The robustness of the generalized Gini index

S. Settepanella (), A. Terni (), M. Franciosi () and L. Li ()
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S. Settepanella: Torino University
A. Terni: Pisa University
M. Franciosi: Pisa University
L. Li: Guangzhou College of Commerce

Decisions in Economics and Finance, 2022, vol. 45, issue 2, No 4, 539 pages

Abstract: Abstract In this paper, we introduce a map $$\varPhi $$ Φ , which we call zonoid map, from the space of all non-negative, finite Borel measures on $${\mathbb {R}}^n$$ R n with finite first moment to the space of zonoids of $${\mathbb {R}}^n$$ R n . This map, connecting Borel measure theory with zonoids theory, allows to slightly generalize the Gini volume introduced, in the context of Industrial Economics, by Dosi (J Ind Econ 4:875–907, 2016). This volume, based on the geometric notion of zonoid, is introduced as a measure of heterogeneity among firms in an industry and it turned out to be a quite interesting index as it is a multidimensional generalization of the well-known and broadly used Gini index. By exploiting the mathematical context offered by our definition, we prove the continuity of the map $$\varPhi $$ Φ which, in turn, allows to prove the validity of a SLLN-type theorem for our generalized Gini index and, hence, for the Gini volume. Both results, the continuity of $$\varPhi $$ Φ and the SLLN theorem, are particularly useful when dealing with a huge amount of multidimensional data.

Keywords: Gini index; Zonoid; Empirical distribution; Hausdorff metric (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10203-022-00378-7

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