Multivariate Measures of Well-Being and an Analysis of Inequality in the Michigan Data

We demonstrate a multidimensional approach for combining several indicators of well-being, including the traditional money-income indicators. This methodology avoids the difficult and much criticized task of computing imputed incomes for such indicators as net worth and schooling. Inequality in the proposed composite measures is computed using relative inequality indexes that permit simple analysis of both the contribution of each welfare indicator (and its factor components) and within and between components of total inequality when the population is grouped by income levels, age, gender, or any other criteria. The analysis is performed on U.S. data using the Michigan Survey of Income Dynamics.     

Contrasting the Gini and Zenga indices of economic inequality

The current financial turbulence in Europe inspires and perhaps requires researchers to rethink how to measure incomes, wealth, and other parameters of interest to policy-makers and others. The noticeable increase in disparities between less and more fortunate individuals suggests that measures based upon comparing the incomes of less fortunate with the mean of the entire population may not be adequate. The classical Gini and related indices of economic inequality, however, are based exactly on such comparisons. It is because of this reason that in this paper we explore and contrast the classical Gini index with a new Zenga index, the latter being based on comparisons of the means of less and more fortunate sub-populations, irrespectively of the threshold that might be used to delineate the two sub-populations. The empirical part of the paper is based on the 2001 wave of the European Community Household Panel data set provided by EuroStat. Even though sample sizes appear to be large, we supplement the estimated Gini and Zenga indices with measures of variability in the form of normal, t-bootstrap, and bootstrap bias-corrected and accelerated confidence intervals.     

中国经济增长的益贫性评价——基于PEGR方法和微观数据的实证

基于2000-2009年中国健康与营养调查（CHNS）微观数据,应用益贫式增长测度方法中唯一满足单调性公理的减贫等值增长率（PEGR）方法,对中国经济增长的益贫性进行系统评价。实证结果显示：中国经济表现出持续的减贫式增长特征,经济增长的减贫效应大于收入分配改善的减贫效应;中国经济增长近期表现出相对益贫性,但仍不具有绝对益贫性。鉴此,结合中国实际,政府益贫式增长政策的关注重点应是改善收入分配的现状。     

Parametric Versus Nonparametric Inference on Zenga Index of Inequality: Issues and Evidence From Survey Data

Recent growing disparities suggests to move from inequality measures based on comparing the incomes of the less fortunate with the overall mean, as the Gini, to the new Zenga index, which instead contrasts the means of the less and the more wealthy subpopulations. After providing a thorough analysis of the theoretical and practical aspects of obtaining parametric and non-parametric confidence intervals for the Zenga inequality measure, we develop a cross-regional study based on the Swiss Income and Consumption Survey, wave 2005. Results show that coverage accuracy and average length of confidence intervals improve when the parametric model offers a good fit to the data.     

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.     

On Stein Identity,Chernoff Inequality,and Orthogonal Polynomials

In this article, we present a general method for deriving Stein-like identity and Chernoff-like inequality based on orthogonal polynomials. In order to illustrate our method, some applications are given with respect to normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution, not only for random variables but also for random vectors, resulting corresponding Stein-like identity and Chernoff-like inequality are obtained consequently. Within our best knowledge, some of our matrix version results are new in the literature. In addition, forward difference formulae of Charlier polynomials, Krawtchouk polynomials and Meixner polynomials, Stein-like identity, and Chernoff-like inequality with respect to Beta distribution, as well as Rodrigues formula of Meixner polynomials are also prepared in the first time within our limited information. Interestingly, as far as normal, Gamma, Beta, Poisson, binomial, and negative binomial distribution are concerned, we found that their Stein-like identity and corresponding Chernoff-like inequality are related closely, by examining their Rodrigues formula.     

贫困测度指标及其评价

一、引言一般来说,研究贫困问题主要涉及三个方面:一是贫困的识别;二是贫困程度测定;三是反贫困效果判定以及相应的战略选择。贫困识别问题主要指贫困的基本范畴及贫困线的合理确定,关系到贫困集合的准确界定;贫困程度的测定主要指贫困指数的定义和计算,关系到贫困程度大小的判     

Some results on dependent random variables and a connection with the multivariate s-increasing convex order

In this paper, some new concepts of dependence are introduced that generalize the concepts of positive and negative association. The new concepts of dependence are linked to the multivariate s-increasing convex order. Furthermore, a Kolmogorov-type inequality and a Hàjek–Rènyi inequality are proven that lead to an asymptotic result for these new random objects.     

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.     

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.     

Stochastic dominance amongst swedish income distributions

:Sweden's income distribution for the whole population and for subgroups, including its immigants, has been extensively studied. The interest in this area has grown with increasing availability of data, including panels. The previous studies are based on indices of inequality or mobility. While indices are useful for complete ordering and have an air of "decisiveness" about them, they lack universal acceptance of the value judgements inherent to the welfare functions that underlay any index. In contrast, uniformpartial order relations are studied in this paper which rank welfare situations over very wide classes of welfare functions. We conduct bootstrap tests for the existence of first and second order stochastic dominance amongst Sweden's income distributions over time and for several subgroups of immigrants. Analysis of immigrant's income is motivated by the fact that the development of income for immigrants has been different and strongly affected by their length of residence and countries of origin. We consider several non-consecutive waves of a panel of incomes in Sweden. Two income definitions are developed. One is pre-transfers and taxes, gross income, the other is a post-transfers and taxes, disposable income. The comparison of the distribution of these two variables affords a partial view of Sweden's welfare system. We have focused on the incomes of Swede's and immigrant groups of single individuals identified by country of origin, length of residence, age, education, gender, marital status and other relevant characteristics. We find that first order dominance is rare, but second order relations hold in several cases, especially amongst disposable income distributions. Sweden's incomes and welfare policies favor the elderly, females, larger families, and longer periods of residency. We find, in general, the higher the educational credentials, the higher is the burden of this equalization policy.     

Diagnostics for glms with linear inequality parameter constraints

This paper proposes an iterative process, that can be implemented using GLIM, for fitting generalized linear models with linear inequality parameter constraints, when the maximum likelihood estimates exist and are unique. A one-step estimate is also introduced and some diagnostic measures are obtained. Finally an example is given for illustration.     

Unbiased estimation of income inequality

In a previous paper Gastwirth shows that a broad family of measures of inequality can be accurately estimated when the tax data are known in groups (more precisely, when we know the number of returns in each of several class intervals and their corresponding average income). In the present paper we show that some measures of the preceding family can be unbiasedly estimated when the tax data are individually known for a sample from the population. Specifically, we construct unbiased estimators of a particular measure of inequality in the samplings with and without replacement, and in the stratified samplings with and without replacement.     

A fuzzy logic approach to poverty analysis based on the Gini and Bonferroni inequality indices

In the poverty analysis framework, a great deal of attention has been paid to the poverty measurement in terms of monetary variables, such as income or consumption. In this context, a relevant open problem is connected with the distinction between poor and non-poor. In fact, the concept of poverty is rather vague and cannot be defined in a clear way. In this respect, following a fuzzy logic approach, some new poverty measures are proposed. In particular, the fuzzy extension of two existing poverty measures based on the Gini and Bonferroni inequality indices is provided. Some synthetic and real applications are given in order to show how the proposed poverty measures work.     

On the importance of nonparametric approaches for a comparison of income inequality in different countries

"A comparison of income inequality in different countries can be made on the basis of parametric or nonparametric approaches. Using nonparametric instead of parametric approaches may raise the validity of such comparisons in some respects. However, new sources of validity reduction may be linked with nonparametric approaches. An important source of validity reduction is the influence of the type of data grouping on inequality measures. Therefore, nonparametric approaches are not strictly preferable to parametric approaches when comparing income inequality in different countries." (SUMMARY IN ENG)     

The New Zenga curve in the context of reliability analysis

ABSTRACT

In this article, we look into the properties and characterizations of the New Zenga curve. The relationship of the curve with other measures of inequality as well as some reliability concepts are examined. Classification of lifetime distributions using the Zenga curve and an illustration for the behaviour of the curve using a survival data are also provided.     

Some Sampling Properties of a Model for Income Distribution

A recently proposed model for describing the distribution of income over a population, based on the Burr distribution, has been shown to fit better than the commonly used lognormal or gamma distributions. The current article extends that analysis by deriving the large-sample properties of the maximum likelihood estimates for this three-parameter model. Consequently, resulting confidence intervals for some measures of income inequality (including the Gini index) are used to further test the model's validity, as well as to examine apparent trends in inequality over time. Since these properties depend on the way the income data are grouped and censored, implications for choosing data-report intervals can be analyzed. Specifically, a choice between two common methods of reporting the data is shown to have an important impact on Gini index estimates.     

A vector valued bivariate gini index for truncated distributions

Gini index is widely used in the study of inequality of income distribution. In the present paper we give a definition of the Gini index in the Bivariate set-up and look into the problem of characterizing probability distributions based on some relationship between this index and various other commonly used measures. We also generalized the Gini index to a situation where several attributes of the population are considered.     

Interdistributional Income Inequality

Incomplete moments are used to characterize income inequality and provide the basis for interdistributional Lorenz curves. Four measures of interdistributional inequality are considered and seen to be related to an interdistributional welfare interpretation. Based upon these measures, there has been a significant secular decline in interdistributional inequality between blacks and whites over the past 30 years.     

Inequality indices and the starshaped principle of transfers

The evaluation of income distributions is usually based on the Pigou-Dalton (PD) principle which says that a transfer from any people to people who have less decreases economic inequality, i.e., increases the social evaluation index. We introduce two weaker principles of transfers which refer to a parameter θ. With the new principles, only those PD transfers increase the social evaluation index which take from the class of incomes above θ and give to the class below θ. The relative positions of individuals remain unchanged, and either no individual may cross the line θ (principle of transfers about θ) or some may do who have been situated next to it (starshaped principle of transfers at θ). θ may be a given constant, a function of mean income, or a quantile of the income distribution. The classes of indices which are consistent with these transfers are completely characterized, and examples are given.