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Gini-stable Lorenz curves and their relation to the generalised Pareto distribution

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  • Lucio Bertoli-Barsotti
  • Marek Gagolewski
  • Grzegorz Siudem
  • Barbara .Zoga{l}a-Siudem

Abstract

We introduce an iterative discrete information production process where we can extend ordered normalised vectors by new elements based on a simple affine transformation, while preserving the predefined level of inequality, G, as measured by the Gini index. Then, we derive the family of empirical Lorenz curves of the corresponding vectors and prove that it is stochastically ordered with respect to both the sample size and G which plays the role of the uncertainty parameter. We prove that asymptotically, we obtain all, and only, Lorenz curves generated by a new, intuitive parametrisation of the finite-mean Pickands' Generalised Pareto Distribution (GPD) that unifies three other families, namely: the Pareto Type II, exponential, and scaled beta distributions. The family is not only totally ordered with respect to the parameter G, but also, thanks to our derivations, has a nice underlying interpretation. Our result may thus shed a new light on the genesis of this family of distributions. Our model fits bibliometric, informetric, socioeconomic, and environmental data reasonably well. It is quite user-friendly for it only depends on the sample size and its Gini index.

Suggested Citation

  • Lucio Bertoli-Barsotti & Marek Gagolewski & Grzegorz Siudem & Barbara .Zoga{l}a-Siudem, 2023. "Gini-stable Lorenz curves and their relation to the generalised Pareto distribution," Papers 2304.07480, arXiv.org, revised Jan 2024.
    • Handle: RePEc:arx:papers:2304.07480
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    References listed on IDEAS

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    1. Cena, Anna & Gagolewski, Marek & Siudem, Grzegorz & Żogała-Siudem, Barbara, 2022. "Validating citation models by proxy indices," Journal of Informetrics, Elsevier, vol. 16(2).
    2. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
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    4. Sarabia, José María, 2008. "A general definition of the Leimkuhler curve," Journal of Informetrics, Elsevier, vol. 2(2), pages 156-163.
    5. Bertoli-Barsotti, Lucio & Lando, Tommaso, 2019. "How mean rank and mean size may determine the generalised Lorenz curve: With application to citation analysis," Journal of Informetrics, Elsevier, vol. 13(1), pages 387-396.
    6. Lucio Bertoli-Barsotti, 2023. "Equivalent Gini coefficient, not shape parameter!," Scientometrics, Springer;Akadémiai Kiadó, vol. 128(1), pages 867-870, January.
    7. Balakrishnan, N. & Sarabia, José María & Kolev, Nikolai, 2010. "A simple relation between the Leimkuhler curve and the mean residual life," Journal of Informetrics, Elsevier, vol. 4(4), pages 602-607.
    8. Marek Gagolewski & Barbara Żogała-Siudem & Grzegorz Siudem & Anna Cena, 2022. "Ockham’s index of citation impact," Scientometrics, Springer;Akadémiai Kiadó, vol. 127(5), pages 2829-2845, May.
    9. Pyatt, Graham, 1976. "On the Interpretation and Disaggregation of Gini Coefficients," Economic Journal, Royal Economic Society, vol. 86(342), pages 243-255, June.
    10. L. Egghe, 2005. "Zipfian and Lotkaian continuous concentration theory," Journal of the American Society for Information Science and Technology, Association for Information Science & Technology, vol. 56(9), pages 935-945, July.
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    1. Lucio Bertoli-Barsotti & Marek Gagolewski & Grzegorz Siudem & Barbara .Zoga{l}a-Siudem, 2023. "Equivalence of inequality indices: Three dimensions of impact revisited," Papers 2304.07479, arXiv.org.

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