Abstract When incomes are ranked in descending order the social-evaluation function corresponding to the Gini relative inequality index can be written as a linear function withthe weights being the odd ā¦
Abstract When incomes are ranked in descending order the social-evaluation function corresponding to the Gini relative inequality index can be written as a linear function withthe weights being the odd numbers in increasing order. We generalize this function by allowing the weights to be an arbitrary non-decreasing sequence of numbers. This results in a class of generalized Gini relative inequality indices and a class of generalized Gini absolute inequality indices. An axiomatic characterization of the latter class is also provided.
Abstract This paper is concerned with the Gini coefficient as a concentration measure of the distribution. The proposed formula for the computation of the Gini coefficient allows its decomposition according ā¦
Abstract This paper is concerned with the Gini coefficient as a concentration measure of the distribution. The proposed formula for the computation of the Gini coefficient allows its decomposition according to different subāgroups of the initial population, either exactly or approximately.
In the present paper, we answer the question: For Ī± + Ī² ā (0,1) , what are the greatest values p,s 1 and the least values q,s 2 such that ā¦
In the present paper, we answer the question: For Ī± + Ī² ā (0,1) , what are the greatest values p,s 1 and the least values q,s 2 such that the inequalities1 (a,b) hold for all a,b > 0 with a = b ?where J p (a,b) , A(a,b) , G(a,b) , H(a,b) and G s,1 (a,b) are the one-parameter mean, arithmetic mean, geometric mean, harmonic mean and Gini mean for two positive numbers a and b , respectively.
The Gini coefficient or index is perhaps one of the most used indicators of social and economic conditions. In this paper we characterize the Gini index as the unique function ā¦
The Gini coefficient or index is perhaps one of the most used indicators of social and economic conditions. In this paper we characterize the Gini index as the unique function that satisfies the properties of scale invariance, symmetry, proportionality and convexity in similar rankings. Furthermore, we discuss a simpler way to compute it.
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We offer new properties of the special Gini mean S ( a, b ) = a a /( a + b ) ā b b /( a + b ) ā¦
We offer new properties of the special Gini mean S ( a, b ) = a a /( a + b ) ā b b /( a + b ) , in connections with other special means of two arguments.
ABSTRACT The purpose of this note is to propose an alternative and intuitively simpler derivation of the Gini coefficient (in Section l), to show how it can be generalized and ā¦
ABSTRACT The purpose of this note is to propose an alternative and intuitively simpler derivation of the Gini coefficient (in Section l), to show how it can be generalized and how then a number of coefficients (concentration coefficient, Kakwani's progressivity index) are obtained directly from this generalization (Section 2), and finally to use this approach to obtain some Gini relationships (Section 3).
In the paper, the monotonicity and logarithmic convexity of Gini means and related functions are investigated.
In the paper, the monotonicity and logarithmic convexity of Gini means and related functions are investigated.
The Gini-Frisch bounds partially identify the constant slope coefficient in a linear equation when the explanatory variable suffers from classical measurement error. This paper generalizes these quintessential bounds to accommodate ā¦
The Gini-Frisch bounds partially identify the constant slope coefficient in a linear equation when the explanatory variable suffers from classical measurement error. This paper generalizes these quintessential bounds to accommodate nonparametric heterogenous effects. It provides suitable conditions under which the main insights that underlie the Gini-Frisch bounds apply to partially identify the average marginal effect of an error-laden variable in a nonparametric nonseparable equation. To this end, the paper puts forward a nonparametric analogue to the standard "forward" and "reverse" linear regression bounds. The nonparametric forward regression bound generalizes the linear regression "attenuation bias" due to classical measurement error.
Abstract For the comparison of inequality and welfare in multiple attributes the use of generalized Gini indices is proposed. Individual endowment vectors are summarized by using attribute weights and aggregated ā¦
Abstract For the comparison of inequality and welfare in multiple attributes the use of generalized Gini indices is proposed. Individual endowment vectors are summarized by using attribute weights and aggregated in a spectral social evaluation function. Such functions are based on classes of spectral functions, ordered by their aversion to inequality. Given a spectrum and a set P of attribute weights, a multivariate Gini dominance ordering, being uniform in weights, is defined. If the endowment vectors are comonotonic, the dominance is determined by their marginal distributions; if not, the dependence structure of the endowment distribution has to be taken into account. For this, a set-valued representative endowment is introduced that characterizes the welfare of a d -dimensioned distribution. It consists of all points above the lower border of a convex compact in $$\mathbb {R}^{d}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , while the set ordering of representative endowments corresponds to uniform Gini dominance. An application is given to the welfare of 28 European countries. Properties of P -uniform Gini dominance are derived, including relations to other orderings of d -variate distributions such as convex and dependence orderings. The multi-dimensioned representative endowment can be efficiently calculated from data. In a sampling context, it consistently estimates its population version.
The goal of many randomized experiments and quasi-experimental studies in economics is to inform policies that aim to raise incomes and reduce economic inequality. A policy maximizing the sum of ā¦
The goal of many randomized experiments and quasi-experimental studies in economics is to inform policies that aim to raise incomes and reduce economic inequality. A policy maximizing the sum of individual incomes may not be desirable if it magnifies economic inequality and post-treatment redistribution of income is infeasible. This article develops a method to estimate the optimal treatment assignment policy based on observable individual covariates when the policy objective is to maximize an equality-minded rank-dependent social welfare function, which puts higher weight on individuals with lower-ranked outcomes. We estimate the optimal policy by maximizing a sample analog of the rank-dependent welfare over a properly constrained set of policies. We show that the average social welfare attained by our estimated policy converges to the maximal attainable welfare at nā1/2 rate uniformly over a large class of data distributions when the propensity score is known. We also show that this rate is minimax optimal. We provide an application of our method using the data from the National JTPA Study.Supplementary materials for this article are available online.
The widely used income inequality measure, Gini index, is extended to form a family of income inequality measures known as Single-Series Gini (S-Gini) indies. In this study, we develop empirical ā¦
The widely used income inequality measure, Gini index, is extended to form a family of income inequality measures known as Single-Series Gini (S-Gini) indies. In this study, we develop empirical likelihood (EL) and jackknife empirical likelihood (JEL) based inference for the S-Gini indices. We prove that the limiting distribution of both EL and JEL ratio statistics are Chi-square distributions with one degree of freedom. Using the asymptotic distribution we construct EL and JEL based confidence intervals for relative S-Gini indices. We also give bootstrap-t and bootstrap calibrated empirical likelihood confidence intervals for the S-Gini indices. A numerical study is carried out to compare the performances of the proposed asymptotic confidence interval and the bootstrap methods. A test for S-Gini indices based on jackknife empirical likelihood ratio is also proposed. Finally, we illustrate the proposed method using an income data.
Widely used income inequality measure, Gini index is extended to form a family of income inequality measures known as Single-Series Gini (S-Gini) indices. In this study, we develop empirical likelihood ā¦
Widely used income inequality measure, Gini index is extended to form a family of income inequality measures known as Single-Series Gini (S-Gini) indices. In this study, we develop empirical likelihood (EL) and jackknife empirical likelihood (JEL) based inference for S-Gini indices. We prove that the limiting distribution of both EL and JEL ratio statistics are Chi-square distribution with one degree of freedom. Using the asymptotic distribution we construct EL and JEL based confidence intervals for realtive S-Gini indices. We also give bootstrap-t and bootstrap calibrated empirical likelihood confidence intervals for S-Gini indices. A numerical study is carried out to compare the performances of the proposed confidence interval with the bootstrap methods. A test for S-Gini indices based on jackknife empirical likelihood ratio is also proposed. Finally we illustrate the proposed method using an income data.
In the context of the binomial decomposition of OWA functions, we investigate the parametric constraints associated with the 3-additive case in n dimensions. The resulting feasible region in two coecients ā¦
In the context of the binomial decomposition of OWA functions, we investigate the parametric constraints associated with the 3-additive case in n dimensions. The resulting feasible region in two coecients is a convex polygon with n vertices and n edges, and is strictly increasing in the dimension n. The orness of the OWA functions within the feasible region is linear in the two coecients, and the vertices associated with maximum and minimum orness are identied.
For analyzing the relationship between two random variables an approach is introduced which is called the Gini method.The approach is based on the Gini mean difference, the Gini covariance, and ā¦
For analyzing the relationship between two random variables an approach is introduced which is called the Gini method.The approach is based on the Gini mean difference, the Gini covariance, and the Gini correlation.The method is then extended to include concentration curves.For two given random variables a condition in terms of their concentration curves (with respect to themselves) is derived which is necessary and sufficient for second degree stochastic dominance between the variables.
The extended Gini is a family of measures of variability which is mainly used in the areas of finance and income distribution. Each index in the family is defined by ā¦
The extended Gini is a family of measures of variability which is mainly used in the areas of finance and income distribution. Each index in the family is defined by specifying one parameter, which reflects the social evaluation of the marginal utility of income. The higher the parameter, the more weight is attached to the lower portion of the cumulative distribution, reflecting higher concern to poverty. In this paper we list and investigate the properties of the equivalents of the correlation coefficient that are associated with the extended Gini family. In addition, we show that the extended Gini of a linear combination of random variables can be decomposed, in a way which is equivalent to the decomposition of the variance, plus additional terms that reflect additional properties of the random variables. The implication of these properties is that any decomposition that is performed with the coefficient of variation can be replicated by an infinite number of indices that are based on the Extended Gini coefficient.
whose creativity and fondness for indices have been matchlessly inspiring L-statistics play prominent roles in various research areas and applications, including development of robust statistical methods, measuring economic inequality and ā¦
whose creativity and fondness for indices have been matchlessly inspiring L-statistics play prominent roles in various research areas and applications, including development of robust statistical methods, measuring economic inequality and insurance risks.In many applications the score functions of L-statistics depend on parameters (e.g., distortion parameter in insurance, risk aversion parameter in econometrics), which turn the L-statistics into functions that we call L-functions.A simple example of an L-function is the Lorenz curve.Ratios of L-functions play equally important roles, with the Zenga curve being a prominent example.To illustrate real life uses of these functions/curves, we analyze a data set from the Bank of Italy year 2006 sample survey on household budgets.Naturally, empirical counterparts of the population L-functions need to be employed and, importantly, adjusted and modified in order to meaningfully capture situations well beyond those based on simple random sampling designs.In the processes of our investigations, we also introduce the L-process on which statistical inferential results about the population L-function hinges.Hence, we provide notes and references facilitating ways for deriving asymptotic properties of the L-process.
Abstract In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We ā¦
Abstract In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such a behavior in terms of the usual stochastic order. For this purpose, we introduce a generalized family of stochastic orders, which is referred to as transform orders, showing that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodnessāofāfit testing procedures.
The goal of many randomized experiments and quasi-experimental studies in economics is to inform policies that aim to raise incomes and reduce economic inequality.A policy maximizing the sum of individual ā¦
The goal of many randomized experiments and quasi-experimental studies in economics is to inform policies that aim to raise incomes and reduce economic inequality.A policy maximizing the sum of individual incomes may not be desirable if it magnifies economic inequality and post-treatment redistribution of income is infeasible.This paper develops a method to estimate the optimal treatment assignment policy based on observable individual covariates when the policy objective is to maximize an equality-minded rank-dependent social welfare function, which puts higher weight on individuals with lower-ranked outcomes.We estimate the optimal policy by maximizing a sample analog of the rank-dependent welfare over a properly constrained set of policies.Although an analytical characterization of the optimal policy under a rankdependent social welfare is not available even with the knowledge of potential outcome distributions, we show that the average social welfare attained by our estimated policy converges to the maximal attainable welfare at n -1/2 rate uniformly over a large class of data distributions.We also show that this rate is minimax optimal.We provide an application of our method using the data from the National JTPA Study.
In this article, we propose the median-of-means type nonparametric estimator for S-Gini index by using the idea of grouping under the massive data framework, which has been widely used in ā¦
In this article, we propose the median-of-means type nonparametric estimator for S-Gini index by using the idea of grouping under the massive data framework, which has been widely used in economics, finance, and insurance. Under certain condition on the growing rate of the number of subgroups, the consistency and asymptotic normality of proposed estimator are investigated. Furthermore, we construct a new method to test S-Gini index based on the empirical likelihood method for median. Our method avoids any prior estimate of variance structure of proposed estimator, which is difficult to estimate and often causes much inaccuracy. Numerical simulations and a real data analysis are designed to show the performance of our estimator. It is shown that the new proposed estimator is quite robust with respect to outliers.
Abstract The principle of maximum entropy is a well-known approach to produce a model for data-generating distributions. In this approach, if partial knowledge about the distribution is available in terms ā¦
Abstract The principle of maximum entropy is a well-known approach to produce a model for data-generating distributions. In this approach, if partial knowledge about the distribution is available in terms of a set of information constraints, then the model that maximizes entropy under these constraints is used for the inference. In this paper, we propose a new three-parameter lifetime distribution using the maximum entropy principle under the constraints on the mean and a general index. We then present some statistical properties of the new distribution, including hazard rate function, quantile function, moments, characterization, and stochastic ordering. We use the maximum likelihood estimation technique to estimate the model parameters. A Monte Carlo study is carried out to evaluate the performance of the estimation method. In order to illustrate the usefulness of the proposed model, we fit the model to three real data sets and compare its relative performance with respect to the beta generalized Weibull family.
We identify an ordinal decomposability property and use it, along with other ordinal axioms, to characterize the Theil inequality ordering.
We identify an ordinal decomposability property and use it, along with other ordinal axioms, to characterize the Theil inequality ordering.
The aim of this paper is to survey and investigate the properties of the extended Gini family of inequality measures. The paper surveys the alternative ways of spelling the extended ā¦
The aim of this paper is to survey and investigate the properties of the extended Gini family of inequality measures. The paper surveys the alternative ways of spelling the extended Gini for continuous distributions (such as via covariance, via Lorenz curves, etc), and the metrics that motivate them. It also offers the adjustments needed for a consistent estimation in the case of discrete distributions. The relationship between the family and welfare dominance is discussed. Then, the equivalent parameters to the covariance and correlations that are required for the decomposition of a sum of random variables into the contribution of components are defined, and the new frontiers it opens for estimating a regression based on those measures is illustrated. Finally, the implications for analyzing the effect of policies intended to change the income distribution are discussed.
In the context of the dual decomposition of the rank dependent social evaluation functions we examine the k-PTS principle introduced by Gajdos and introduce a new property with balanced sensitivity ā¦
In the context of the dual decomposition of the rank dependent social evaluation functions we examine the k-PTS principle introduced by Gajdos and introduce a new property with balanced sensitivity to both tails of the distribution. In particular we analyse its implications for the S-Gini family.
It is well-known that a Lorenz curve derived from the distribution function of a random variable can be viewed itself as a probability distribution function of a new random variable ā¦
It is well-known that a Lorenz curve derived from the distribution function of a random variable can be viewed itself as a probability distribution function of a new random variable [3]. We prove the surprising result that a sequence of consecutive iterations of this map leads to a non-corner case convergence independent of the starting random variable. In the primal case, the limiting distribution has a power law with coefficient equal to the golden section. In the reflected case, the limiting distribution is classical Pareto with a conjugate coefficient. Possible directions for applications are discussed.
We propose a complete parametrical class of redistribution measures that satisfy desirable properties: S-convexity, monotonicity on the normative parameter, and equivalence with the Lorenz dominance criterion. The last property is ā¦
We propose a complete parametrical class of redistribution measures that satisfy desirable properties: S-convexity, monotonicity on the normative parameter, and equivalence with the Lorenz dominance criterion. The last property is not satisfied by the common redistribution indices. Moreover, we prove that, under these conditions, redistribution cannot be decomposed into the difference between two S-convex inequality indices. A particular parameterization class is proposed, in which we can always find a critical parameter value such that the index adopts a zero value if there is one (several) intersection(s) between the Lorenz curves. A parallel progressivity class is proposed, with the usual decomposition property.
The Gini index signals only the dispersion of the distribution and is not very sensitive to income differences at the tails of the distribution, where it would matter most. In ā¦
The Gini index signals only the dispersion of the distribution and is not very sensitive to income differences at the tails of the distribution, where it would matter most. In the current work, novel inequality measures are proposed that address these limitations. In addition, two related measures of skewness are established. They all are based on a pair of functions that is obtained by attaching simple weights to the distances between the Lorenz curve and the 45-degree line, both in ascending and descending order, resulting in a pair of alternative inequality curves. The inequality measures derived from these two alternative inequality curves either complement the information about distributional dispersion measured by the Gini coefficient with information about distributional asymmetry, or are more sensitive to income differences at both tails of the distribution. The novel tools measure inequality appropriately and their Lorenz-based graphical representations give them intuitive appeal. Beyond socioeconomics, they can be applied in other disciplines of science.
For the comparison of inequality in multiple attributes the use of generalized Gini indices is proposed. Spectral social evaluation functions are used in the multivariate setting, and Gini dominance orderings ā¦
For the comparison of inequality in multiple attributes the use of generalized Gini indices is proposed. Spectral social evaluation functions are used in the multivariate setting, and Gini dominance orderings are introduced that are uniform in attribute weights. Classes of spectral evaluators are considered that are parameterized by their aversion to inequality. Then a set-valued representative endowment is defined that characterizes $d$-dimensioned inequality. It consists of all points above the lower border of a convex compact in $R^d$, while the pointwise ordering of such endowments corresponds to uniform Gini dominance. Properties of uniform Gini dominance are derived, including relations to other orderings of $d$-variate distributions such as usual multivariate stochastic order and convex order. The multi-dimensioned representative endowment can be efficiently calculated from data; in a sampling context, it consistently estimates its population version.