Rethinking Generalized Beta family of distributions
Abstract
We approach the Generalized Beta (GB) family of distributions using a meanreverting stochastic differential equation (SDE) for a power of the variable, whose steadystate (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 longrange powerlaw 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.
 Publication:

European Physical Journal B
 Pub Date:
 February 2023
 DOI:
 10.1140/epjb/s10051023004853
 arXiv:
 arXiv:2209.05225
 Bibcode:
 2023EPJB...96...24L
 Keywords:

 Quantitative Finance  Statistical Finance;
 Economics  Econometrics;
 Quantitative Finance  Mathematical Finance
 EPrint:
 21 pages, 11 figyes, 2 tables