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From Gini to Bonferroni to Tsallis: an inequality-indices trek

Author

Listed:
  • Iddo Eliazar

    (Independent Researcher)

  • Giovanni M. Giorgi

    (“Sapienza” University of Rome)

Abstract

Arguably, the Gini index is the best known and the most widely applied inequality index in socioeconomics in particular, and across the sciences in general. On the other hand, far less known and less applied is the Bonferroni index. Addressed via Lorenz curves, the Gini index can be formulated as the average of two continuums of inequality indices that, respectively, stem from two sets of Lorenz-based distances: vertical and horizontal. These Gini-index formulations use one natural type of Lorenz-based distances. However, there is another natural type of Lorenz-based distances, and when using this type: (1) averaging the vertical continuum of inequality indices yields the known Bonferroni index; (2) averaging the horizontal continuum of inequality indices yields a new Bonferroni index. This paper explores comprehensively the two Bonferroni indices, and presents the many analogies between these indices and the Gini index. This paper also unifies the Bonferroni indices and the Gini index via two families of inequality induces: (1) a “vertical family” of which the known Bonferroni index and the Gini index are special cases; (2) a “horizontal family” of which the new Bonferroni index and the Gini index are special cases. These two families are shown to be counterparts of the Tsallis family of entropy measures, and the two Bonferroni indices are shown to be counterparts of the Shannon entropy. Written in an entirely self-contained manner, this paper is accessible also to audiences with no prior knowledge of inequality indices.

Suggested Citation

  • Iddo Eliazar & Giovanni M. Giorgi, 2020. "From Gini to Bonferroni to Tsallis: an inequality-indices trek," METRON, Springer;Sapienza Università di Roma, vol. 78(2), pages 119-153, August.
    • Handle: RePEc:spr:metron:v:78:y:2020:i:2:d:10.1007_s40300-020-00171-9
    DOI: 10.1007/s40300-020-00171-9
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    References listed on IDEAS

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    1. Eliazar, Iddo, 2015. "The sociogeometry of inequality: Part II," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 426(C), pages 116-137.
    2. Eliazar, Iddo, 2015. "The sociogeometry of inequality: Part I," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 426(C), pages 93-115.
    3. Eliazar, Iddo, 2018. "Average is Over," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 123-137.
    4. Aaberge, Rolf & Mogstad, Magne & Peragine, Vito, 2011. "Measuring long-term inequality of opportunity," Journal of Public Economics, Elsevier, vol. 95(3), pages 193-204.
    5. Elena Bárcena-Martin & Jacques Silber, 2013. "On the generalization and decomposition of the Bonferroni index," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(4), pages 763-787, October.
    6. Luis J. Imedio‐Olmedo & Elena Bárcena‐Martín & Encarnación M. Parrado‐Gallardo, 2011. "A Class of Bonferroni Inequality Indices," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 13(1), pages 97-124, February.
    7. Giovanni Maria Giorgi & Chiara Gigliarano, 2017. "The Gini Concentration Index: A Review Of The Inference Literature," Journal of Economic Surveys, Wiley Blackwell, vol. 31(4), pages 1130-1148, September.
    8. Eliazar, Iddo I. & Sokolov, Igor M., 2012. "Measuring statistical evenness: A panoramic overview," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1323-1353.
    9. Luis Imedio-Olmedo & Encarnación Parrado-Gallardo & Elena Bárcena-Martín, 2012. "Income Inequality Indices Interpreted as Measures of Relative Deprivation/Satisfaction," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 109(3), pages 471-491, December.
    10. Giovanni Maria Giorgi & Saralees Nadarajah, 2010. "Bonferroni and Gini indices for various parametric families of distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 23-46.
    11. Eliazar, Iddo & Cohen, Morrel H., 2014. "Hierarchical socioeconomic fractality: The rich, the poor, and the middle-class," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 402(C), pages 30-40.
    12. Cowell, Frank, 2011. "Measuring Inequality," OUP Catalogue, Oxford University Press, edition 3, number 9780199594047.
    13. Shlomo Yitzhaki & Peter Lambert, 2013. "The relationship between the absolute deviation from a quantile and Gini’s mean difference," METRON, Springer;Sapienza Università di Roma, vol. 71(2), pages 97-104, September.
    14. Giovanni Maria Giorgi & Michele Crescenzi, 2001. "A proposal of poverty measures based on the Bonferroni inequality index," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3-4), pages 3-16.
    15. Giovanni M. Giorgi & Stefania Gubbiotti, 2017. "Celebrating the Memory of Corrado Gini: a Personality Out of the Ordinary," International Statistical Review, International Statistical Institute, vol. 85(2), pages 325-339, August.
    16. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
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