Gini Family Research Articles

Public education expenditure and food inequality in rural India: an analysis in major states of India during 1983-2012

Economic inequality in the distribution of wealth or income continues to be a very serious problem for any community. Economic inequality can be captured by different families of inequality measures of which two are very popular and convenient - the Gini family and the SD-CV family. Inequality measures in any family may be of three types - relative, absolute and index measures of inequality. A serious question is that, what type of inequality do we want to reduce - relative or absolute? If we really want to reduce inequality, we should consider both relative and absolute measures of inequality. For India, inequality is measured for consumer expenditure of households with the help of data collected by National Sample Survey Organisation (NSSO) which divides household's total expenditure into two broad categories - food and non-food items. Income/expenditure inequality occurs due to many reasons/factors. Education expenditure is one of the important factors. Using panel data for 15 major states of rural India for the period 1983-2012, this paper empirically examines whether public expenditure on education succeeds to reduce food inequality or not.

Can a single indicator measure both attainment and shortfall inequality?

When a distribution can be described either in terms of attainment or in terms of shortfall, the classical inequality measures appear to be one-sided. I show that it is possible to find indicators which simultaneously measure attainment and shortfall inequality. I derive one indicator belonging to the Gini family, and another belonging to Coefficient of Variation family. I also indicate how they are connected to other measures.

Who Does Not Respond in the Household Expenditure Survey: An Exercise in Extended Gini Regressions

The aim of this paper is to suggest a new, nonparametric multiple regression method, based on the Extended Gini (EG) measures of dispersion, which enables the user to follow the curvature of the regression curve. The method is capable of estimating a series of linear approximations of the regression curve, allowing the investigator to stress different sections along the range of one independent variable while keeping the treatment of other independent variables intact. The method is based on the extended Gini family which depends on one parameter, ν. The choice of this parameter enables the user to produce infinite alternative estimators of the regression curve. The difference between them lies in the weighting schemes applied to the slopes of the regression curve. By investigating the patterns of changes in those regression coefficients, the curvature of the regression curve can be traced. As an application, we investigate nonresponse patterns in the survey of household expenditures in Israel. We are mainly interested in whether nonresponse increases or decreases with income, and the kind of functional relationship one can find between income and nonresponse. The empirical illustration shows that the higher the income, the larger the response rate, and the larger the household, the higher the response rate. Also, the Arab population tends to respond more than the Jewish one, while the ultra religious group tends to respond less than the rest of the population. Those last two results hold with and without adjustment for income and household size. The implications on the bias in the estimates are discussed.

The Properties of the Extended Gini Measures of Variability and Inequality

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.

A Family of Correlation Coefficients Based on Extended Gini

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.

A Family of Correlation Coefficients Based on the Extended Gini Index

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 for 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, with, in addition, 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.

Axiomatic Characterization of the Gini Coefficient and Lorenz Curve Orderings

This paper is concerned with distributions of income and the ordering of related Lorenz curves. By introducing appropriate preference relations on the set of Lorenz curves, two alternative axiomatic characterizations of Lorenz curve orderings are proposed. Moreover, the Gini coefficient is recognized to be rationalizable under both axiom sets; as a result, a complete axiomatic characterization of the Gini coefficient is obtained. Furthermore, axiomatic characterizations of the extended Gini family and an alternative “generalized” Gini family of inequality measures are proposed. Journal of Economic Literature Classification Numbers: C43, D31, D63.

Confidence Interval Estimation for Inequality Indices of the GiniFamily

In this paper we present some nonparametric bootstrap methods to construct distribution-free confidence intervals for inequality indices belonging to the Gini family. These methods have a coverage accuracy better than that obtained with the asymptotic distribution of their natural estimators, typically the standard normal. The coverage performances of these methods are evaluated for the index R by Gini with a Monte Carlo experiment on samples simulated from the Dagum income model (Type I), which is usually used to describe the income distribution.

Approximated Distributions of Sampling Inequality Indices

Often, in finite samples, the true level of the confidence intervals for natural estimators of inequality indices belonging to the Gini family differs greatly from their nominal level, which is based on the asymptotic confidence limits. This paper shows how the Gram-Charlier series can be used to obtain improved finite-sample confidence intervals. Our work focuses on the implementation in Mathematica 3.0 of computational procedures to compute the Gram-Charlier distribution for the following sampling inequality indices: R by Gini, P by Piesch and M by Mehran for the Dagum (Type I) distribution. The results of a Monte Carlo experiment confirm that, for the cases investigated, the Gram-Charlier distribution largely eliminates the problem of incorrect finite-sample level.

$L$-Statistics in Complex Survey Problems

We study linear combinations of order statistics ($L$-statistics) in survey problems with a stratified multistage sampling design. Two general types of $L$-statistics are considered: smooth $L$-statistics with weights generated by a smooth function and nonsmooth $L$-statistics (sample quantiles). The trimmed sample mean, the decile mean and variance, the sample Lorenz curve and the sample Gini's family parameters are examples of smooth $L$-statistics or functions of smooth $L$-statistics used in survey problems. It is shown that under weak conditions the smooth $L$-statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing. For the sample quantiles, their asymptotic normality requires more conditions on the finite population distribution functions. Consistent estimators for the asymptotic variances of the sample quantiles are derived. Asymptotic validity of the Woodruff's confidence intervals for population quantiles is also proved.

Income inequality measures based on sample surveys

By relating the Gini coefficient to a general family of inequality measures, the Gini family, methods of estimating inequality parameters from samples are reviewed. Three approaches to making inferences about unknown inequality parameters are discussed and some large-sample results are presented. Alternative variance estimates are presented and compared in the case of the Gini coefficient.