IDEAS home Printed from https://ideas.repec.org/a/spr/metron/v78y2020i2d10.1007_s40300-020-00171-9.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s40300-020-00171-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s40300-020-00171-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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, 2018. "Average is Over," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 123-137.
    3. Aaberge, Rolf & Mogstad, Magne & Peragine, Vito, 2011. "Measuring long-term inequality of opportunity," Journal of Public Economics, Elsevier, vol. 95(3), pages 193-204.
    4. 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.
    5. 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.
    6. Cowell, Frank, 2011. "Measuring Inequality," OUP Catalogue, Oxford University Press, edition 3, number 9780199594047.
    7. 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.
    8. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. Eliazar, Iddo, 2015. "The sociogeometry of inequality: Part I," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 426(C), pages 93-115.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ziqing Dong & Yves Tillé & Giovanni M. Giorgi & Alessio Guandalini, 2021. "Linearization and variance estimation of the Bonferroni inequality index," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 184(3), pages 1008-1029, July.
    2. Mariateresa Ciommi & Chiara Gigliarano & Giovanni Maria Giorgi, 2019. "Bonferroni And De Vergottini Are Back: New Subgroup Decompositions And Bipolarization Measures," Working Papers 439, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    3. Iddo Eliazar, 2024. "Beautiful Gini," METRON, Springer;Sapienza Università di Roma, vol. 82(3), pages 293-313, December.
    4. Banerjee, Suchismita & Chakrabarti, Bikas K. & Mitra, Manipushpak & Mutuswami, Suresh, 2020. "On the Kolkata index as a measure of income inequality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    5. Enora Belz, 2019. "Estimating Inequality Measures from Quantile Data," Economics Working Paper Archive (University of Rennes & University of Caen) 2019-09, Center for Research in Economics and Management (CREM), University of Rennes, University of Caen and CNRS.
    6. Palestini, Arsen & Pignataro, Giuseppe, 2016. "A graph-based approach to inequality assessment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 455(C), pages 65-78.
    7. Alessio Guandalini, 2022. "Things you should know about the Gini index," RIEDS - Rivista Italiana di Economia, Demografia e Statistica - The Italian Journal of Economic, Demographic and Statistical Studies, SIEDS Societa' Italiana di Economia Demografia e Statistica, vol. 76(4), pages 4-12, October-D.
    8. Amparo Ba'illo & Javier C'arcamo & Carlos Mora-Corral, 2021. "Extremal points of Lorenz curves and applications to inequality analysis," Papers 2103.03286, arXiv.org.
    9. Suchismita Banerjee & Bikas K. Chakrabarti & Manipushpak Mitra & Suresh Mutuswami, 2020. "Inequality Measures: The Kolkata index in comparison with other measures," Papers 2005.08762, arXiv.org, revised Oct 2020.
    10. Joseph L. Gastwirth & Richard Luo & Qing Pan, 2024. "A statistical examination of wealth inequality within the Forbes 400 richest families in the United States from 2000 to 2023," METRON, Springer;Sapienza Università di Roma, vol. 82(3), pages 329-344, December.
    11. Eliazar, Iddo, 2024. "From Zipf to Price and beyond," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 647(C).
    12. Francesca Greselin & Ričardas Zitikis, 2018. "From the Classical Gini Index of Income Inequality to a New Zenga-Type Relative Measure of Risk: A Modeller’s Perspective," Econometrics, MDPI, vol. 6(1), pages 1-20, January.
    13. Elena Bárcena-Martin & Jacques Silber, 2017. "The Bonferroni index and the measurement of distributional change," METRON, Springer;Sapienza Università di Roma, vol. 75(1), pages 1-16, April.
    14. Shweta Bahl & Ajay Sharma, 2021. "Education–Occupation Mismatch and Dispersion in Returns to Education: Evidence from India," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 153(1), pages 251-298, January.
    15. Loi, Andrea & Matta, Stefano, 2021. "Minimal entropy and uniqueness of price equilibria in a pure exchange economy," Journal of Mathematical Economics, Elsevier, vol. 97(C).
    16. Alvaredo, Facundo & Cogneau, Denis & Piketty, Thomas, 2021. "Income inequality under colonial rule. Evidence from French Algeria, Cameroon, Tunisia, and Vietnam and comparisons with British colonies 1920–1960," Journal of Development Economics, Elsevier, vol. 152(C).
    17. Yonatan Berman, 2022. "The Long-Run Evolution of Absolute Intergenerational Mobility," American Economic Journal: Applied Economics, American Economic Association, vol. 14(3), pages 61-83, July.
    18. Fontanari Andrea & Cirillo Pasquale & Oosterlee Cornelis W., 2020. "Lorenz-generated bivariate Archimedean copulas," Dependence Modeling, De Gruyter, vol. 8(1), pages 186-209, January.
    19. Tatjana Miljkovic & Ying-Ju Chen, 2021. "A new computational approach for estimation of the Gini index based on grouped data," Computational Statistics, Springer, vol. 36(3), pages 2289-2311, September.
    20. Fabio Clementi & Mauro Gallegati & Lisa Gianmoena & Simone Landini & Joseph E. Stiglitz, 2019. "Mis-measurement of inequality: a critical reflection and new insights," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 14(4), pages 891-921, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metron:v:78:y:2020:i:2:d:10.1007_s40300-020-00171-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.