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76 3. Distributions exponentielles étirées contre distributions régulièrement variables
Nevertheless, an optimistic view point is that, discarding the largest quantile of 0.995, the GPD estimator is
compatible with a value ξ = 0.32 ± 0.1 for the iid data with Pareto distribution while it is compatible with
ξ = (−0.21) − (0.19) ± 0.6 − 0.8 for c = 0.3 and with ξ = (−0.13) − (0.11) ± 0.04 − 0.5 for c = 0.7. The
consistency of the GPD estimator in the case of the Pareto distribution in contrast with the wild variations for
the Stretched-Exponential distributions suggests that one could conclude that the first result qualifies (correctly)
a regularly varying function while the second one either (correctly) disqualifies a regularly varying
function or more conservatily is unable to conclude. In other words, when the sample data is truely Pareto,
the GPD estimator seems to be able to retrieve this information reliably, in constrast with the GEV estimator
which is quite unreliable in all cases.
Panel (b) of table 2 presents the same results for dependent data. The GEV estimates exhibit in each case a
significant bias, either positive or negative and a huge increase of the standard deviation in the case of the
Stretched-Exponential with exponent c = 0.7. Interestingly, the GEV estimator for the Pareto distribution is
utterly wrong. The situation is different for the GPD estimates which show a weak bias not really sensitive to
the quantile. In constrast, the standard deviations of the GPD estimators strongly increase with the quantile,
which is natural since the number of obervations decreases accordingly. The GPD behaves surprisingly well
and seems to be the only one able to perfom a reasonable estimation of the tail index.
To summarize, the Maximum Likelihood estimators derived form the GEV or GPD distributions are not very
efficient in the presence of dependence in the data and of non-asymptotic effects due to the slow convergence
toward the asymptotic GEV or GPD distributions. The only positive note is that the GPD estimator correctly
recovers the range of the index ξ with an uncertainty smaller than 20% for data with a pure Pareto distribution
while it is cannot reject the hypothesis that ξ = 0 when the data is generated with a Stretched-Exponential
distribution, albeit with a very large uncertainty, in other words with very little power.
Table 3 focuses on the results given by Pickands’ estimator for the tail index of the GPD. For each threshods
u, corresponding to the quantiles 90%, 95%, 99% and 99.5% respectively, the results of our simulations are
given for two particular values of k (defined in 9) corresponding to N/k = 4, which is the largest admissible
value, and N/k = 10 corresponding to be sufficiently far in the tail of the GPD. Table 3 provides the mean
value and the numerically estimated as well as the theoretical (given by (10)) standard deviation of ˆ ξk,N.
Panel (a) gives the result for iid data. The mean values do not exhibit a significant bias for the Pareto
distribution and the Stretched-Exponential with c = 0.7, but are utterly wrong in the case c = 0.3 since
the estimates are comparable with those given for the Pareto distribution. In each case, we note a very
good agreement between the empirical and theoretical standard deviations, even for the larger quantiles
(and thus the smaller samples). Panel (b) presents the results for dependent data. The estimated standard
deviations remains of the same order as the theoretical ones, contrarily to results reported by Kearns and
Pagan (1997) for GARCH processes. However, like these authors, we find that the bias, either positive or
negative, becomes very significant and leads on to misclassify a Stretched-Exponential distribution with
c = 0.3 for a Pareto distribution with b = 3. Thus, in presence of dependence, Pickands’ estimator is
unreliable.
To summarize, the impact of the dependence can add a severe scatter of estimators which increase their
standard deviation. The determination of the maximum domain of attraction with usual estimators does
not appear to be a very efficient way to study the extreme properties of dependent times series. Almost all
the previous studies which have investigated the tail behavior of asset returns distributions have focused on
these methods (see the influencial works of Longin (1996) for instance) and may thus have led to spurious
results on the determination of the tail behavior. In particular, our simulations show that rapidly varying
function may be mistaken for regularly varying functions. Thus, according to our simulations, this casts
doubts on the strength of the conclusion of previous works that the distributions of returns are regularly
varying as seems to have been the consensus until now and suggests to re-examine the possibility that the
distribution of returns may be rapidly varying as suggested by Gouriéroux and Jasiak (1998) or Laherrère
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