论文标题
重新思考广义beta分销家族
Rethinking Generalized Beta Family of Distributions
论文作者
论文摘要
我们使用均值的随机微分方程（SDE）接近广义分布族的广义分布家族，以实现变量的幂，其稳态（固定）概率密度函数（PDF）是修改的GB（MGB）分布。 SDE方法允许对GB分布的广义beta Prime（GB2）和广义β（GB1）限制进行清晰的解释，并在更进一步的情况下对广义逆伽马（GIGA）和广义伽玛（GGA）限制进行了清晰的解释，并描述了后两者之间的过渡。我们为“传统” GB PDF提供了一种替代形式，以强调GB分布的大量实用性在于它允许最终以有限的价值终止远程幂律行为。我们得出了“传统” GB的累积分布函数（CDF），该函数属于正规β函数产生的家族，对于分析分布的尾巴的分析至关重要。我们分析了有关实现市场波动的五十年历史数据，特别是S \＆P500，作为使用GB/MGB分布的案例研究，并表明其行为与负龙王的行为一致。
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.
