statistique, théorie et gestion de portefeuille - Docs at ISFA

Another problem lies in the determination of the optimal threshold u of the GPD, which is in fact related to
the optimal determination of the sub-samples size q in the case of the estimation of the parameters of the
distribution of maximum.
In sum, none of these methods seem really satisfying and each one presents severe drawbacks. The estimation
of the parameters of the GEV distribution and of the GPD may be less sensitive to the dependence of
the data, but this property is only asymptotic, thus it requires a bootstrap investigation to be able to compare
the real power of each estimation method.
As a first simple example illustrating the possibly very slow convergence to the limit distributions of extreme
value theory mentioned above, let us consider a simulated sample of iid Weibull random variables (we thus
fulfill the most basic assumption of extreme values theory, i.e, iid-ness). For such a distribution, one must
obtain in the limit N → ∞ the exponential distribution (ξ = 0). We took two values for the exponent of
the Weibull distribution: c = 0.7 and c = 0.3, with d = 1 (scale parameter). Such the estimation of ξ by
the distribution of the GPD of exceedance should give estimated values of ξ close to zero in the limit of
large N. In order to use the GPD, we have taken the conditional Weibull distribution under condition X >
Uk,k = 1...15, where the thresholds Uk are choosen as: U1 = 0.1; U2 = 0.3; U3 = 1; U4 = 3; U5 = 10; U6 =
30; U7 = 100; U8 = 300; U9 = 1000; U10 = 3000; U11 = 10 4 ; U12 = 3 · 10 4 ; U13 = 10 5 ; U14 = 3 · 10 5 and
U15 = 10 6 .
For each simulation, the size of the sample above the considered threshold Uk is choosen equal to 50,000
in order to get small standard deviations. The Maximum-Likelihood estimates of the GPD form parameter
ξ are shown in figure 3. For c = 0.7, the threshold U7 gives an estimate ξ = 0.0123 with standard deviation
equal to 0.0045, i.e., the estimate for ξ differs significantly from zero (which is the correct theoretical limit
value). This occurs notwithstanding the huge size of the implied data set; indeed, the probability PrX > U7
for c = 0.7 is about 10 −9 , so that in order to obtain a data set of conditional samples from an unconditional
data set of the size studied here (50,000 realizations above U7), the size of such unconditional sample should
be approximately 10 9 times larger than the number of “peaks over threshold”, i.e., it is practically impossible
to have such a sample. For c = 0.3 the convergence to the theoretical value zero is even slower. Indeed,
even the largest financial datasets for a single asset, drawn from high frequency data, are no larger than or
of the order of one million points 3 . The situation does not change even for data sets one or two orders of
magnitudes larger as considered in (Gopikrishnan et al. 1998), obtained by aggregating thousands of stocks
4 . Thus, although the GPD form parameter should be zero theoretically in the limit of large sample for the
Weibull distribution, this limit cannot be reached for any available sample sizes.
This is a clear illustration that a rapidly varying distribution, like the Weibull distribution with exponent
smaller than one, i.e., a Stretched-Exponential distribution, can be mistaken for a Pareto or any other regularly
varying distribution for any practical applications.
3.3 Generation of a long memory process with a well-defined stationary distribution
In order to study the performance of the various estimators of the tail index ξ and the influence of interdependence
of sample values, we have generated six samples with distinct properties. The first three samples
are made of iid realizations drawn respectively from a Pareto Distribution with tail index b = 3 and from
a Stretched-Exponential distribution with exponent c = 0.3 and c = 0.7. The three other samples contain
realizations exhibiting long-range memory with the same three distributions as for the first three samples:
a regularly varying distribution with tail index b = 3 and a Stretched-Exponential distribution with expo-
3 One year of data sampled at the 1 minute time scale gives approximately 1.2 · 10 5 data points
4 In this case, another issue arises concerning the fact that the aggregation of returns from differents assets may distord the
information and the very structure of the tails of the pdfs, if they exhibit some intrinsic variability (Matia et al. 2002).
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