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 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 t. Notice that the location
parameter is equal to the truncation point t, 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 yj.
Secondly, a fit to the original data
is achieved
by selecting location and scale
parameters
and
such that
Figure 1 exemplifies this procedure. The left-hand plot shows the empirical df (solid line) based on the exceedances above the threshold 1.13 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 k of exceedances. Then, the threshold t=xn-k+1:n is utilized.
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MD*TECH Method and Data Technologies |
http://www.mdtech.de mdtech@mdtech.de |