|
In this section, we introduce a parametric model for maxima
,
where the
may not be observable. Such data occur,
e.g. when annual flood maxima or monthly maxima of temperatures
are recorded. In the first case, we have
while
is the number of observed years. We mention
a limit theorem which suggests a parametric modeling
of such maxima using extreme value (EV) distributions.
Assume that
are independent random
variables with common distribution function (df) F,
and let
Then,
is the df of
If, for an appropriate choice of constants
and
, the dfs of the
maxima converge to a continuous limiting distribution
function, i.e.
(i) | Gumbel (EV0) |
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(ii) | Fréchet (EV1) |
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(iii) | Weibull (EV2) |
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By employing the reparametrization
and
appropriate scale and
location parameters, one can unify these models in the
von Mises parameterization
by
Notice that
for
Moreover, the relation
holds
with
,
for
, and
for
.
The quantlets concerning densities, distribution and quantile functions (qfs) of extreme value distributions as well as the generation of pseudorandom variables are listed at the beginning of this section. The routines belonging to the von Mises parameterization are merely displayed. One can address the three submodels by providing the names "ev0", "ev1" and "ev2" in place of "ev". Notice the shape parameter is not required within the EV0 model. For example, use
pdfx("ev2", x, alpha)to calculate the Weibull density with shape parameter
r = mu + sigma * randx("ev", n, gamma)to generate a data set with