The Bonferroni index and the measurement of distributional change
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The Bonferroni index and the measurement of distributional change Elena Bárcena-Martin1 · Jacques Silber2,3
Received: 4 November 2016 / Accepted: 15 December 2016 © Sapienza Università di Roma 2017
Abstract This paper extends previous work that derived a matricial approach to the definition of the Bonferroni index. The present study is devoted to the decomposition of a change over time in the Bonferroni index into two components, one measuring re-ranking and the other the progressivity of income change. An extension is then proposed, based on a generalization of the Bonferroni index. The paper ends with an empirical illustration, based on the EU-SILC database, which analyzes the change in income inequality that was observed in 23 European countries between 2007 and 2010. Keywords Bonferroni index · Decomposition · Progressivity · Re-ranking
1 Introduction The Gini [18] index has been for close to a century the most popular measure of income inequality. Many studies have proposed the use of other indices such as the Theil indices or their extension (generalized entropy indices) or the Atkinson index but the Gini index remains, without any doubt, the most widespread measure of inequality. Some other inequality indices, such as those proposed by Bonferroni [10] and De Vergottini [15], have been until very recently completely ignored in the literature, most probably because these articles were written in Italian although, as suggested by Giorgi and Nadarajah [26], one cannot exclude the possibility that Corrado Gini who was very active and influent in the domain of inequality measurement made sure his index would be the most commonly used measure of inequality. According to Giorgi and Nadarajah [26], Piesch [32] and Nygard and Sandström [33] were
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Jacques Silber [email protected] Elena Bárcena-Martin [email protected]
1
Facultad de Ciencias Económicas y Empresariales, Universidad de Málaga, Málaga, Spain
2
Department of Economics, Bar-Ilan University, 52900 Ramat-Gan, Israel
3
CEPS/INSTEAD, Esch-sur-Alzette, Luxembourg
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E. Bárcena-Martin, J. Silber
the first to mention again the Bonferroni and De Vergottini indices. Later on many other papers were devoted to these forgotten indices, in particular to the Bonferroni index. Some of these studies emphasized the statistical properties of the Bonferroni index such as those by Giorgi [19], Giorgi and Crescenzi [20–22], Giorgi and Guandalini [23], Giorgi and Mondani [24,25] and Giorgi and Nadarajah [26]. Other papers, such as those by Aristondo et al. [3], Bárcena and Imedio [4], Bárcena Martín and Silber [6], Chakravaty [12], Tarsitano [37] and Zenga [39] stressed rather the properties of the Bonferroni index which should make it an attractive measure of income inequality. Aaberge [2] who introduced the concept of “scaled conditional mean curve”, which is just another name for the Bonferron [10] curve, showed that this curve may give important information on poverty, assuming the poverty line has been determined. Aaberge [2] showed also that the Bonferro
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