金融收益分布的新统计：幂律还是指数？
New statistic for financial return distributions: power-law or exponential?
摘要：
我们引入了一个新的统计工具（TP-统计量和TE-统计量），专门用于比较幂律分布和指数分布的样本尾部作为阈值下限u的函数的行为。这些统计量的一个重要性质是，它们收敛到零的幂律或指数相应的，无论值的指数或形式参数。这对于独立于量化参数值的可能性来测试分布的结构（幂律或非幂律，指数或非指数）特别有用。我们将这些统计数据应用于道琼斯工业平均指数一个世纪的每日数据和纳斯达克综合指数一年多的5分钟数据的回报率分布。我们的分析证实了以前的作品显示的趋势，尾部越来越像一个幂律的最高分位数，但我们可以检测到明显的偏差，这表明尾部的结构的回报率分布比通常假设的更复杂;它显然是更复杂，只是一个幂律。
英文摘要：
We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u. One important property of these statistics is that they converge to zero for power laws or for exponentials correspondingly, regardless of the value of the exponent or of the form parameter. This is particularly useful for testing the structure of a distribution (power law or not, exponential or not) independently of the possibility of quantifying the values of the parameters. We apply these statistics to the distribution of returns of one century of daily data for the Dow Jones Industrial Average and over one year of 5-minutes data of the Nasdaq Composite index. Our analysis confirms previous works showing the tendency for the tails to resemble more and more a power law for the highest quantiles but we can detect clear deviations that suggest that the structure of the tails of the distributions of returns is more complex than usually assumed; it is clearly more complex that just a power law.
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