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96 3. Distributions exponentielles étirées contre distributions régulièrement variables
where L ′ (x) = ln(x) L(x). The lower panel of figure 10 shows the local index β(x) for a simulated Stretched-
Exponential distribution with c = 0.3. In this case, the local index β(x) continuously increases, which corresponds
to our intuition that an Stretched-Exponential behaves like a power-law with an always increasing
exponent.
Figure 11 shows the local index β(x) for a distribution constructed by joining two Pareto distributions with
exponents b1 = 0.70 and b2 = 1.5 at the cross-over point u1 = 10. In this case, the local index β(x) again
increases but not so quickly as in previous Stretched-Exponential case. Even for such large sample size
(n = 15000), the “final” β(x) is about 1.3 which is still less than the true b-value for the second Pareto part
(b = 1.5), showing the existence of strong cross-over effects. Such very strong cross-over effects occurring
in the presence of a transition from a power law to another power law have already been noticed in (Sornette
et al. 1996).
Similarly to the local index β(x) based on the Pareto distribution taken as a reference, we define the notion
of a local exponent c(x) taking Stretched-Exponential distributions as a reference. Indeed, given any
sufficiently smooth positive function g(x), one can always find a function c(x) such that
g(x) = exp 1 − x c(x)
, x ≥ 1, (80)
with c(x) = ln(1 − ln(g(x)))/ln(x). Obviously, for any Stretched-Exponential distribution with exponent c,
the local exponent c(x) converges to c as x goes to infinity. This property is the same as for the local index
β(x) which goes to the true tail index β for regularly varying distributions.
Figure 12 shows the sample tail (continuous line), the local index β(x) (dashed line) and the local exponent
c(x) (dash-dotted line) for the negative tail of the Nasdaq five-minutes returns. The local exponent c(x)
clearly reaches an asymptotic value ∼ = 0.32 for large enough values of the returns. In contrast, the local exponent
β(x) remains continuously increasing. The lower panel shows in double logarithmic scale the local
index β(x). Over a large range, β(x) increases approximately a power law of index 0.77 while beyond the
quantile 99% (see the inset) it behaves like a power law with smaller index equal to 0.54 implying a decelerating
growth. The goodness of fit of the regression of lnβ(x) on lnx has been qualified by a χ 2 test which
does not allow to reject this model at any usual confidence level. Note that a power law dependence of the
local Pareto exponent β(x) as a function of x qualifies a Stretched-Exponential distribution, according to (78)
and (79). The second regime fitted with the exponent c = 0.54 seems still perturbed by a cross-over effect
as it does not retrieve the value c ∼ = 0.32 which characterizes the tail of the distributions of returns according
to the Stretched-Exponential model. These fits quantifying the growth of the local Pareto exponent are not
in contradiction with figure 7 and expression (33). Indeed, a decreasing positive exponent is an alternative
description for a logarithmic growth and vice-versa. These fits provide an improved quantification of the
previously rough characterization of the growth of the exponent estimated per quantile shown with 7 and
expression (33) but using a better characterization of the “local” exponent.
The positive tail of the Nasdaq and both tails of the Dow Jones exhibit exactly the same continuously increasing
behaviour with the same characteristics and we are not showing them. Taken together, these obervations
suggest that the Stretched-Exponential representation provides a better model of the tail behavior of large
returns than does a regularly varying distribution.
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