The GP approach concerns a local, parametric modeling and estimation of the underlying distribution in the upper tail. By using the estimation procedures in Section 13.6, one may fit a GP distribution in two steps.
Firstly, one estimates a
GP distribution, say
, within
the GP submodel
,
based on the exceedances
over a selected threshold
. Notice that the location
parameter is equal to the truncation point
, which is also
the left endpoint of the estimated GP distribution.
The estimated GP df, density, quantile function
and mean excess function can be fitted to the empirical df,
density, quantile function and mean excess function
based on the exceedances
.
Secondly, a fit to the original data
is achieved
by selecting location and scale
parameters
and
such that
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Figure 13.1 exemplifies this procedure. The left-hand plot
shows the empirical df (solid line) based on
the exceedances above the threshold
and a fitted GP df (dotted).
The plot on the right-hand side shows the empirical df
of the original data set and the reparametrized GP df that
fits to the upper tail.
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Within the GP1-submodel of Pareto dfs
with location parameter
, we have
In our implementation, one has to select the number of exceedances.
Then, the threshold
is utilized.