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J Econ Inequal (2015) 13:321324
DOI 10.1007/s10888-014-9277-8
Published online: 21 May 2014 Springer Science+Business Media New York 2014
In this book, Shlomo Yitzhaki and Edna Schechtman investigate the Gini methodology. Both of the authors have dedicated an important part of their life to the development and the understanding of the Gini coefficient and its applications in Statistics, Econometrics, and Economics (welfare and inequality measurement). More generally, the authors have developed (sometimes with other co-authors) a family of parameters based on Ginis mean difference: the Gini covariance (the co-Gini, which is the expression of GMD with the use of the covariance), the Gini correlation (G-correlation), the Extended Gini coefficient (EG), and the Gini parameters of a regression, among others. Those theoretical concepts, including inference purposes as well as the absolute Lorenz curves (i.e. the generalized Lorenz curve) are presented in the first part of the book (Chapters 2-11). The second part (Chapters 12-22) is devoted to the applications.
In the first part, the well-known variance and covariance are replaced by the GMD and its variants. It is shown that everything can be done with the GMD as a measure of variability. The variance analysis (ANOVA) may be viewed as too restrictive in many applications because it imposes restrictions not supported by the data. The Gini analysis (ANOGI) becomes an important feature since less demanding assumptions are required to deal with empirical applications.
The GMD was first introduced by Corrado Gini as an alternative measure of variability. It shares many properties with the variance, but can be more informative about the properties of distributions that depend on normality. The fundamental difference between those two measures lies in the metric. The variance is based on the Euclidean distance while the Gini is defined on the 1 norm, i.e. the city block distance. It turns out that there are more than 12 alternative expressions of the GMD (or the Gini coefficient, Chapter 2).
Let x, y be two random variables. Traditionally, two main approaches are considered for analyzing the relationship between x and y. The first one is concerned with the covariance and Pearsons correlation coefficient that depends on the variance analysis. The second one is based on the covariance between the cumulative distribution function (cdf) of x...