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

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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 x-axis 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] -
a fitted object of class "gpdFit".

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 (TRUE) or number of extremes (FALSE).

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 "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".

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".

...

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 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 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

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## 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)