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The Gini Methodology pp 11–31Cite as

More Than a Dozen Alternative Ways of Spelling Gini

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Part of the book series: Springer Series in Statistics ((SSS,volume 272))

Abstract

Gini’s mean difference (GMD) as a measure of variability has been known for over a century. It has more than 14 alternative representations. Some of them hold only for continuous distributions while others hold only for nonnegative variables. It seems that the richness of alternative representations and the need to distinguish among definitions that hold for different types of distributions are the main causes for its sporadic reappearances in the statistics and economics literature as well as in other areas of research. An exception is the area of income inequality, where it is holding the position as the most popular measure of inequality. GMD was “rediscovered” several times (see, for example, Chambers 6; Quiggin, 2007; David, 1968; Jaeckel, 1972; Jurečková, 1969; Olkin Yitzhaki, 1992; Kőszegi Rabin, 2007; Simpson, 1949) and has been used by investigators who did not know that they were using a statistic which was a version of the GMD. This is unfortunate, because by recognizing the fact that a GMD is being used the researcher could save time and research effort and use the already known properties of GMD.

This chapter is based on Yitzhaki (1998) and Yitzhaki (2003).

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Notes

  1. 1.

    For a description of its early development see Dalton (1920), Gini (1921, 1936), David (1981, p. 192), and several entries in Harter (1978). Unfortunately we are unable to survey the Italian literature which includes, among others, several papers by Gini, Galvani, and Castellano. A survey on those contributions can be found in Wold (1935). An additional comprehensive survey of this literature can be found in Giorgi (1990, 1993). See Yntema (1933) on the debate between Dalton and Gini concerning the relevant approach to inequality measurement.

  2. 2.

    Ceriani and Verme (2012) present several additional forms in Gini’s original writing that as observed by Lambert (2011) do not correspond to the presentations used in this book.

  3. 3.

    For the use of the GMD in categorical data see the bibliography in Dennis, Patil, Rossi, Stehman, and Taille (1979) and Rao (1982) in biology, Lieberson (1969) in sociology, Bachi (1956) in linguistic homogeneity, and Gibbs and Martin (1962) for industry diversification. Burrell (2006) uses it in informetrics, while Druckman and Jackson (2008) use it in resource usage, Puyenbroeck (2008) uses it in political science while Portnov and Felsenstein (2010) in regional diversity.

  4. 4.

    One way of writing the Gini is based on vectors and matrices. This form is clearly restricted to discrete variables and hence it is not covered in this book. For a description of the method see Silber (1989).

  5. 5.

    See also Pyatt (1976) for an interesting interpretation based on a view of the Gini as the equilibrium of a game.

  6. 6.

    The GMD is based on the difference of two such formulae, so this restriction on the range (to be bounded from below) does not affect the GMD. See Dorfman (1979).

  7. 7.

    This formula, which is a special case of the statistic suggested by Cramer, plays an important role in his composition of elementary errors although it seems that he did not identify the implied GMD (see Cramer, 1928, pp. 144–147). Von Mises (1931) made an independent equivalent suggestion and developed additional properties of the statistic. Smirnov (1937) modified the statistic to be

    \( {{\text{w}}^{{2}}} = {\text{n}}\int {{{\left[ {{{\text{F}}_{\rm{n}}}\left( {\text{x}} \right) - {\text{F}}\left( {\text{x}} \right)} \right]}^{{2}}}{\text{dF}}\left( {\text{x}} \right)} \).

    Changing the integration from dx to dF(x) eliminates the connection to the GMD and creates a distribution-free statistic. The above description of the non-English literature is based on the excellent review in Darling (1957). Further insight about the connection between the Cramér–Von Mises test can be found in Baker (1997) which also corrects for the discrepancy in calculating the GMD in discrete distributions.

  8. 8.

    This “duality” resembles the alternative approach to the expected utility theory as suggested by Yaari (1988) and others. While expected utility theory is linear in the probabilities and nonlinear in the income, Yaari’s approach is linear in the income and nonlinear in the probabilities. In this sense, one can argue that the relationship between “dual” approach and the GMD resembles the relationship between expected utility theory and the variance. Both indices can be used to construct a specific utility function for the appropriate approach (the quadratic utility function is based on the mean and the variance while the mean minus the GMD is a specific utility function of the dual approach).

  9. 9.

    Wold (1935) used a slightly different presentation, based on Stieltjes integrals.

  10. 10.

    See Lerman and Yitzhaki (1984) for the derivation and interpretation of the formula, see Jenkins (1988) and Milanovic (1997) on actual calculations using available software, and see Lerman and Yitzhaki (1989) on using this equation to calculate the GMD in stratified samples. As far as we know, Stuart (1954) was the first to notice that the GMD can be written as a covariance. However, his findings were confined to normal distributions. Pyatt, Chen and Fei (1980) also write the GMD as a covariance. Sen (1973) uses the covariance formula for the Gini, but without noticing that he is dealing with a covariance. Hart (1975) argues that the moment-generating function was at the heart of the debate between Corrado Gini and the western statisticians. Hence, it is a bit ironic to find that one can write the GMD as some kind of a central moment.

  11. 11.

    The term “generalized Lorenz curve” (GLC) was coined by Shorrocks (1983). Lambert and Aronson (1993) give an excellent description of the properties of GLC. However, it seems that the term “absolute” is more intuitive because it distinguishes the absolute curve from the relative one. Hart (1975) presents inequality indices in terms of the distribution of first moments, which is related to the GLC.

  12. 12.

    In the case of the GMD, the weights are not functions of Δxk so that it is reasonable to refer to them as weights. In the case of the variance, the “weights” are also functions of Δxk which makes the reference to them as weights to be incorrect. We refer to them as weights in order to compare with the GMD. See Yitzhaki (1996).

  13. 13.

    This phenomenon seems to be a characteristic of the literature on the GMD from its early development. Gini (1921) argues: “probably these papers have escaped Mr. Dalton’s attention owing to the difficulty of access to the publications in which they appeared.” (Gini, 1921, p. 124).

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Authors and Affiliations

  1. Department of Economics, The Hebrew University, Mount Scopus, Jerusalem, Israel

    Shlomo Yitzhaki

  2. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Edna Schechtman

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  1. Shlomo Yitzhaki
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Yitzhaki, S., Schechtman, E. (2013). More Than a Dozen Alternative Ways of Spelling Gini. In: The Gini Methodology. Springer Series in Statistics, vol 272. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4720-7_2

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