V.Pisarenko  and D.Sornette

Abstract.
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.