基于分组数据基尼系数的一种估算方法

基尼系数是测量收入差距的最常用指标,自提出以来学者们演绎出了很多种测算方法,对于离散数据主要有平均差法、几何方法、矩阵法和协方差法等,但这些方法实质上是一致的,只是形式有所不同。文章以上述方法为基础,以分组数据为研究对象,假定各阶层的收入同分布且与总体分布相似,利用“逼近法”来拟合洛伦兹曲线,并在此基础上提出了一种较为简洁的测算基尼系数的方法。对我国部分年份不同群体基尼系数的测算结果表明,所提方法是可行的。     

Inference for Income Distributions Using Grouped Data

We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. We derive generic moment conditions and an optimal weight matrix that can be used for generalized method-of-moments (GMM) estimation of any parametric income distribution. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, which incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution, which uses only population proportions. In contrast to the earlier work of Chotikapanich, Griffiths, and Rao, and Chotikapanich, Griffiths, Rao, and Valencia, which did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind (GB2), and its special-case distributions, the beta-2, Singh–Maddala, Dagum, generalized gamma, and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided. Supplementary materials for this article are available online.     

基于分组数据的样本基尼系数范围估计

 由于很多收入抽样数据只是公布了相对简约的分组数据,如何依据信息不完整的分组数据估计样本基尼系数的范围是非常重要的。本文针对分组数据中各组收入的取值范围以及各组人均收入是否已知,从多个方面探讨了样本基尼系数的取值范围,并给出了相应的估算公式。最后,结合我国城乡居民收入的分组数据,实证分析了城乡收入基尼系数的范围。     

Histogram-Based Interpolation of the Lorenz Curve and Gini Index for Grouped Data

In grouped data, the estimation of the Lorenz curve without taking into account the within-class variability leads to an overestimation of the curve and an underestimation of the Gini index. We propose a new strictly convex estimator of the Lorenz curve derived from a linear interpolation-based approximation of the cumulative distribution function. Integrating the Lorenz curve, a correction can be derived for the Gini index that takes the intraclass variability into account.     

Large Sample Theory for the Bounds on the Gini and Related Indices of Inequality Estimated From Grouped Data

The joint asymptotic distribution of the upper and lower bounds for the Gini index derived by Gastwirth for grouped data are obtained. From them a conservative asymptotically distribution-free confidence interval for the population Gini index is presented. The methods also yield similar results for other indices of inequality (e.g., Theil's and Atkinson's).     

On the Posterior Consistency of Mixtures of Dirichlet Process Priors with Censored Data

Mixtures of Dirichlet process priors offer a reasonable compromise between purely parametric and purely non‐parametric models, and are popularly used in survival analysis and for testing problems with non‐parametric alternatives. In this paper, we study large sample properties of the posterior distribution with a mixture of Dirichlet process priors. We show that the posterior distribution of the survival function is consistent with right censored data.     

从统计实施角度对基尼系数的再认识

文章从收集收入样本数据、到采用各种方法估算基尼系数、再到具体运用基尼系数进行收入差异评价的整个环节中,探讨了各个部分可能会出现的问题,以期能帮助我们如何用基尼系数客观地评判现实的经济生活。     

城乡混合基尼系数分解方法研究

本文从分布函数出发,设计了一种新的城乡混合基尼系数分解方法,较好破解了传统方法中约束条件严格、分解结果不清晰等技术难题,同时还通过分布函数的交叉对比提出了一种新的测度城乡差距的相对指标,该指标与基尼系数的内涵是一致的,有着明确的经济涵义和优良的理论性质。最后应用新方法对2009至2011年中国城乡混合基尼系数进行了计算和分解,发现收入差距有减小的趋势,而城乡差距的贡献率为60%左右。本文提出的方法也适用于其他类型不同群体之间的收入差距分析。     

Posterior Analysis for Normalized Random Measures with Independent Increments

Abstract.  One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell–MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.     

Improved Measures of the Spread of Data for Some Unknown Complex Distributions Using Saddlepoint Approximations

Measures of the spread of data for random sums arise frequently in many problems and have a wide range of applications in real life, such as in the insurance field (e.g., the total claim size in a portfolio). The exact distribution of random sums is extremely difficult to determine, and normal approximation usually performs very badly for this complex distributions. A better method of approximating a random-sum distribution involves the use of saddlepoint approximations.

Saddlepoint approximations are powerful tools for providing accurate expressions for distribution functions that are not known in closed form. This method not only yields an accurate approximation near the center of the distribution but also controls the relative error in the far tail of the distribution.

In this article, we discuss approximations to the unknown complex random-sum Poisson–Erlang random variable, which has a continuous distribution, and the random-sum Poisson-negative binomial random variable, which has a discrete distribution. We show that the saddlepoint approximation method is not only quick, dependable, stable, and accurate enough for general statistical inference but is also applicable without deep knowledge of probability theory. Numerical examples of application of the saddlepoint approximation method to continuous and discrete random-sum Poisson distributions are presented.     

Approximate Marginal Posterior Distributions Using Asymptotic Modes

This article gives asymptotic expansions for marginal posterior distributions with asymptotic modes of order n ^?2, and shows their validity. In addition, by using the asymptotic expansion, an approximate central posterior credible interval is derived.     

A Unified Approach to Estimating and Testing Income Distributions With Grouped Data

We propose a unified approach that is flexibly applicable to various types of grouped data for estimating and testing parametric income distributions. To simplify the use of our approach, we also provide a parametric bootstrap method and show its asymptotic validity. We also compare this approach with existing methods for grouped income data, and assess their finite-sample performance by a Monte Carlo simulation. For empirical demonstrations, we apply our approach to recovering China's income/consumption distributions from a sequence of income/consumption share tables and the U.S. income distributions from a combination of income shares and sample quantiles. Supplementary materials for this article are available online.     

The Asymptotic Distribution of the S–Gini Index

Several generalizations of the classical Gini index, placing smaller or greater weights on various portions of income distribution, have been proposed by a number of authors. For purposes of statistical inference, the large sample distribution theory of the estimators of those measures of economic inequality is required. The present paper was stimulated by the use of bootstrap by Xu (2000) to estimate the variance of the estimator of the S –Gini index. It shows that the theory of L –statistics (Chernoff, Gastwirth & Johns, 1967; Shorack & Wellner, 1986) makes possible the construction of a consistent estimator for the S –Gini index and proof of its asymptotic normality. The paper also presents an explicit formula for the asymptotic variance. The formula should be helpful in planning the size of samples from which the S –Gini index can be estimated with a prescribed margin of error.     

我国社会保障基尼系数的讨论与估算

 内容提要：我国收入基尼系数的扩大已被大多数学者研究与认同,但基于收入“二次分配”的社会保障基尼系数的研究还十分欠缺。本文认为社会保障作为收入再分配的重要手段之一是对人均社会福利收入分配的补充与调整,在社会经济资源配置过程中起着重要作用。本文采用相对平均差的基尼系数方法对我国社会保障差异化程度进行离散型数据分析,发现我国社会保障基尼系数值较高,但总体上有缩小趋势;其中,主要表现在养老保险与其他社会保险项目发展的不平衡性与区域差异性,所以应进一步加强全民养老保险与医疗保险制度建设,促进工伤、失业保险制度的协同发展。建立全国统筹的社会保障制度,协调各社会保险项目的平衡发展关系对于缩小社会保险基尼系数有重要现实意义。     

Inference Using Latent Variables for Mixtures of Distributions for Censored Data with Partial Identification

Abstract

In this article two methods are proposed to make inferences about the parameters of a finite mixture of distributions in the context of partially identifiable censored data. The first method focuses on a mixture of location and scale models and relies on an asymptotic approximation to a suitably constructed augmented likelihood; the second method provides a full Bayesian analysis of the mixture based on a Gibbs sampler. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to other methods which deal directly with the likelihood of the mixture. This may be crucial if the number of components in the mixture is not small. Our proposals are illustrated on a classical example on failure times for communication devices first studied by Mendenhall and Hader (Mendenhall, W., Hader, R. J. (1958 Mendenhall, W. and Hader, R. J. 1958. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika, 45: 504–520. [Crossref], [Web of Science ®] , [Google Scholar]). Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520.). In addition, we study the coverage of the confidence intervals obtained from each of the methods by means of a small simulation exercise.     

Case-deletion Influence Measures for the Data from Multivariate t Distributions

For the data from multivariate t distributions, it is very hard to make an influence analysis based on the probability density function since its expression is intractable. In this paper, we present a technique for influence analysis based on the mixture distribution and EM algorithm. In fact, the multivariate t distribution can be considered as a particular Gaussian mixture by introducing the weights from the Gamma distribution. We treat the weights as the missing data and develop the influence analysis for the data from multivariate t distributions based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Several case-deletion measures are proposed for detecting influential observations from multivariate t distributions. Two numerical examples are given to illustrate our methodology.     

Expected Posterior Priors in Selecting the Largest Mean of the Exponential Distributions

In reliability theory or survival analysis, selecting the largest mean among many exponential distributions is an important issue. Such a problem can also be viewed as a model selection problem via the Bayesian approach. It is well known that Bayes factors under proper priors have been very successful in Bayesian model selection or testing problems. However, Bayes factors are typically invalid with respect to improper noninformative priors. Objective Bayesian criteria are thus desired. In this work, we consider to use the expected posterior priors originally proposed by Pérez and Berger (2002 Pérez , J. M. , Berger , J. ( 2002 ). Expected posterior prior distributions for model selection . Biometrika 89 : 491 – 512 .[Crossref], [Web of Science ®] , [Google Scholar]) to select the largest exponential mean. Specific expected posterior priors are derived in recursive formulas. Some simulation results are also given to illustrate the method.     

Small Sample Tests for Shape Parameters of Gamma Distributions

The introduction of shape parameters into statistical distributions provided flexible models that produced better fit to experimental data. The Weibull and gamma families are prime examples wherein shape parameters produce more reliable statistical models than standard exponential models in lifetime studies. In the presence of many independent gamma populations, one may test equality (or homogeneity) of shape parameters. In this article, we develop two tests for testing shape parameters of gamma distributions using chi-square distributions, stochastic majorization, and Schur convexity. The first one tests hypotheses on the shape parameter of a single gamma distribution. We numerically examine the performance of this test and find that it controls Type I error rate for small samples. To compare shape parameters of a set of independent gamma populations, we develop a test that is unbiased in the sense of Schur convexity. These tests are motivated by the need to have simple, easy to use tests and accurate procedures in case of small samples. We illustrate the new tests using three real datasets taken from engineering and environmental science. In addition, we investigate the Bayes’ factor in this context and conclude that for small samples, the frequentist approach performs better than the Bayesian approach.     

Censored Exponential Data: Large Deviations for MLEs and Posterior Distributions

In this article, we consider several statistical models for censored exponential data. We prove a large deviation result for the maximum likelihood estimators (MLEs) of each model, and a unique result for the posterior distributions which works well for all the cases. Finally, comparing the large deviation rate functions for MLEs and posterior distributions, we show that a typical feature fails for one model; moreover, we illustrate the relation between this fact and a well-known result for curved exponential models.     

Estimation of Wood Fibre Length Distributions from Censored Data through an EM Algorithm

Abstract.  An expectation maximization (EM) algorithm is proposed to find fibre length distributions in standing trees. The available data come from cylindric wood samples (increment cores). The sample contains uncut fibres as well as fibres cut once or twice. The sample contains not only fibres, but also other cells, the so-called 'fines'. The lengths are measured by an automatic fibre-analyser, which is not able to distinguish fines from fibres and cannot tell if a cell has been cut. The data thus come from a censored version of a mixture of the fine and fibre length distributions in the tree. The parameters of the length distributions are estimated by a stochastic version of the EM algorithm, and an estimate of the corresponding covariance matrix is derived. The method is applied to data from northern Sweden. A simulation study is also presented. The method works well for sample sizes commonly obtained from increment cores.