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Rethinking Generalized Beta family of distributions

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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.

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Data availability statement

We obtained S &P500 data at Yahoo! Finance. Our datasets are available upon request.

Notes

  1. There exist numerous extensions of GB as well—see, e.g., [6].

  2. Notice, that in [22] we used the standard B2 PDF,

    $$\begin{aligned} f_{B2}(y; \beta _2,p,q)=\frac{(1+\frac{y}{\beta _2})^{-p-q}(\frac{y}{\beta _2})^{-1+p}}{\beta _2 B(p,q)}, \end{aligned}$$
    (16)

    whence \(q=1+\frac{2 \gamma }{\kappa _2^2}\). The reason behind this ambiguity in the definition of q stems from the fact that p and q are independent at the B2/GB2 level in the steady-state PDF of the respective SDE, which is not the case for B/GB as will be shown below. Similar ambiguity with respect to the value of q exists for B1/GB1 for the same reason as for B2/GB2—that p and q are independent at that level and, consequently, B1/GB1 PDF can be written either in standard or modified version.

  3. This can also be done, albeit less efficiently, using Maximum Likelihood Estimation (MLE).

  4. Since parameters of GB/mGB distributions where obtained by fitting the data, the applicability of KS statistic may be of concern. However, a “Monte Carlo” analysis, based on generation of random variates, appears to support goodness of fit.

  5. Since realizes variance is a square realized volatility, if one is described by an GB/mGB so will be the other—with renormalized parameters.

  6. An alternative approach to modeling intra-day and daily returns is by jump processes [43], rather than continuous models, such as considered here.

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Acknowledgements

We used Wolfram Mathematica in a subset of analytical calculations and MathWorks Matlab for much of the numerical work. We also wish to thank Jeffrey Mills for helpful discussions.

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  1. Department of Physics, University of Cincinnati, Cincinnati, OH, 45221-0011, USA

    Jiong Liu & R. A. Serota

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  1. Jiong Liu
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Authors contributed equally to the paper, with Jiong Liu lead on numerical and R.A. Serota on analytical part.

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Correspondence to R. A. Serota.

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Liu, J., Serota, R.A. Rethinking Generalized Beta family of distributions. Eur. Phys. J. B 96, 24 (2023). https://doi.org/10.1140/epjb/s10051-023-00485-3

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