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In order to obtain a process with Stretched-Exponential distribution with long range dependence, we apply
to {rt}t≥1 the following increasing mapping G : r → y
⎧
⎪⎨ (x0 + ln
G(r) =
⎪⎩
r
r0 )1/c r > r0
sgn(r) · |r| 1/c |r| ≤ r0
−(r0 + ln|r/r0|) 1/c (16)
r < −r0 .
This transformation gives a stretched exponential of index c for all values of the return larger than the scale
factor r0. This derives from the fact that the process {rt}t≥1 admits a regularly varying distribution function,
characterized by ¯Fr(r) = 1 − Fr(r) = L(r)|r| −b , for some slowly varying function L. As a consequence, the
stationary distribution of {Yt}t≥1 is given by
¯FY (y) = L r0e −x0 exp(y c ) e br0
which is a Stretched-Exponential distribution.
x b 0
75
· e −b|y|c
, ∀|y| > r0, (17)
= L ′ (y) · e −b|y|c
, L ′ is slowly varying at infinity, (18)
To summarize, starting with a long memory Gaussian process, we have defined a long memory process
characterized by a stationary distribution function of our choice, thanks to the invariance of the temporal
dependence structure (the copula) under strictly increasing change of variable. In particular, this approach
gives long memory processes with a regularly varying marginal distribution and with a stretched-exponential
distribution. Notwithstanding the difference in their marginals, these two processes possess by construction
exactly the same time dependence. This allows us to compare the impact of the same dependence on these
two classes of marginals.
3.4 Results of numerical simulations
We have generated 1000 samples of each kind (iid Stretched-Exponential, iid Pareto, long memory process
with a Pareto distribution and with a Stretched-Exponential distribution). Each sample contains 10,000
realizations, which is approximately the number of points in each tail of our real samples. In order to
generate the Gaussian process with correlation function (11), we have used the algorithm based on Fast
Fourier Transform described in Beran (1994). The parameter T has been set to 250 and α to 0.5 (it can be
checked that for α = 0.5 the lower bound for T is equal to 23).
Panel (a) of table 2 presents the mean values and standard deviations of the Maximum Likelihood estimates
of ξ, using the Generalized Extreme Value distribution and the Generalized Pareto Distribution for the three
samples of iid data. To estimate the parameters of the GEV distribution and study the influence of the
sub-sample size, we have grouped the data in clusters of size q = 10,50,100 and 200. For the analysis in
terms of the GPD, we have considered four different large thresholds u, corresponding to the quantiles 90%,
95%, 99% and 99.5%. The estimates obtained from the distribution of maxima are significantly different
from the theoretical ones: 0.2 in average over the four different size of sub-samples intead of 0.0 for the
Stretched-Exponential distribution with c = 0.7, 1.0 instead of 0.0 for c = 0.3 for the Stretched-Exponential
distribution and 0.40 instead of 0.33 for the Pareto Distribution. At the same time, the standard deviation of
these estimator remains very low. This significant bias of the estimator is a clear sign that the distribution of
the maximum has not yet converged to the asymptotic GEV distribution, even for subsamples of size 200.
The results are better with smaller biases for the Maximum Likelihood estimates obtained from the GPD.
However, the standard deviations are significantly larger than in the previous case, which testifies of the high
variability of this estimator. Thus, for such sample sizes, the GEV and GPD Maximum Likelihood estimates
seem not very reliable due to an important bias for the former and large statistical fluctuations for the later.
11