Extremal Index Estimation

Description

A collection and description of functions to simulate time series with a known extremal index, and to estimate the extremal index by four different kind of methods, the blocks method, the reciprocal mean cluster size method, the runs method, and the method of Ferro and Segers.

The functiona are:

thetaSim Simulates a time Series with known theta,
blockTheta Computes theta from Block Method,
clusterTheta Computes theta from Reciprocal Cluster Method,
runTheta Computes theta from Run Method,
ferrosegersTheta Computes Theta according to Ferro and Seegers,
exindexPlot Calculate and Plot Theta(1,2,3),
exindexesPlot Calculate Theta(1,2) and Plot Theta(1).

Usage

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## S4 method for signature 'fTHETA'
show(object)

thetaSim(model = c("max", "pair"), n = 1000, theta = 0.5)

blockTheta(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
    title = NULL, description = NULL)
clusterTheta(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
    title = NULL, description = NULL)
runTheta(x, block = 22, quantiles = seq(0.950, 0.995, length = 10),
    title = NULL, description = NULL)
ferrosegersTheta(x, quantiles = seq(0.950, 0.995, length = 10),
    title = NULL, description = NULL)
    
exindexPlot(x, block = c("monthly", "quarterly"), start = 5, end = NA, 
    doplot = TRUE, plottype = c("thresh", "K"), labels = TRUE, ...)
    
exindexesPlot(x, block = 22, quantiles = seq(0.950, 0.995, length = 10), 
    doplot = TRUE, labels = TRUE, ...)

Arguments

block

[*Theta] -
an integer value, the block size. Currently only integer specified block sizes are supported.
[exindex*Plot] -
the block size. Either "monthly", "quarterly" or an integer value. An integer value is interpreted as the number of data values in each successive block. The default value is "monthly" which correpsond for daily data to an approximately 22-day periods.

description

a character string which allows for a brief description.

doplot

a logical, should the results be plotted?

labels

whether or not axes should be labelled. If set to FALSE then user specified lables can be passed through the "..." argument.

model

[thetaSim] -
a character string denoting the name of the model. Either "max" or "pair", the first representing the maximimum Frechet series, and the second the paired exponential series.

n

[thetaSim] -
an integer value, the length of the time series to be generated.

object

an object of class "fTHETA" as returned by the functions *Theta.

plottype

[exindexPlot] -
whether plot is to be by increasing threshold (thresh) or increasing K value (K).

quantiles

[exindexesPlot] -
a numeric vector of quantile values.

start, end

[exindexPlot] -
start is the lowest value of K at which to plot a point, and end the highest value; K is the number of blocks in which a specified threshold is exceeded.

theta

[thetaSim] -
a numeric value between 0 and 1 setting the value of the extremal index for the maximum Frechet time series. (Not used in the case of the paired exponential series.)

title

a character string which allows for a project title.

x

a 'timeSeries' object or any other object which can be transformed by the function as.vector into a numeric vector. "monthly" and "quarterly" blocks require x to be an object of class "timeSeries".

...

additional arguments passed to the plot function.

Value

exindexPlot
returns a data frame of results with the following columns: N, K, un, theta2, and theta. A plot with K on the lower x-axis and threshold Values on the upper x-axis versus the extremal index is displayed.

exindexesPlot
returns a data.frame with four columns: thresholds, theta1, theta2, and theta3. A plot with quantiles on the x-axis and versus the extremal indexes is displayed.

Author(s)

Alexander McNeil, for parts of the exindexPlot function, and
Diethelm Wuertz for the exindexesPlot function.

References

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer. Chapter 8, 413–429.

See Also

hillPlot, gevFit.

Examples

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## Extremal Index for the right and left tails 
## of the BMW log returns:
   data(bmwRet)
   par(mfrow = c(2, 2), cex = 0.7)
   exindexPlot( as.timeSeries(bmwRet), block = "quarterly")
   exindexPlot(-as.timeSeries(bmwRet), block = "quarterly")   
   
## Extremal Index for the right and left tails 
## of the BMW log returns:
   exindexesPlot( as.timeSeries(bmwRet), block = 65)
   exindexesPlot(-as.timeSeries(bmwRet), block = 65)