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On the decomposition by subpopulations of the point and synthetic Bonferroni inequality measures

digital On the decomposition by subpopulations of the point and synthetic
Bonferroni inequality measures
Article
journal STATISTICA & APPLICAZIONI
issue STATISTICA & APPLICAZIONI - 2016 - 1
title On the decomposition by subpopulations of the point and synthetic Bonferroni inequality measures
authors
publisher Vita e Pensiero
format Article | Pdf
online since 04-2017
issn 18246672 (print)
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This paper, by using the ‘‘two-step’’ approach proposed in Radaelli (2008, 2010) and in Zenga (2016) for the decomposition of the Zenga (2007) index, obtains the decomposition of the Bonferroni (1930) inequality measure. In the first step the Bonferroni point measure Vh(Y) is decomposed in a weighted mean of k x k relative differences between the mean Mg(Y) of subpopulation g and the lower mean Mhl(Y) of the subpopulation l; the weights are the product of their relative frequencies. From this decomposition, we obtain two decompositions of Vh(Y) into the within and the between components, and into the sum of the k contributions of each subpopulation. In the second step, the decompositions of the Bonferroni point index are extended, in a simple way, to the Bonferroni synthetic measure V(Y). We remark that the decomposition obtained in this paper is rather different from those proposed by Tarsitano (1990), and Barcena-Martin and Silber (2013). Actually, they provide only a decomposition by subpopulations of the Bonferroni synthetic index.