texmex (version 1.3)

chi: Measures of extremal dependence

Description

Compute measures of extremal dependence for 2 variables.

Usage

chi(data, nq = 100, qlim = NULL, alpha = 0.05, trunc = TRUE)
## S3 method for class 'chi':
summary(object, digits=3, ...)
## S3 method for class 'chi':
plot(x, show=c("Chi"=TRUE,"ChiBar"=TRUE), lty=1, cilty=2, col=1, spcases=TRUE,
    cicol=1, xlim=c(0, 1), ylimChi = c(-1, 1), ylimChiBar = c(-1, 1),
    mainChi = "Chi", mainChiBar = "Chi Bar", xlab = "Quantile", 
    ylabChi    = expression(chi(u)),#"Chi(u)"
    ylabChiBar = expression(bar(chi)(u)), #"Chi Bar(u)"
    ask, ...)

Arguments

data

A matrix containing 2 numeric columns.

nq

The number of quantiles at which to evaluate the dependence measures.

qlim

The minimum and maximum quantiles at which to do the evaluation.

alpha

The size of the confidence interval to be used. Defaults to alpha = 0.05.

trunc

Logical flag indicating whether the estimates should be truncated at their theoretical bounds. Defaults to trunc = TRUE.

x, object

An object of class chi.

digits

Number of digits for printing.

show

Logical, of length 2, names "Chi" and "ChiBar". Defaults to c("Chi" = TRUE, "ChiBar" = TRUE).

lty, cilty, col, cicol

Line types and colours for the the estimated quantities and their confidence intervals.

xlim, ylimChi, ylimChiBar

Limits for the axes.

mainChi, mainChiBar

Main titles for the plots.

xlab, ylabChi, ylabChiBar

Axis labels for the plots.

spcases

Whether or not to plot special cases of perfect (positive and negative) dependence and indpenendence. Defaults to FALSE.

ask

Whether or not to ask before reusing the graphics device.

...

Further arguments to be passed to methods.

Value

• An object of class chi containing the following.
• chi
  Values of chi and their esitmated upper and lower confidence limits.
• chibar
  Values of chibar and their esitmated upper and lower confidence limits.
• quantile
  The quantiles at which chi and chi-bar were evaluated.
• chiulb, chibarulb
  Upper and lower bounds for chi and chi-bar.

Details

Computes the functions chi and chi-bar described by Coles, Heffernan and Tawn. The limiting values of these functions as the Quantile approaches 1 give an empirical measure of the type and strength of tail dependendce exhibited by the data. A limiting value of ChiBar equal to 1 indicates Asymptotic Dependence, in which case the limiting value of Chi gives a measure of the strength of dependence in this class. A limiting value of ChiBar of less than 1 indicates Asymptotic Independence in which case the limiting value of ChiBar gives a measure of the strength of dependence. The plot method shows the ChiBar and Chi functions. In the case of the confidence interval for ChiBar excluding the value 1 for all of the largest values of Quantile, the plot of the Chi function is shown in grey.

References

S. Coles, J. E. Heffernan and J. A. Tawn, Dependence measures for extreme values analyses, Extremes, 2, 339 -- 365, 1999.

See Also

MCS, rank

Examples

D <- liver[liver$dose == "D",]
chiD <- chi(D[, 5:6])
par(mfrow=c(1,2))
plot(chiD)

A <- liver[liver$dose == "A",]
chiA <- chi(A[, 5:6])
# here the limiting value of chi bar(u) lies away from one so the chi plot is not relevant and is plotted in grey
plot(chiA)