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On the Generalization and Decomposition of the Bonferroni Index

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  • Elena Barcena-Martin
  • Jacques Silber

Abstract

A simple algorithm is proposed which defines the Bonferroni index as the product of a row vector of individual population shares, a linear mathematical operator called the Bonferroni matrix and a column vector of income shares. This algorithm greatly simplifies the decomposition of the Bonferroni index by income sources or classes and population subgroups. The proposed algorithm also links the Bonferroni index to the concepts of relative deprivation and social welfare and leads to a generalization where the traditional Bonferroni and Gini indices are special cases. The paper ends with an empirical illustration based on EU-SILC data for the year 2008.

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  • Elena Barcena-Martin & Jacques Silber, 2012. "On the Generalization and Decomposition of the Bonferroni Index," OPHI Working Papers 51, Queen Elizabeth House, University of Oxford.
    • Handle: RePEc:qeh:ophiwp:ophiwp051
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    Cited by:

    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. Chakravarty, Satya R. & Sarkar, Palash, 2022. "A synthesis of local and effective tax progressivity measurement," MPRA Paper 115180, University Library of Munich, Germany.
    3. 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.
    4. Satya R. Chakravarty & Palash Sarkar, 2022. "Inequality minimising subsidy and taxation," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(1), pages 53-67, May.
    5. Giovanni Maria Giorgi & Alessio Guandalini, 2016. "Bonferroni Index Decomposition And The Shapley Method," 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. 70(4), pages 67-78, October-D.
    6. Giovanni M. Giorgi & Alessio Guandalini, 2018. "Decomposing the Bonferroni Inequality Index by Subgroups: Shapley Value and Balance of Inequality," Econometrics, MDPI, vol. 6(2), pages 1-16, April.
    7. Satya R. Chakravarty & Rama Pal & Rupayan Pal & Palash Sarkar, 2022. "Minimum inequality taxation, average and minimally progressive taxations and depolarization," Indira Gandhi Institute of Development Research, Mumbai Working Papers 2022-010, Indira Gandhi Institute of Development Research, Mumbai, India.
    8. Satya R. Chakravarty & Palash Sarkar, 2021. "Designing income distributions with specified inequalities," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 9(2), pages 297-311, October.
    9. 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.
    10. Satya R. Chakravarty & Palash Sarkar, 2023. "Notes on the postulate of the monotonicity in distance in inequality," Bulletin of Economic Research, Wiley Blackwell, vol. 75(2), pages 312-322, April.
    11. Giovanni Maria Giorgi & Alessio Guandalini, 2013. "A sampling estimator of the Bonferroni inequality 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. 67(3-4), pages 151-158, July-Dece.
    12. Satya R. Chakravarty & Palash Sarkar, 2021. "An inequality paradox: relative versus absolute indices?," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 241-254, August.
    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.

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