statistique, théorie et gestion de portefeuille - Docs at ISFA

and Sornette (1999) for instance. We now turn to this question using the framework of GEV and GDP
estimators just described.
3.5 GEV and GPD estimators of the Dow Jones and Nasdaq data sets
We have applied the same analysis as in the previous section on the real samples of the Dow Jones and
Nasdaq (raw and corrected) returns. To this aim, we have randomly generated one thousand sub-samples,
each sub-sample being constituted of ten thousand data points in the positive or negative parts of the samples
respectively (without replacement). Obviously, among the one thousand sub-samples, many of them are
interdependent as they contain parts of the same observed values. With this database, we have estimated the
mean value and standard deviations of Pickands’ estimator for the GPD derived from the upper quantiles of
these distributions, and of ML-estimators for the distribution of maximum and for the GPD. The results are
given in tables 4 and 5.
These results confirm the confusion about the tail behavior of the returns distributions and it seems impossible
to exclude a rapidly varying behavior of their tails. Indeed, even the estimations performed by Maximum
Likelihood with the GPD tail index, which have appeared as the less unreliable estimator in our previous
tests, does not allow us to reject the hypothesis that the tails of the empirical distributions of returns are
rapidly varying.
For the Nasdaq dataset, accounting for the lunch effect does not yield a significant change in the estimations,
except for a very strong increase of the standard variation of the GPD Maximum Likelihood estimator. This
results from the fact that extremes are no more dominated by the few largest realizations of the returns at
the begining or the end of trading days. Indeed, panel (b) of table 4 shows that the sample variance of the
GPD maximum likelihood estimate vanishes for quantile 99% and 99.5%. It is due to the important overlap
of the sub-samples together with the impact of the extreme realizations of the returns at the open or close
trading days. In panel (c), which corresponds to the Nasdaq data corrected for the lunch effect, the sample
variance vanishes only in one case (instead of four), clearly showing that extremes are less dominated by
the large returns of the beginning and at the end of each day.
As a last non-parametric attempt to distinguish between a regularly varying tail and a rapidly varying tail
of the exponential or Stretched-Exponential families, we study the Mean Excess Function which is one of
the known methods that often can help in deciding what parametric family is appropriate for approximation
(see for details Embrechts et al. (1997)). The Mean Excess Function MEF(u) of a random value X (also
called “shortfall” when applied to negative returns in the context of financial risk management) is defined as
MEF(u) = E(X − u|X > u) . (19)
The Mean Excess Function MEF(u) is obviously related to the GPD for sufficiently large threshold u and
its behavior can be derived in this limit for the three maximum domains of attraction. In addition, more
precise results can be given for particular radom variables, even in a non-asymptotic regime. Indeed, for an
exponential random variable X, the MEF(u) is just a constant. For a Pareto random variable, the MEF(u)
is a straight increasing line, whereas for the Streched-Exponential and the Gauss distributions the MEF(u)
is a decreasing function. We evaluated the sample analogues of the MEF(u) (Embrechts et al. 1997, p.296)
which are shown in figure 4. All attempts to find a constant or a linearly increasing behavior of the MEF(u)
on the main central part of the range of returns were ineffective. In the central part of the range of negative
returns (|X| > 0.002; q ∼ = 98% for ND data, and |X| > 0.025 ; q ∼ = 96% for DJ data), the MEF(u) behaves
like a convex function which exclude both exponential and power (Pareto) distributions. Thus, the MEF(u)
tool does not support using any of these two distributions.
In view of the stalemate reached with the above non-parametric approaches and in particular with the stan-
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