RTDE (version 0.2-2)

zvalueRTDE: The Z-value random variable

Description

Compute the Z-value variable from a bivariate dataset.

Usage

zvalueRTDE(obs, omega, nbpoint, output=c("orig", "relexcess"), 
    marg=c("upareto", "ufrechet", "uunif"))
# S3 method for zvalueRTDE
print(x, ...)
# S3 method for zvalueRTDE
summary(object, ...)

relexcess(x, nbpoint, ...)
# S3 method for default
relexcess(x, nbpoint, ...)
# S3 method for zvalueRTDE
relexcess(x, nbpoint, ...)

Value

zvalueRTDE computes the Z-variable and returns an object of class "zvalueRTDE"

having the following components type (either

"orig" or "relexcess"), omega,

Ztilde or Z, n, possibly m.

relexcess computes the relative excesses from a Z-variable and returns an object of class "zvalueRTDE"

of type "relexcess".

Arguments

obs

bivariate numeric dataset.

omega

a numeric for omega, see Details.

nbpoint

a numeric for the number of largest points to be selected.

output

a character string for the output: either "orig" for original value or "relexcess" for relative excess.

marg

a character string for the empirical margin transformation: either "upareto" for unit Pareto, "ufrechet" for unit Frechet or "uunif" for unit uniform margin.

x, object

an R object inheriting from "zvalueRTDE".

...

arguments to be passed to subsequent methods.

Author

Christophe Dutang

Details

Given a bivariate dataset \((X_i, Y_i)_i\) of \(n\) points, two variables are defined: (1) for output="orig", the \(\tilde Z_{\omega,i}\) variable $$\tilde Z_{\omega,i} = \min \left( f\left(\frac{R_i^X}{n+1}\right), \frac{\omega}{1-\omega} f\left(\frac{R_i^Y}{n+1}\right) \right) $$ where \(f(x)\) is the margin transformation and \(i=1,...,n\); (2) for output="relexcess", the \(Z_{j}\) variable $$ \frac{\widetilde Z_{\omega,n-m+j,n}}{\widetilde Z_{\omega,n-m,n}} $$ where \(m\) equals nbpoint, \(j=1,\dots, m\), and \(\widetilde Z_{\omega,1,n},..., \widetilde Z_{\omega,n,n}\) are the order statistics of \(\widetilde Z_{\omega,1},...,\widetilde Z_{\omega,n}\). The margin transformation is $$ f(x) = \frac{1}{1-x}, f(x) = \frac{1}{-\log(x)}, f(x) = x, $$ respectively for unit Pareto (marg="upareto"), unit Frechet (marg="ufrechet") and unit uniform margin (marg="uunif").

References

C. Dutang, Y. Goegebeur, A. Guillou (2014), Robust and bias-corrected estimation of the coefficient of tail dependence, Volume 57, Insurance: Mathematics and Economics

This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416).

See Also

See fitRTDE for the fitting process and dataRTDE for the data-handling process.

Examples

#####
# (1) example

omega <- 1/2
m <- 10
n <- 100
obs <- cbind(rupareto(n), rupareto(n)) + rupareto(n)

#unit Pareto transform
zvalueRTDE(obs, omega, output="orig")

relexcess(zvalueRTDE(obs, omega, output="orig"), m)
zvalueRTDE(obs, omega, nbpoint=m, output="relexcess")