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From Pareto to Weibull – A Constructive Review of Distributions on ℝ+

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  • Corinne Sinner
  • Yves Dominicy
  • Julien Trufin
  • Wout Waterschoot
  • Patrick Weber
  • Christophe Ley

Abstract

Power laws and power laws with exponential cut‐off are two distinct families of distributions on the positive real half‐line. In the present paper, we propose a unified treatment of both families by building a family of distributions that interpolates between them, which we call Interpolating Family (IF) of distributions. Our original construction, which relies on techniques from statistical physics, provides a connection for hitherto unrelated distributions like the Pareto and Weibull distributions, and sheds new light on them. The IF also contains several distributions that are neither of power law nor of power law with exponential cut‐off type. We calculate quantile‐based properties, moments and modes for the IF. This allows us to review known properties of famous distributions on ℝ+ and to provide in a single sweep these characteristics for various less known (and new) special cases of our Interpolating Family.

Suggested Citation

  • Corinne Sinner & Yves Dominicy & Julien Trufin & Wout Waterschoot & Patrick Weber & Christophe Ley, 2023. "From Pareto to Weibull – A Constructive Review of Distributions on ℝ+," International Statistical Review, International Statistical Institute, vol. 91(1), pages 35-54, April.
    • Handle: RePEc:bla:istatr:v:91:y:2023:i:1:p:35-54
    DOI: 10.1111/insr.12508
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    References listed on IDEAS

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    1. Xavier Gabaix, 2016. "Power Laws in Economics: An Introduction," Journal of Economic Perspectives, American Economic Association, vol. 30(1), pages 185-206, Winter.
    2. Benoit Mandelbrot, 1964. "Random Walks, Fire Damage Amount and Other Paretian Risk Phenomena," Operations Research, INFORMS, vol. 12(4), pages 582-585, August.
    3. Thomas Piketty & Gabriel Zucman, 2014. "Capital is Back: Wealth-Income Ratios in Rich Countries 1700–2010," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 129(3), pages 1255-1310.
    4. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
    5. Erzo G. J. Luttmer, 2007. "Selection, Growth, and the Size Distribution of Firms," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 122(3), pages 1103-1144.
    6. Alexis Akira Toda & Kieran Walsh, 2015. "The Double Power Law in Consumption and Implications for Testing Euler Equations," Journal of Political Economy, University of Chicago Press, vol. 123(5), pages 1177-1200.
    7. Alexis Akira Toda & Kieran Walsh, 2015. "The Double Power Law in Consumption and Implications for Testing Euler Equations," Journal of Political Economy, University of Chicago Press, vol. 123(5), pages 1177-1200.
    8. M. C. Jones, 2015. "On Families of Distributions with Shape Parameters," International Statistical Review, International Statistical Institute, vol. 83(2), pages 175-192, August.
    9. Shogo Kato & M. C. Jones, 2015. "A tractable and interpretable four-parameter family of unimodal distributions on the circle," Biometrika, Biometrika Trust, vol. 102(1), pages 181-190.
    10. Aban, Inmaculada B. & Meerschaert, Mark M. & Panorska, Anna K., 2006. "Parameter Estimation for the Truncated Pareto Distribution," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 270-277, March.
    11. repec:hal:pseose:halshs-01109372 is not listed on IDEAS
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