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New statistic for financial return distributions: power-law or exponential?


  • V. F. Pisarenko

    (Russian Acad. Sci.)

  • D. Sornette

    (UCLA and CNRS-Univ. Nice)


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.

Suggested Citation

  • V. F. Pisarenko & D. Sornette, 2004. "New statistic for financial return distributions: power-law or exponential?," Papers physics/0403075,
  • Handle: RePEc:arx:papers:physics/0403075

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    References listed on IDEAS

    1. Parameswaran Gopikrishnan & Vasiliki Plerou & Luis A. Nunes Amaral & Martin Meyer & H. Eugene Stanley, 1999. "Scaling of the distribution of fluctuations of financial market indices," Papers cond-mat/9905305,
    2. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    3. Y. Malevergne & V. F. Pisarenko & D. Sornette, 2003. "Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?," Papers physics/0305089,
    4. Y. Malevergne & V. Pisarenko & D. Sornette, 2006. "On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of stock returns," Applied Financial Economics, Taylor & Francis Journals, vol. 16(3), pages 271-289.
    5. Carmela Quintos & Zhenhong Fan & Peter C. B. Phillips, 2001. "Structural Change Tests in Tail Behaviour and the Asian Crisis," Review of Economic Studies, Oxford University Press, vol. 68(3), pages 633-663.
    6. P. Gopikrishnan & M. Meyer & L.A.N. Amaral & H.E. Stanley, 1998. "Inverse cubic law for the distribution of stock price variations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 3(2), pages 139-140, July.
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