fExtremes (version 201.10060)

MdaPlots: Hill and Shape Parameter Plots

Description

A collection and description of functions for the estimation of the tail index of extreme data using the maximum domain of attraction, MDA, method. Included are the Pickands, Einmal-Decker-deHaan, and Hill estimators together with several plot variants. The functions are: ll{ hillPlot shape parameter and Hill estimate of the tail index, shaparmPlot variation of shape parameter with tail depth. }

Usage

hillPlot(x, option = c("alpha", "xi", "quantile"), start = 15,
    end = NA, reverse = FALSE, p = NA, ci = 0.95, autoscale = TRUE, 
    labels = TRUE, ...)
    
shaparmPlot(x, revert = FALSE, standardize = FALSE, tails = 0.01*(1:10), 
    doplot = c(FALSE, FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE), 
    which = c(TRUE, TRUE, TRUE), doprint = TRUE, both.tails = TRUE, 
    xi.range = c(-0.5, 1.5), alpha.range = c(0, 10))

Arguments

x

[hillPlot][shaparmPlot] - the data from which to calculate the shape parameter, a numeric vector.

autoscale

[hillPlot] - whether or not plot should be automatically scaled; if not, xlim and ylim graphical parameters may be entered.

ci

[hillPlot] - probability for asymptotic confidence band; for no confidence band set ci to zero.

labels

[hillPlot] - whether or not axes should be labelled.

option

[hillPlot] - whether alpha, xi (1/alpha) or quantile (a quantile estimate) should be plotted.

p

[hillPlot] - probability required when option quantile is chosen.

reverse

[hillPlot] - whether plot is to be by increasing threshold, TRUE, or increasing number of order statistics FALSE.

start, end

[hillPlot] - lowest and highest number of order statistics at which to plot a point.

...

[hillPlot] - other graphics parameters.

alpha.range, xi.range

[saparmPlot] - plotting ranges.

both.tails

[shaparmPlot] - a logical, decides whether or not both tails should be investigated. By default TRUE. If FALSE only the lower tail will be investigated.

doplot

[shaparmPlot] - a vector of logicals of the same lengths as tails defining for wich tail depths plots should be created, by default plots will be generated for a tail depth of 5 percent. By default c(FALSE, FALSE, FALSE,

doprint

[shaparmPlot] - a logical, decides whether or not for all tail depths the result for the shape parameter 1/alpha should be printed.

revert

[shaparmPlot] - a logical value, by default FALSE, if set to TRUE the sign of the vector will be reverted: x = -x.

standardize

[shaparmPlot] - a logical value, by default FALSE, if set to TRUE the vector x will be standardized: x = (x-mean(x))/sqrt(var(x)).

tails

[shaparmPlot] - a numeric vector of tail depths to be considered; by default ten values ranging from 0.1 to 1.0 in steps of 0.1 corresponding to values ranging from 1 to 10 percent.

which

[shaparmPlot] - a vector of 3 logicals indicating which plots from the three methods will be created. The first entry decides for the Pickands estimator, the second for the Hill estimator, and the last for the Deckers-Einmahl

Value

• 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.

Details

Hill Plot: The function hillPlot investigates the shape parameter and plots the Hill estimate of the tail index of heavy-tailed data, or of an associated quantile estimate. This plot is usually calculated from the alpha perspective. For a generalized Pareto analysis of heavy-tailed 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 Decker-Einmal-deHaan estimator.

References

Coles S. (2001); Introduction to Statistical Modelling of Extreme Values, Springer. Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer.

See Also

gpdFit.

Examples

## hillPlot -
   xmpExtremes("Start: Hill Estimator >")
   # Hill plot of heavy-tailed Danish fire insurance data 
   # and BMW stock data for estimated 0.999 quantile
   par(mfrow = c(2, 2))
   data(bmw)
   hillPlot(bmw)
   hillPlot(bmw, option = "quantile", end = 500, p = 0.999)
   data(danish)
   hillPlot(danish)
   hillPlot(danish, option = "quantile", end = 500, p = 0.999)
   
## shaparmPlot -
   xmpExtremes("Next: Shape Parameter Plots >")
   par(mfcol = c(3, 2), cex = 0.6)
   data(bmw)
   shaparmPlot(bmw)

## shaparmPlot -
   xmpExtremes("Next: Simulated Frechet Data >")
   par(mfcol = c(3, 2), cex = 0.6)
   set.seed(4711)
   x = rgev(10000, xi = 1/4)
   shaparmPlot(x, revert = TRUE, both.tails = FALSE)
   lines(c(0.01, 0.1), c(4, 4), col = "steelblue3") # True Value