GPD Distributions for Extreme Value Theory
Description
A collection and description to functions to fit and to simulate
processes that are generated from the generalized Pareto distribution.
Two approaches for parameter estimation are provided: Maximum
likelihood estimation and the probability weighted moment method.
The GPD modelling functions are:
gpdSim  generates data from the GPD, 
gpdFit  fits empirical or simulated data to the distribution, 
print  print method for a fitted GPD object of class ..., 
plot  plot method for a fitted GPD object, 
summary  summary method for a fitted GPD object. 
Usage
1 2 3 4 5 6 7 8 9 10 11  gpdSim(model = list(xi = 0.25, mu = 0, beta = 1), n = 1000,
seed = NULL)
gpdFit(x, u = quantile(x, 0.95), type = c("mle", "pwm"), information =
c("observed", "expected"), title = NULL, description = NULL, ...)
## S4 method for signature 'fGPDFIT'
show(object)
## S3 method for class 'fGPDFIT'
plot(x, which = "ask", ...)
## S3 method for class 'fGPDFIT'
summary(object, doplot = TRUE, which = "all", ...)

Arguments
description 
a character string which allows for a brief description. 
doplot 
a logical. Should the results be plotted? 
information 
whether standard errors should be calculated with

model 
[gpdSim]  
n 
[rgpd][gpdSim\  
object 
[summary]  
seed 
[gpdSim]  
title 
a character string which allows for a project title. 
type 
a character string selecting the desired estimation mehtod, either

u 
the threshold value. 
which 
if 
x 
[dgpd]  
xi, mu, beta 

... 
control parameters and plot parameters optionally passed to the
optimization and/or plot function. Parameters for the optimization
function are passed to components of the 
Details
Generalized Pareto Distribution:
Compute density, distribution function, quantile function and
generates random variates for the Generalized Pareto Distribution.
Simulation:
gpdSim
simulates data from a Generalized Pareto
distribution.
Parameter Estimation:
gpdFit
fits the model parameters either by the probability
weighted moment method or the maxim log likelihood method.
The function returns an object of class "gpd"
representing the fit of a generalized Pareto model to excesses over
a high threshold. The fitting functions use the probability weighted
moment method, if method method="pwm"
was selected, and the
the general purpose optimization function optim
when the
maximum likelihood estimation, method="mle"
or method="ml"
is chosen.
Methods:
print.gpd
, plot.gpd
and summary.gpd
are print,
plot, and summary methods for a fitted object of class gpdFit
.
The plot method provides four different plots for assessing fitted
GPD model.
gpd* Functions:
gpdqPlot
calculates quantile estimates and confidence intervals
for high quantiles above the threshold in a GPD analysis, and adds a
graphical representation to an existing plot. The GPD approximation in
the tail is used to estimate quantile. The "wald"
method uses
the observed Fisher information matrix to calculate confidence interval.
The "likelihood"
method reparametrizes the likelihood in terms
of the unknown quantile and uses profile likelihood arguments to
construct a confidence interval.
gpdquantPlot
creates a plot showing how the estimate of a
high quantile in the tail of a dataset based on the GPD approximation
varies with threshold or number of extremes. For every model
gpdFit
is called. Evaluation may be slow. Confidence intervals
by the Wald method may be fastest.
gpdriskmeasures
makes a rapid calculation of point estimates
of prescribed quantiles and expected shortfalls using the output of the
function gpdFit
. This function simply calculates point estimates
and (at present) makes no attempt to calculate confidence intervals for
the risk measures. If confidence levels are required use gpdqPlot
and gpdsfallPlot
which interact with graphs of the tail of a loss
distribution and are much slower.
gpdsfallPlot
calculates expected shortfall estimates, in other
words tail conditional expectation and confidence intervals for high
quantiles above the threshold in a GPD analysis. A graphical
representation to an existing plot is added. Expected shortfall is
the expected size of the loss, given that a particular quantile of the
loss distribution is exceeded. The GPD approximation in the tail is used
to estimate expected shortfall. The likelihood is reparametrised in
terms of the unknown expected shortfall and profile likelihood arguments
are used to construct a confidence interval.
gpdshapePlot
creates a plot showing how the estimate of shape
varies with threshold or number of extremes. For every model
gpdFit
is called. Evaluation may be slow.
gpdtailPlot
produces a plot of the tail of the underlying
distribution of the data.
Value
gpdSim
returns a vector of datapoints from the simulated
series.
gpdFit
returns an object of class "gpd"
describing the
fit including parameter estimates and standard errors.
gpdQuantPlot
returns invisible a table of results.
gpdShapePlot
returns invisible a table of results.
gpdTailPlot
returns invisible a list object containing
details of the plot is returned invisibly. This object should be
used as the first argument of gpdqPlot
or gpdsfallPlot
to add quantile estimates or expected shortfall estimates to the
plot.
Author(s)
Alec Stephenson for the functions from R's evd
package,
Alec Stephenson for the functions from R's evir
package,
Alexander McNeil for the EVIS functions underlying the evir
package,
Diethelm Wuertz for this Rport.
References
Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer.
Hosking J.R.M., Wallis J.R., (1987); Parameter and quantile estimation for the generalized Pareto distribution, Technometrics 29, 339–349.
Examples
1 2 3 4 5 6 7 