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have been chosen to converge to 1 approximately exponentially as
1 − qn = 3.08 e −0.342n , (32)
the linear fit of ˆbu as a function of n shown as the dashed line in figure 7 corresponds to
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ˆbu(qn) = 0.08 + 0.626ln 3.08
. (33)
1 − qn
Expression (33) suggests an unbound logarithmic growth of ˆbu as the quantile approaches 1. For instance,
for a quantile 1 −q = 0.1%, expression (33) predicts ˆbu(1 −q = 10 −3 ) = 5.1. For a quantile 1 −q = 0.01%,
expression (33) predicts ˆbu(1 − q = 10 −4 ) = 6.5, and so on. Each time the quantile 1 − q is divided by
a factor 10, the apparent exponent ˆbu(q) is increased by the additive constant ∼ = 1.45: ˆbu((1 − q)/10) =
ˆbu(1 − q) + 1.45. This very slow growth uncovered here may be an explanation for the belief and possibly
mistaken conclusion that the Hill and other estimators of the tail index tends to a constant for high quantiles.
Indeed, it is now clear that the slowdowns of the growth of ˆbu seen in figures 6 decorated by large fluctuations
due to small size effects is mostly the result of a dilatation of the data expressed in terms of threshold u.
When recast in the more natural logarithm scale of the quantiles q1 ...q18, this slowdown disappears. Of
course, it is impossible to know how long this growth given by (33) may go on as the quantile q tends to 1.
In other words, how can we escape from the sample range when estimating quantiles? How can we estimate
the so-called “high quantiles” at the level q > 1 − 1/T where T is the total number of sampled points.
Embrechts et al. (1997) have summarized the situation in this way: “there is no free lunch when it comes to
high quantiles estimation!” It is possible that ˆbu(q) will grow without limit as would be the case if the true
underlying distribution was rapidly varying. Alternatively, ˆbu(q) may saturate to a large value, as predicted
for instance by the traditional GARCH model which yields tails indices which can reach 10 − 20 (Engle
and Patton 2001, Starica and Pictet 1999) or by the recent multifractal random walk (MRW) model which
gives an asymptotic tail exponent in the range 20 − 50 (Muzy et al. 2000, Muzy et al. 2001). According to
(33), a value ˆbu ≈ 20 (respectively 50) would be attained for 1 − q ≈ 10 −13 (respectively 1 − q ≈ 10 −34 )! If
one believes in the prediction of the MRW model, the tail of the distribution of returns is regularly varying
but this insight is completely useless for all practical purposes due to the astronomically high statistics that
would be needed to sample this regime. In this context, we cannot hope to get access to the true nature of the
pdf of returns but only strive to define the best effective or apparent most parsimonious and robust model.
We do not discuss here the new class of estimation issues raised by the MRW model, which is interesting in
itself but requires a specific analysis of its own left for another work.
The question of the exhaustion of the growth of the tail index is really crucial. Indeed, if it is unboundedly
increasing, it is the signature that the tails of the distributions of returns decay faster than any power-law, and
thus cannot be regularly varying. We revisit this question of the growth of the apparent exponent b, using
the notion of local exponent, in Appendix C as an attempt to better constraint this growth. The analysis
developed in this Appendix C basically confirms the first indication shown in figure 7 .
4.3.2 Weibull distributions
Let us now fit our data with the Weibull (SE) distribution (23). The Anderson-Darling statistics (ADS) for
this case are shown in table 7. The ML-estimates and AD-estimates of the form parameter c are represented
in table 9. Table 7 shows that, for the higest quantiles, the ADS for the Stretched-Exponential is the smallest
of all ADS, suggesting that the SE is the best model of all. Moreover, for the lowest quantiles, it is the sole
model not systematically rejected at the 95% level.
The c-estimates are found to decrease when increasing the order q of the threshold uq beyond which the
estimations are performed. In addition, the c-estimate is identically zero for u18. However, this does not
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