拟合无穷可分分布：Gamma方差模型的情形
Fitting Infinitely divisible distribution: Case of Gamma-Variance Model
摘要：
本文研究了基于分数阶傅里叶变换（FRFT）的概率密度函数及其导数的获取方法，主要用于拟合具有无穷可除性的基本概率关系的随机模型。计算了方差-伽玛（VG）模型和复合泊松正态复合模型的概率密度函数，讨论了它们的尖峰、尖峰和不对称等分布特性。计算VG模型的概率密度函数的一阶和二阶导数，以建立极大似然方法的费舍尔信息矩阵。VG模型越来越多地被用作经典对数正态模型（CLM）的替代品，用于建模资产价格。用分数阶傅里叶变换估计了五参数VG模型。数据来自SPY ETF的每日价格数据。Kolmogorov-Smirnov（KS）拟合优度表明VG模型比CLM模型更好地拟合经验累积分布。最好的VG模型来自FRFT估计。
英文摘要：
The paper examines the Fractional Fourier Transform (FRFT) based technique as a tool for obtaining probability density function and its derivatives, and mainly for fitting stochastic model with the fundamental probabilistic relationships of infinite divisibility. The probability density functions are computed and the distributional proprieties such as leptokurtosis, peakedness, and asymmetry are reviewed for Variance-Gamma (VG) model and Compound Poisson with Normal Compounding model. The first and second derivatives of probability density function of the VG model are also computed in order to build the Fisher information matrix for the Maximum likelihood method. The VG model has been increasingly used as an alternative to the Classical Lognormal Model (CLM) in modelling asset price. The VG model with fives parameters was estimated by the FRFT. The data comes from the daily SPY ETF price data. The Kolmogorov-Smirnov (KS) goodness-of-fit shows that the VG model fits better the empirical cumulative distribution than the CLM. The best VG model comes from the FRFT estimation.
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