On the generalization and decomposition of the Bonferroni index

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.Keywords: Gini, inequality decomposition, population subgroups, relative deprivation, social welfare JEL classification: D31, D63 1 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. Barcena-Martin and SilberGeneralization This publication is copyright, however it may be reproduced without fee for teaching or non-profit purposes, but not for resale. Formal permission is required for all such uses, and will normally be granted immediately. For copying in any other circumstances, or for re-use in other publications, or for translation or adaptation, prior written permission must be obtained from OPHI and may be subject to a fee.

Bonferroni and De Vergottini are back: New subgroup decompositions and bipolarization measures

“…Recently, the Bonferroni index has been revalued since its features and new interesting applications in social and economic contexts have been studied (Barcena-Martin and Silber [8], [9], [10]; Chakravarty and Muliere [18]; Chakravarty [13]).…”

Inequality, Polarization and Conflict

“…The Gini index for such a partitioned distribution can be written as Ebert (1988b) in their characterizations, it is not included as a member of the generalized entropy family and the Ebert family. Barcena-Martin and Silber (2013) decomposed the Bonferroni index for a population partitioned according to income classes and showed that in addition to between-and within-group components, a residual term, reflecting the effect of ranking and the index's failure to fulfill DPP, appears in the decomposition.…”

Decomposing the Bonferroni Inequality Index by Subgroups: Shapley Value and Balance of Inequality

“…To measure the effect of the differences in size (n), we compare the case when the subgroups have different sizes ( f j = 1/J) to the case when the sizes are equal ( f j = 1/J). To make the subgroups have the same size, the least common multiple (lcm) is 2 For the expressions of B w , B b and B i in the case of income classes, see Tarsitano (1990), while for the matrix decomposition of B see Bárcena-Martin and Silber (2013). calculated for the sizes of the analyzed subgroups and the values x ji are repeated lcm times; this leads to equality in size between the subgroups. When applying such an approach to B, the objection is usually raised that B, as opposed to R, does not satisfy the Dalton (1925) principle of being replication invariant.…”

“…Piesch (1975) and Nygård and Sandström (1981) were the first to investigate B in depth. New and interesting interpretations and extensions of B have been just recently proposed: its welfare implications have been studied by Benedetti (1986), Aaberge (2000), Chakravarty (2007) and Bárcena-Martin and Silber (2013). Giorgi and Crescenzi (2001c) proposed a poverty measure based on B, while other socio-economic aspects have been studied by Bárcena-Martin and Olmedo (2008), Silber and Son (2010), Silber (2011, 2013), and Imedio Olmedo et al (2012).…”

“…Many contributions are related to the decomposition of R (for a deep investigation see, e.g., Kakwani 1980;Nygård and Sandström 1981;Giorgi 2011a). Tarsitano (1990) introduced several standard results that can be used for the decomposition of B, while Bárcena-Martin and Silber (2013) derived an algorithm that greatly simplifies such a decomposition.…”