GPD Distributions for Extreme Value Theory
Description
A collection and description to functions to compute
tail risk under the GPD approach.
The GPD modelling functions are:
gpdQPlot  estimation of high quantiles, 
gpdQuantPlot  variation of high quantiles with threshold, 
gpdRiskMeasures  prescribed quantiles and expected shortfalls, 
gpdSfallPlot  expected shortfall with confidence intervals, 
gpdShapePlot  variation of shape with threshold, 
gpdTailPlot  plot of the tail, 
tailPlot  , 
tailSlider  , 
tailRisk  . 
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  gpdQPlot(x, p = 0.99, ci = 0.95, type = c("likelihood", "wald"),
like.num = 50)
gpdQuantPlot(x, p = 0.99, ci = 0.95, models = 30, start = 15, end = 500,
doplot = TRUE, plottype = c("normal", "reverse"), labels = TRUE,
...)
gpdSfallPlot(x, p = 0.99, ci = 0.95, like.num = 50)
gpdShapePlot(x, ci = 0.95, models = 30, start = 15, end = 500,
doplot = TRUE, plottype = c("normal", "reverse"), labels = TRUE,
...)
gpdTailPlot(object, plottype = c("xy", "x", "y", ""), doplot = TRUE,
extend = 1.5, labels = TRUE, ...)
gpdRiskMeasures(object, prob = c(0.99, 0.995, 0.999, 0.9995, 0.9999))
tailPlot(object, p = 0.99, ci = 0.95, nLLH = 25, extend = 1.5, grid =
TRUE, labels = TRUE, ...)
tailSlider(x)
tailRisk(object, prob = c(0.99, 0.995, 0.999, 0.9995, 0.9999), ...)

Arguments
ci 
the probability for asymptotic confidence band; for no confidence band set to zero. 
doplot 
a logical. Should the results be plotted? 
extend 
optional argument for plots 1 and 2 expressing how far xaxis should extend as a multiple of the largest data value. This argument must take values greater than 1 and is useful for showing estimated quantiles beyond data. 
grid 
... 
labels 
optional argument for plots 1 and 2 specifying whether or not axes should be labelled. 
like.num 
the number of times to evaluate profile likelihood. 
models 
the number of consecutive gpd models to be fitted. 
nLLH 
... 
object 
[summary]  
p 
a vector of probability levels, the desired probability for the quantile estimate (e.g. 0.99 for the 99th percentile). 
reverse 
should plot be by increasing threshold ( 
prob 
a numeric value. 
plottype 
a character string. 
start, end 
the lowest and maximum number of exceedances to be considered. 
type 
a character string selecting the desired estimation mehtod, either

x 
[dgpd]  
... 
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 graphicalx
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 8 9 10 11 12 13  ## Load Data:
danish = as.timeSeries(data(danishClaims))
## Tail Plot:
x = as.timeSeries(data(danishClaims))
fit = gpdFit(x, u = 10)
tailPlot(fit)
## Try Tail Slider:
# tailSlider(x)
## Tail Risk:
tailRisk(fit)
