Abstract
A quick, alternate proof is given for a previously known inequality relating the standard deviation and the Gini mean difference. The inequality is sharpened and generalized to higher, even moments. Further inequalities are derived that involve the standard deviation, higher Ginis and order statistics.
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References
Farris, F.: The Gini index and measures of inequality. Am. Math. Mon. 12, 851–864 (2010)
Gastwirth, J.: A general definition of the Lorenz curve. Econometrica 39, 1037–1039 (1971)
Gini, C.: Variabilita* e mutabilita*: contributo allo studio delle distribuzioni e delle relazioni statistiche, in Studi Economico-giuridici della Regia Facolta* Giusirsprudenza, anno III, parte II. Cuppini, Bologna (1912)
Glasser, G.: Variance formulas for the mean difference and coefficient of concentration. J. Am. Stat. Assoc. 57, 648–654 (1962)
Kakwani, N.: On a class of poverty measures. Econometrica 48, 437–446 (1980)
Kendall, M., Stuart, A., Ord, J.: Kendall’s Advanced Theory of Statistics, vol. 1, 5th edn, pp. 39–71. Oxford University Press, New York (1987)
Piesch, W.: A look at the structure of some extended Ginis. Metron LXIII, 263–296 (2005)
Yitzhaki, S.: Gini’s mean difference: a superior measure of variability for non-normal distributions. Metron LXI, 285–316 (2003). https://doi.org/10.2139/ssrn.301740
Yitzhaki, S.: Gini’s mean difference offers a response to Leamer’s critique. Metron LXXIII, 31–43 (2015). https://doi.org/10.1007/s40300-014-0057-9
Yitzhaki, S., Schechtman, E.: The Gini Methodology: a Primer on a Statistical Methodology. Series in Statistics, pp. 11–31. Springer, New York (2017)
Zizler, P.: Gini indices and the moments of the share density function. Appl. Math. 59, 167–175 (2014)
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La Haye, R., Zizler, P. The Gini mean difference and variance. METRON 77, 43–52 (2019). https://doi.org/10.1007/s40300-019-00149-2
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DOI: https://doi.org/10.1007/s40300-019-00149-2