GPD Distributions for Extreme Value Theory

Share:

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 "observed" or "expected" information. This only applies to the maximum likelihood method; for the probability-weighted moments method "expected" information is used if possible.

model

[gpdSim] -
a list with components shape, location and scale giving the parameters of the GPD distribution. By default the shape parameter has the value 0.25, the location is zero and the scale is one.

n

[rgpd][gpdSim\ -
the number of observations to be generated.

object

[summary] -
a fitted object of class "gpdFit".

seed

[gpdSim] -
an integer value to set the seed for the random number generator.

title

a character string which allows for a project title.

type

a character string selecting the desired estimation mehtod, either "mle" for the maximum likelihood mehtod or "pwm" for the probability weighted moment method. By default, the first will be selected. Note, the function gpd uses "ml".

u

the threshold value.

which

if which is set to "ask" the function will interactively ask which plot should be displayed. By default this value is set to FALSE and then those plots will be displayed for which the elements in the logical vector which ar set to TRUE; by default all four elements are set to "all".

x

[dgpd] -
a numeric vector of quantiles.
[gpdFit] -
the data vector. Note, there are two different names for the first argument x and data depending which function name is used, either gpdFit or the EVIS synonyme gpd.
[print][plot] -
a fitted object of class "gpdFit".

xi, mu, beta

xi is the shape parameter, mu the location parameter, and beta is the scale parameter.

...

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 control argument of optim.

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 R-port.

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
## gpdSim  -
   x = gpdSim(model = list(xi = 0.25, mu = 0, beta = 1), n = 1000)
## gpdFit - 
   par(mfrow = c(2, 2), cex = 0.7)  
   fit = gpdFit(x, u = min(x), type = "pwm") 
   print(fit)
   summary(fit)