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The robustness of the generalized Gini index

Author

Listed:
  • S. Settepanella

    (Torino University)

  • A. Terni

    (Pisa University)

  • M. Franciosi

    (Pisa University)

  • L. Li

    (Guangzhou College of Commerce)

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.

Suggested Citation

  • S. Settepanella & A. Terni & M. Franciosi & L. Li, 2022. "The robustness of the generalized Gini index," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(2), pages 521-539, December.
  • Handle: RePEc:spr:decfin:v:45:y:2022:i:2:d:10.1007_s10203-022-00378-7
    DOI: 10.1007/s10203-022-00378-7
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    References listed on IDEAS

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