A simple algorithm is proposed which defines the Bonferroni 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 links also 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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A preliminary version of this paper was presented by Jacques Silber at the forty eighth meeting of the Italian Society of Economics, Demography and Statistics, which took place in Rome on May 26-28 2011. Jacques Silber completed this paper while he was visiting the Department of Economic Sciences of the University of Geneva, Switzerland, which he thanks for its very warm hospitality. He also gratefully acknowledges the financial support of the Adar Foundation of the Department of Economics of Bar-Ilan University. Elena Bárcena gratefully acknowledges the financial support provided by the Spanish Ministry of Education through Grant ECO2012-33993. Both authors thank three anonymous referees for their very helpful suggestions. There are also very grateful to Dr. Luis Imedio-Olmedo for his very useful comments.