statistique, théorie et gestion de portefeuille - Docs at ISFA

84 3. Distributions exponentielles étirées contre distributions régulièrement variables
automatically imply that the SE model is not the correct model for the data even for these highest quantiles.
Indeed, numerical simulations show that, even for synthetic samples drawn from genuine Stretched-
Exponential distributions with exponent c smaller than 0.5 and whose size is comparable with that of our
data, in about one case out of three (depending on the exact value of c) the estimated value of c is zero. This
a priori surprising result comes from condition (51) in appendix A which is not fulfilled with certainty even
for samples drawn for SE distributions.
Notwithstanding this cautionary remark, note that the c-estimate of the positive tail of the Nasdaq data equal
zero for all quantiles higher than q14 = 0.97%. In fact, in every cases, the estimated c is not significantly
different from zero - at the 95% significance level - for quantiles higher than q12-q14. In addition, table 10
gives the values of the estimated scale paremeter d, which are found very small - particularly for the Nasdaq
- beyond q12 = 95%. In constrast, the Dow Jones keeps significant scale factors until q16 − q17.
These evidences taken all together provide a clear indication on the existence of a change of behavior of
the true pdf of these four distributions: while the bulks of the distributions seem rather well approximated
by a SE model, a fatter tailed distribution than that of the (SE) model is required for the highest quantiles.
Actually, the fact that both c and d are extremely small may be interpreted according to the asymptotic
correspondence given by (26) and (27) as the existence of a possible power law tail.
4.3.3 Exponential and incomplet Gamma distribution
Let us now fit our data with the exponential distribution (24). The average ADS for this case are shown in
table 7. The maximum likelihood- and Anderson-Darling estimates of the scale parameter d are given in
table 11. Note that they always decrease as the threshold uq increases. Comparing the mean ADS-values
of table 7 with the standard AD quantiles, we can conclude that, on the whole, the exponential distribution
(even with moving scale parameter d) does not fit our data: this model is systematically rejected at the 95%
confidence level for the lowest and highest quantiles - excepted for the negative tail of the Nasdaq.
Finally, we fit our data by the IG-distribution (25). The mean ADS for this class of functions are shown in
table 7. The Maximum likelihood and Anderson Darling estimates of the power index b are represented in
table 12. Comparing the mean ADS-values of table 7 with the standard AD quantiles, we can again conclude
that, on the whole, the IG-distribution does not fit our data. The model is rejected at the 95% confidence
level excepted for the negative tail of the Nasdaq for which it is not rejected marginally (significance level:
94.13%). However, for the largest quantiles, this model becomes again revelant since it cannot be rejected
at the 95% level.
4.4 Summary
At this stage, two conclusions can be drawn. First, it appears that none of the considered distributions fit the
data over the entire range, which is not a surprise. Second, for the highest quantiles, three models seem to
be able to represent to data, the Gamma model, the Pareto model and the Stretched-Exponential model. This
last one has the lowest Anderson-Darling statistic and thus seems to be the most reasonable model among
the three models compatible with the data.
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