  A large consensus now seems to take for granted that the distributions of
empirical returns of financial time series are regularly varying, with a tail
exponent close to 3. We revisit this results and use standard tests as well as
develop a battery of new non-parametric and parametric tests (in particular
with stretched exponential (SE) distributions) to characterize the
distributions of empirical returns of financial time series, with application
to the 100 years of daily return of the Dow Jones Industrial Average and over 1
years of 5-minutes returns of the Nasdaq Composite index. Based on the
discovery that the SE distribution tends to the Pareto distribution in a
certain limit such that the Pareto (or power law) distribution can be
approximated with any desired accuracy on an arbitrary interval by a suitable
adjustment of the parameters of the SE distribution, we demonstrate that Wilks'
test of nested hypothesis still works for the non-exactly nested comparison
between the SE and Pareto distributions. The SE distribution is found
significantly better over the whole quantile range but becomes unnecessary
beyond the 95% quantiles compared with the Pareto law. Similar conclusions hold
for the log-Weibull model with respect to the Pareto distribution. Our main
result is that the tails ultimately decay slower than any stretched exponential
distribution but probably faster than power laws with reasonable exponents.
Implications of our results on the ``moment condition failure'' and for risk
estimation and management are presented.
