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New multivariate Gini’s indices

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  • Capaldo, Marco
  • Navarro, Jorge

Abstract

The Gini’s mean difference was defined as the expected absolute difference between a random variable and its independent copy. The corresponding normalized version, namely Gini’s index, denotes two times the area between the egalitarian line and the Lorenz curve. Both are dispersion indices because they quantify how far a random variable and its independent copy are. Aiming to measure dispersion in the multivariate case, we define and study new Gini’s indices. For the bivariate case we provide several results and we point out that they are “dependence-dispersion” indices. Covariance representations are exhibited, with an interpretation also in terms of conditional distributions. Further results, bounds and illustrative examples are discussed too. Multivariate extensions are defined, aiming to apply both indices in more general settings. Then, we define efficiency Gini’s indices for any semi-coherent system and we discuss about their interpretation. Empirical versions are considered as well in order to apply multivariate Gini’s indices to data.

Suggested Citation

  • Capaldo, Marco & Navarro, Jorge, 2025. "New multivariate Gini’s indices," Journal of Multivariate Analysis, Elsevier, vol. 206(C).
    • Handle: RePEc:eee:jmvana:v:206:y:2025:i:c:s0047259x24001015
    DOI: 10.1016/j.jmva.2024.105394
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    References listed on IDEAS

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    1. Georgios Psarrakos & Jorge Navarro, 2013. "Generalized cumulative residual entropy and record values," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(5), pages 623-640, July.
    2. Lidia Ceriani & Paolo Verme, 2012. "The origins of the Gini index: extracts from Variabilità e Mutabilità (1912) by Corrado Gini," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 10(3), pages 421-443, September.
    3. Jorge Navarro & Fabio Spizzichino, 2010. "Comparisons of series and parallel systems with components sharing the same copula," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(6), pages 775-791, November.
    4. Fuchs, Sebastian & Tschimpke, Marco, 2024. "A novel positive dependence property and its impact on a popular class of concordance measures," Journal of Multivariate Analysis, Elsevier, vol. 200(C).
    5. Gastwirth, Joseph L, 1972. "The Estimation of the Lorenz Curve and Gini Index," The Review of Economics and Statistics, MIT Press, vol. 54(3), pages 306-316, August.
    6. Xin Dang & Hailin Sang & Lauren Weatherall, 2019. "Gini covariance matrix and its affine equivariant version," Statistical Papers, Springer, vol. 60(3), pages 641-666, June.
    7. Koshevoy, G. A. & Mosler, K., 1997. "Multivariate Gini Indices," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 252-276, February.
    8. Xuehua Yin & Narayanaswamy Balakrishnan & Chuancun Yin, 2023. "Bounds for Gini’s mean difference based on first four moments, with some applications," Statistical Papers, Springer, vol. 64(6), pages 2081-2100, December.
    9. Franco Pellerey & Jorge Navarro, 2022. "Stochastic monotonicity of dependent variables given their sum," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 543-561, June.
    10. Francisco J. Samaniego, 2007. "System Signatures and their Applications in Engineering Reliability," International Series in Operations Research and Management Science, Springer, number 978-0-387-71797-5, June.
    11. Marco Capaldo & Antonio Di Crescenzo & Alessandra Meoli, 2024. "Cumulative information generating function and generalized Gini functions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 87(7), pages 775-803, October.
    12. Bernard Carole & Müller Alfred, 2020. "Dependence uncertainty bounds for the energy score and the multivariate Gini mean difference," Dependence Modeling, De Gruyter, vol. 8(1), pages 239-253, January.
    13. Patricia Ortega-Jiménez & Miguel A. Sordo & Alfonso Suárez-Llorens, 2021. "Stochastic Comparisons of Some Distances between Random Variables," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
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