13.4 Estimation in EV Models
Different parametric estimation procedures are provided for
estimating the shape, scale and location parameter of an extreme
value distribution.
13.4.1 Linear Combination of Ratios of Spacings (LRS)
- {gamma, mu, sigma} =
lrseev
(x)
- applies the LRS estimator to
the vector x and returns the estimated shape, location and
scale parameter of an EV distribution
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Let
where
with
and
The previous construction becomes plausible by noting that
is an estimator of the
-quantile
, and
therefore the relation
holds.
A natural estimator of
is given by the
linear combination
Estimates of the location and scale parameters are
derived from a
least square line
fitted to the
QQ-plot
One obtains
and
as estimates.
13.4.2 ML Estimator in the EV Model
- {gamma, mu, sigma} =
mleev
(x, k)
- applies the ML estimator in the EV model
to the vector x and returns the estimated shape, location and
scale parameter of an EV distribution
|
The maximum likelihood estimator in the EV model is numerically
evaluated by
using an iteration procedure. The LRS estimator described
in Subsection 13.4.1 serves as an initial value.
Note that the maximum likelihood estimator fails for
because in that case no global maximum of the likelihood function exists.
Yet it seems to be that a local maximum close to the initial
value is attained.
13.4.3 ML Estimator in the Gumbel Model
- {mu, sigma} =
mleev0
(x)
- applies the ML estimator in the Gumbel (EV0) model to
the vector x and returns the estimated location and
scale parameters
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The ML estimator in the Gumbel (EV0) model
must be evaluated numerically.
A certain moment estimator is utilized as the initial
value in the iteration procedure.