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Rethinking Generalized Beta family of distributions

Author

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  • Jiong Liu

    (University of Cincinnati)

  • R. A. Serota

    (University of Cincinnati)

Abstract

We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the “traditional” GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the “traditional” GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S &P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings. Graphic Abstract

Suggested Citation

  • Jiong Liu & R. A. Serota, 2023. "Rethinking Generalized Beta family of distributions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 96(2), pages 1-14, February.
    • Handle: RePEc:spr:eurphb:v:96:y:2023:i:2:d:10.1140_epjb_s10051-023-00485-3
    DOI: 10.1140/epjb/s10051-023-00485-3
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    Cited by:

    1. Nowak, Przemysław & Santolini, Marc & Singh, Chakresh & Siudem, Grzegorz & Tupikina, Liubov, 2024. "Beyond Zipf’s law: Exploring the discrete generalized beta distribution in open-source repositories," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 649(C).
    2. Jiong Liu & M. Dashti Moghaddam & R. A. Serota, 2023. "Are there Dragon Kings in the Stock Market?," Papers 2307.03693, arXiv.org.
    3. Jiong Liu & Hamed Farahani & R. A. Serota, 2023. "Exploring Distributions of House Prices and House Price Indices," Papers 2312.14325, arXiv.org.

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