precisely off-centredness. Importantly, these parameters are globally orthogonal. An illustrative example of fitting the model to data is given. Conditional and marginal distributions are considered. The new distributions are compared favourably with an earlier suggestion of the same author.     

Copula analysis of mixture models

Contemporary computers collect databases that can be too large for classical methods to handle. The present work takes data whose observations are distribution functions (rather than the single numerical point value of classical data) and presents a computational statistical approach of a new methodology to group the distributions into classes. The clustering method links the searched partition to the decomposition of mixture densities, through the notions of a function of distributions and of multi-dimensional copulas. The new clustering technique is illustrated by ascertaining distinct temperature and humidity regions for a global climate dataset and shows that the results compare favorably with those obtained from the standard EM algorithm method.     

Stochastic comparisons of distorted variability measures

In this paper, we consider the dispersive order and the excess wealth order to compare the variability of distorted distributions. We know from Sordo (2009a) that the excess wealth order can be characterized in terms of a class of variability measures associated to the tail conditional distribution which includes, as a particular measure, the tail variance. Given that the tail conditional distribution is a particular distorted distribution, a natural question is whether this result can be extended to include other classes of variability measures associated to general distorted distributions. As we show in this paper, the answer is yes, by focusing on distorted distributions associated to concave distortion functions. For distorted distributions associated to more general distortions, the characterizations are stated in terms of the stronger dispersive order.     

The exponentiated generalized power Lindley distribution: Properties and applications

In this paper, we introduce a new extension of the power Lindley distribution, called the exponentiated generalized power Lindley distribution. Several mathematical properties of the new model such as the shapes of the density and hazard rate functions, the quantile function, moments, mean deviations, Bonferroni and Lorenz curves and order statistics are derived.Moreover, we discuss the parameter estimation of the new distribution using the maximum likelihood and diagonally weighted least squares methods. A simulation study is performed to evaluate the estimators. We use two real data sets to illustrate the applicability of the new model. Empirical findings show that the proposed model provides better fits than some other well-known extensions of Lindley distributions.     

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’.     

Relative difference in diversity between populations

An entropy is conceived as a functional on the space of probability distributions. It is used as a measure of diversity (variability) of a population. Cross entropy leads to a measure of dissimilarity between populations. In this paper, we provide a new approach to the construction of a measure of dissimilarity between two populations, not depending on the choice of an entropy function, measuring diversity. The approach is based on the principle of majorization which provides an intrinsic method of comparing the diversities of two populations. We obtain a general class of measures of dissimilarity and show some interesting properties of the proposed index. In particular, it is shown that the measure provides a metric on a probability space. The proposed measure of dissimilarity is essentially a measure of relative difference in diversity between two populations. It satisfies an invariance property which is not shared by other measures of dissimilarity which are used in ecological studies. A statistical application of the new method is given.     

Bayesian Influence Assessment in the Growth Curve Model with Unstructured Covariance

From a Bayesian point of view, in this paper we discuss the influence of a subset of observations on the posterior distributions of parameters in a growth curve model with unstructured covariance. The measure used to assess the influence is based on a Bayesian entropy, namely Kullback-Leibler divergence (KLD). Several new properties of the Bayesian entropy are studied, and analytically closed forms of the KLD measurement both for the matrix-variate normal distribution and the Wishart distribution are established. In the growth curve model, the KLD measurements for all combinations of the parameters are also studied. For illustration, a practical data set is analyzed using the proposed approach, which shows that the diagnostics measurements are useful in practice.