From Gini to Bonferroni to Tsallis: an inequality-indices trek

Abstract

Arguably, the Gini index is the best known and the most widely applied inequality index in socioeconomics in particular, and across the sciences in general. On the other hand, far less known and less applied is the Bonferroni index. Addressed via Lorenz curves, the Gini index can be formulated as the average of two continuums of inequality indices that, respectively, stem from two sets of Lorenz-based distances: vertical and horizontal. These Gini-index formulations use one natural type of Lorenz-based distances. However, there is another natural type of Lorenz-based distances, and when using this type: (1) averaging the vertical continuum of inequality indices yields the known Bonferroni index; (2) averaging the horizontal continuum of inequality indices yields a new Bonferroni index. This paper explores comprehensively the two Bonferroni indices, and presents the many analogies between these indices and the Gini index. This paper also unifies the Bonferroni indices and the Gini index via two families of inequality induces: (1) a “vertical family” of which the known Bonferroni index and the Gini index are special cases; (2) a “horizontal family” of which the new Bonferroni index and the Gini index are special cases. These two families are shown to be counterparts of the Tsallis family of entropy measures, and the two Bonferroni indices are shown to be counterparts of the Shannon entropy. Written in an entirely self-contained manner, this paper is accessible also to audiences with no prior knowledge of inequality indices.