英文标题：
《Skewness and kurtosis analysis for non-Gaussian distributions》
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作者：
Ahmet Celikoglu and Ugur Tirnakli
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最新提交年份：
2014
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英文摘要：
  In a recent paper [\\textit{M. Cristelli, A. Zaccaria and L. Pietronero, Phys. Rev. E 85, 066108 (2012)}], Cristelli \\textit{et al.} analysed relation between skewness and kurtosis for complex dynamical systems and identified two power-law regimes of non-Gaussianity, one of which scales with an exponent of 2 and the other is with $4/3$. Finally the authors concluded that the observed relation is a universal fact in complex dynamical systems. Here, we test the proposed universal relation between skewness and kurtosis with large number of synthetic data and show that in fact it is not universal and originates only due to the small number of data points in the data sets considered. The proposed relation is tested using two different non-Gaussian distributions, namely $q$-Gaussian and Levy distributions. We clearly show that this relation disappears for sufficiently large data sets provided that the second moment of the distribution is finite. We find that, contrary to the claims of Cristelli \\textit{et al.} regarding a power-law scaling regime, kurtosis saturates to a single value, which is of course different from the Gaussian case ($K=3$), as the number of data is increased. On the other hand, if the second moment of the distribution is infinite, then the kurtosis seems to never converge to a single value. The converged kurtosis value for the finite second moment distributions and the number of data points needed to reach this value depend on the deviation of the original distribution from the Gaussian case. We also argue that the use of kurtosis to compare distributions to decide which one deviates from the Gaussian more can lead to incorrect results even for finite second moment distributions for small data sets, whereas it is totally misleading for infinite second moment distributions where the difference depends on $N$ for all finite $N$.
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中文摘要：
在最近的一篇论文[\\textit{M.Cristelli，a.Zaccaria和L.Pietronero，Phys.Rev.E 85，066108（2012）]中，Cristelli\\textit{et al.}分析了复杂动力系统的偏度和峰度之间的关系，并确定了两个非高斯性的幂律区，其中一个指数为2，另一个指数为4/3$。最后，作者得出结论，所观察到的关系是复杂动力系统中的普遍事实。在这里，我们用大量的合成数据测试了偏度和峰度之间的普遍关系，并表明它实际上不是普遍的，只是由于所考虑的数据集中的数据点很少。使用两种不同的非高斯分布，即$q$-高斯分布和Levy分布，对所提出的关系进行了测试。我们清楚地表明，如果分布的二阶矩是有限的，对于足够大的数据集，这种关系将消失。我们发现，与Cristelli等人关于幂律标度制度的主张相反，随着数据数量的增加，峰度饱和为单一值，这当然不同于高斯情况（$K=3$）。另一方面，如果分布的二阶矩是无限的，那么峰度似乎永远不会收敛到一个值。有限二阶矩分布的收敛峰度值以及达到该值所需的数据点数量取决于原始分布与高斯情况的偏差。我们还认为，使用峰度来比较分布，以确定哪一个更偏离高斯分布，即使对于小数据集的有限二阶矩分布，也可能导致不正确的结果，而对于无限二阶矩分布，这是完全误导的，其中差分取决于所有有限的$N$。
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Physics        物理学
二级分类：Data Analysis, Statistics and Probability        数据分析、统计与概率
分类描述：Methods, software and hardware for physics data analysis: data processing and storage; measurement methodology; statistical and mathematical aspects such as parametrization and uncertainties.
物理数据分析的方法、软硬件：数据处理与存储；测量方法；统计和数学方面，如参数化和不确定性。
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一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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