Generalized Extreme Value Modelling
Description
A collection and description functions to estimate
the parameters of the GEV distribution. To model
the GEV three types of approaches for parameter
estimation are provided: Maximum likelihood
estimation, probability weighted moment method,
and estimation by the MDA approach. MDA includes
functions for the Pickands, EinmalDeckerdeHaan,
and Hill estimators together with several plot
variants.
The GEV modelling functions are:
gevSim  generates data from the GEV distribution, 
gumbelSim  generates data from the Gumbel distribution, 
gevFit  fits data to the GEV distribution, 
gumbelFit  fits data to the Gumbel distribution, 
print  print method for a fitted GEV object, 
plot  plot method for a fitted GEV object, 
summary  summary method for a fitted GEV object, 
gevrlevelPlot  kblock return level with confidence intervals. 
Usage
1 2 3 4 5 6 7 8 9 10 11 12  gevSim(model = list(xi = 0.25, mu = 0, beta = 1), n = 1000, seed = NULL)
gumbelSim(model = list(mu = 0, beta = 1), n = 1000, seed = NULL)
gevFit(x, block = 1, type = c("mle", "pwm"), title = NULL, description = NULL, ...)
gumbelFit(x, block = 1, type = c("mle", "pwm"), title = NULL, description = NULL, ...)
## S4 method for signature 'fGEVFIT'
show(object)
## S3 method for class 'fGEVFIT'
plot(x, which = "ask", ...)
## S3 method for class 'fGEVFIT'
summary(object, doplot = TRUE, which = "all", ...)

Arguments
block 
block size. 
description 
a character string which allows for a brief description. 
doplot 
a logical. Should the results be plotted?

model 
[gevSim][gumbelSim]  
n 
[gevSim][gumbelSim]  
object 
[summary][grlevelPlot]  
seed 
[gevSim]  
title 
[gevFit]  
type 
a character string denoting the type of parameter estimation,
either by maximum likelihood estimation 
which 
[plot][summary]  
x 
[dgev][devd]  
xi, mu, beta 
[*gev]  
... 
[gevFit]  
Details
Parameter Estimation:
gevFit
and gumbelFit
estimate the parameters either
by the probability weighted moment method, method="pwm"
or
by maximum log likelihood estimation method="mle"
. The
summary method produces diagnostic plots for fitted GEV or Gumbel
models.
Methods:
print.gev
, plot.gev
and summary.gev
are
print, plot, and summary methods for a fitted object of class
gev
. Concerning the summary method, the data are
converted to unit exponentially distributed residuals under null
hypothesis that GEV fits. Two diagnostics for iid exponential data
are offered. The plot method provides two different residual plots
for assessing the fitted GEV model. Two diagnostics for
iid exponential data are offered.
Return Level Plot:
gevrlevelPlot
calculates and plots the kblock return level
and 95% confidence interval based on a GEV model for block maxima,
where k
is specified by the user. The kblock return level
is that level exceeded once every k
blocks, on average. The
GEV likelihood is reparameterized in terms of the unknown return
level and profile likelihood arguments are used to construct a
confidence interval.
Hill Plot:
The function hillPlot
investigates the shape parameter and
plots the Hill estimate of the tail index of heavytailed data, or
of an associated quantile estimate. This plot is usually calculated
from the alpha perspective. For a generalized Pareto analysis of
heavytailed data using the gpdFit
function, it helps to
plot the Hill estimates for xi
.
Shape Parameter Plot:
The function shaparmPlot
investigates the shape parameter and
plots for the upper and lower tails the shape parameter as a function
of the taildepth. Three approaches are considered, the Pickands
estimator, the Hill estimator, and the
DeckerEinmaldeHaan estimator.
Value
gevSim
returns a vector of data points from the simulated series.
gevFit
returns an object of class gev
describing the fit.
print.summary
prints a report of the parameter fit.
summary
performs diagnostic analysis. The method provides two different
residual plots for assessing the fitted GEV model.
gevrlevelPlot
returns a vector containing the lower 95% bound of the confidence
interval, the estimated return level and the upper 95% bound.
hillPlot
displays a plot.
shaparmPlot
returns a list with one or two entries, depending on the
selection of the input variable both.tails
. The two
entries upper
and lower
determine the position of
the tail. Each of the two variables is again a list with entries
pickands
, hill
, and dehaan
. If one of the
three methods will be discarded the printout will display zeroes.
Note
GEV Parameter Estimation:
If method "mle"
is selected the parameter fitting in gevFit
is passed to the internal function gev.mle
or gumbel.mle
depending on the value of gumbel
, FALSE
or TRUE
.
On the other hand, if method "pwm"
is selected the parameter
fitting in gevFit
is passed to the internal function
gev.pwm
or gumbel.pwm
again depending on the value of
gumbel
, FALSE
or TRUE
.
Author(s)
Alec Stephenson for R's evd
and evir
package, and
Diethelm Wuertz for this Rport.
References
Coles S. (2001); Introduction to Statistical Modelling of Extreme Values, Springer.
Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  ## gevSim 
# Simulate GEV Data, use default length n=1000
x = gevSim(model = list(xi = 0.25, mu = 0 , beta = 1), n = 1000)
head(x)
## gumbelSim 
# Simulate GEV Data, use default length n=1000
x = gumbelSim(model = list(xi = 0.25, mu = 0 , beta = 1))
## gevFit 
# Fit GEV Data by Probability Weighted Moments:
fit = gevFit(x, type = "pwm")
print(fit)
## summary 
# Summarize Results:
par(mfcol = c(2, 2))
summary(fit)

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