We describe the estimators for the GP models which are implemented in XploRe. The output always concerns the reparametrized GP or Pareto (GP1) distributions which were introduced in Section 5.
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The moment estimator (Dekkers, Einmal and de Haan; 1989) for the shape parameter
in the von Mises parameterization, based on the
k largest values of the sample, is given by
The scale parameter
is estimated by fitting a
least squares line to the GP QQ-plot under the estimated
shape parameter
,
i.e. to the points
x = 1 + 2 * gpdata (1, 500) momentgp (x, 500)Then, XploRe displays in its output window
Contents of _tmp.gamma [1,] 1.0326 Contents of _tmp.mu [1,] 1.0024 Contents of _tmp.sigma [1,] 1.9743
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The maximum likelihood estimator in the GP model is numerically evaluated by using an iteration procedure. The moment estimator described in Subsection 6.1 serves as an initial value.
The remarks about the ML estimator in the EV model (see Subsection 4.2) also apply to this estimator.
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The Pickands estimator (Pickands; 1975) of the shape parameter
is given by
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A refinement of the Pickands estimator was introduced
by Drees (1995). It uses a convex combination
of Pickands estimates
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The celebrated Hill estimator is a maximum likelihood estimator for the
GP1 submodel of Pareto dfs
with left endpoint
t. It is given by
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The maximum likelihood estimator for the exponential distributions, based on the k largest values, is given by
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A visual tool to facilitate the selection of a threshold t
(or, likewise, the number of upper extremes) is the
diagram of estimates
or
For small values of k one recognizes a high random fluctuation
of the estimator, while for large values of k, there is typically a bias
due to a model deviation. Within an intermediate range, the
estimates often stabilize around a value which gives a hint
for the selection of the number of extremes. Of course, one
should also apply QQ-plots and empirical mean excess functions
to justify the choice of the threshold. In the statistical
literature one can also find the advice to take the upper ten
percent of a sample. Hydrologists take the 3-4 highest flood
peaks in a water year. The automatic choice of a threshold
is presently a hot research topic.
A diagram option is provided for each
estimator of the shape parameter of a GP distribution.
The corresponding calls are listed above.
These quantlets return a vector with the estimates
for each value of k. Thus, to plot a diagram of
the Hill estimates based on a simulated Fréchet data set with
shape parameter ,
one can use the commands
x = ev1data (1, 500) line (2:500~hillgp1diag (x,2:250))The output of the above commands is shown in Figure 2. One can see that after a strong fluctuation for small values of k the estimates are close to the true parameter
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