标题： 重新思考广义贝塔分布族 显示英文标题

标题： Rethinking Generalized Beta Family of Distributions

摘要： 我们通过一个均值回归的随机微分方程（SDE）来研究广义贝塔（GB）分布族，该方程针对变量的幂进行建模，其稳态（平稳）概率密度函数（PDF）是一个修改的广义贝塔（mGB）分布。 SDE方法允许对广义贝塔第二型（GB2）和广义贝塔第一型（GB1）的GB分布极限进行清晰的解释，并进一步向下扩展到广义逆伽马（GIGa）和广义伽马（GGa）极限，同时描述后两者之间的转换过程。 我们提供了“传统”GB PDF的另一种形式，以强调GB分布的大量实用性在于它能够最终在有限值处终止长范围的幂律行为。 我们推导了“传统”GB的累积分布函数（CDF），该函数属于由正则化贝塔函数生成的族，对于分析分布尾部至关重要。 我们分析了过去五十年关于实际市场波动的历史数据，特别是标普500的数据，作为GB/mGB分布应用的案例研究，并表明其行为与负向龙王的行为一致。 显示英文摘要

摘要： 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.