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In this section, we introduce a parametric model for those
data
which exceed a threshold t, where
are the original data.
Likewise, one may consider the k largest values
of the data xi.
Generalized Pareto (GP) distributions constitute adequate models for
such data (Reiss and Thomas; 1997). They consist of the following three submodels:
(i) | Exponential (GP0) | W0(x) = 1-e-x, | ![]() |
(ii) | Pareto (GP1) |
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(iii | Beta (GP2) |
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Again, one can unify these distributions by using the
parameterization with
.
We have
A mathematical justification of the modeling is obtained by a limit theorem.
Assume that X is a random variable with df
,
and
consider the conditional distribution
GP distributions are the only continuous dfs such that
The quantlets concerning densities, distribution and quantile functions of generalized Pareto distributions as well as the generation of pseudorandom variables are shown at the beginning of this section. Again, the routines belonging to the von Mises parameterization are merely displayed. One can address the three submodels by providing the names "gp0", "gp1" and "gp2" instead of "gp". Within the GP0 model, the shape parameter is not required.
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