13.6 Parametric Estimators for GP Models
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 13.5.
13.6.1 Moment Estimator
- {gamma, mu, sigma} =
momentgp
(x, k)
- applies the moment estimator to the k largest order statistics
of the vector x and returns the estimated shape, location and
scale parameter of the GP model
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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
largest values of the sample, is given by
where
and
with
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
where the
are the
largest values of the data set
The location parameter is equal to the threshold
,
and the transformation to the tail described in Section 13.5 is applied.
As an example,
we simulate a sample with 500 data points under
and
apply the moment estimator, based on all values of the sample:
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
13.6.2 ML Estimator in the GP Model
- {gamma, mu, sigma} =
mlegp
(x, k)
- applies the ML estimator in the GP model
to the k largest order statistics
of the vector x and returns the estimated shape, location and
scale parameter of the GP model
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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 13.6.1 serves as an initial value.
The remarks about the ML estimator in the EV model
(see Subsection 13.4.2) also apply to
this estimator.
13.6.3 Pickands Estimator
- {gamma, mu, sigma} =
pickandsgp
(x, k)
- applies the Pickands estimator to the k largest order statistics
of the vector x and returns the estimated shape, location and
scale parameter of the GP model
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The Pickands estimator (Pickands; 1975) of the shape parameter
is given by
where
This construction is similar to the one for the LRS estimator for the EV model
described in Subsection 13.4.1.
Scale and location parameter are estimated as described in
Subsection 13.6.1.
13.6.4 Drees-Pickands Estimator
- {gamma, mu, sigma} =
dpgp
(x, k)
- applies the Drees-Pickands estimator to the k largest order statistics
of the vector x and returns the estimated shape, location and
scale parameter of the GP model
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A refinement of the Pickands estimator was introduced
by Drees (1995). It uses a convex combination
of Pickands estimates
with
and
where
if
and
if
13.6.5 Hill Estimator
- {alpha, sigma} =
hillgp1
(x, k)
- applies the Hill estimator to the k largest order statistics
of the vector x and returns the estimated shape and scale
parameter of the GP1 model
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The celebrated Hill estimator is a maximum likelihood estimator for the
GP1 submodel of Pareto dfs
with left endpoint
. It is given by
Recall that
is used as the threshold.
Notice that
are further Pareto dfs
with left endpoint equal to
. When
, then we are
in the above-mentioned submodel. When
, then the
Hill estimator can be inaccurate. Therefore, one should be
cautious when the Hill estimator is applied to real data.
13.6.6 ML Estimator for Exponential Distributions
- {mu, sigma} =
mlegp0
(x, k)
- applies the ML estimator for the exponential distributions (GP0)
to the k largest order statistics
of the vector x and returns the estimated location and scale
parameter of the GP0 model
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The maximum likelihood estimator for the exponential distributions,
based on the
largest values, is given by
13.6.7 Selecting a Threshold by Means of a Diagram
- r =
momentgpdiag
(x, k)
- evaluates the moment estimator for all number of extremes given
in the vector k
- r =
mlegpdiag
(x, k)
- evaluates the MLE of the GP model for all number of extremes
given in the vector k
- r =
pickandsgpdiag
(x, k)
- evaluates the Pickands estimator for all number of extremes
given in the vector k
- r =
dpgpdiag
(x, k)
- evaluates the Drees-Pickands estimator for all number of
extremes given in the vector k
- r =
hillgp1diag
(x, k)
- evaluates the Hill estimator for all number of extremes
given in the vector k
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A visual tool to facilitate the selection of a threshold
(or, likewise, the number of upper extremes) is the
diagram of estimates
or
For small values of
one recognizes a high random fluctuation
of the estimator, while for large values of
, 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
. 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 13.2.
One can see that after a strong fluctuation for small values
of
the estimates are close to the true parameter
.
Moreover, a bias for large values of
is visible.
Figure 13.2:
Diagram of Hill estimates applied to
Fréchet data
.
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