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72 3. Distributions exponentielles étirées contre distributions régulièrement variables
The form parameter ξ is of paramount importance for the form of the limiting distribution. Its sign determines
three possible limiting forms of the distribution of maximua: If ξ > 0, then the limit distribution is
the Fréchet power-like distribution; If ξ = 0, then the limit distribution is the Gumbel (double-exponential)
distribution; If ξ < 0, then the limit distribution has a support bounded from above. All these three distributions
are united in eq.(2) by this parameterization. The determination of the parameter ξ is the central
problem of extreme value analysis. Indeed, it allows one to determine the maximum domain of attraction
of the underling distribution. When ξ > 0, the underlying distribution belongs to the Fréchet maximum
domain of attraction and is regularly varying (power-like tail). When ξ = 0, it belongs to the Gumbel MDA
and is rapidly varying (exponential tail), while if ξ < 0 it belongs to the Weibull MDA and has a finite right
endpoint.
3.2 Examples of slow convergence to limit GEV and GPD distributions
There exist two ways of estimating ξ. First, if there is a sample of maxima (taken from sub-samples of
sufficiently large size), then one can fit to this sample the GEV distribution, thus estimating the parameters
by Maximum Likelihood method. Alternatively, one can prefer the distribution of exceedance over a large
threshold given by the GPD (7), whose tail index can be estimated with Pickands’ estimator or by Maximum
Likelihood, as previously. Hill’s estimator cannot be used since it assumes ξ > 0, while the essence
of extreme value analysis is, as we said, to test for the class of limit distributions without excluding any
possibility, and not only to determine the quantitative value of an exponent. Each of these methods has its
advantages and drawbacks, especially when one has to study dependent data, as we show below.
Given a sample of size N, one consider the q-maxima drawn from q sub-samples of size p (such that p · q =
N) to estimate the parameters (µ,ψ,ξ) in (4) by Maximum Likelihood. This procedure yields consistent and
asymptotically Gaussian estimators, provided that ξ > −1/2 (Smith 1985). The properties of the estimators
still hold approximately for dependent data, provided that the interdependence of data is weak. However, it
is difficult to choose an optimal value of q of the sub-samples. It depends both on the size N of the entire
sample and on the underlying distribution: the maxima drawn from an Exponential distribution are known
to converge very quickly to Gumbel’s distribution (Hall and Wellnel 1979), while for the Gaussian law,
convergence is particularly slow (Hall 1979).
The second possibility is to estimate the parameter ξ from the distribution of exceedances (the GPD) or
Pickand’s estimator. For this, one can use either the Maximum Likelihood estimator or Pickands’ estimator.
Maximum Likelihood estimators are well-known to be the most efficient ones (at least for ξ > −1/2 and for
independent data) but, in this particular case, Pickands’ estimator works reasonably well. Given an ordered
sample x1 ≤ x2 ≤ ···xN of size N, Pickands’ estimator is given by
ˆξk,N = 1
ln2 ln xk − x2k
. (9)
x2k − x4k
For independent and identically distributed data, this estimator is consistent provided that k is chosen so that
k → ∞ and k/N → 0 as N → ∞. Morover, ˆ ξk,n is asymptotically normal with variance
σ( ˆ ξk,N) 2 · k = ξ2 (22ξ+1 + 1)
(2(2ξ . (10)
− 1)ln2) 2
In the presence of dependence between data, one can expect an increase of the standard deviation, as reported
by Kearns and Pagan (1997). For time dependence of the GARCH class, Kearns and Pagan (1997) have
indeed demonstrated a significant increase of the standard deviation of the tail index estimator, such as
Hill’s estimator, by a factor more than seven with respect to their asymptotic properties for iid samples. This
leads to very inaccurate index estimates for time series with this kind of temporal dependence.
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