RTDEzvalue() R function from [RTDE]

The Z-value random variable

Compute the Z-value variable from a bivariate dataset.


zvalueRTDE(obs, omega, nbpoint, output=c("orig", "relexcess"), 
    marg=c("upareto", "ufrechet", "uunif"))

## S3 method for class 'zvalueRTDE'
print(x, ...)
## S3 method for class 'zvalueRTDE'
summary(object, ...)

relexcess(x, nbpoint, ...)
## Default S3 method:
relexcess(x, nbpoint, ...)
## S3 method for class 'zvalueRTDE'
relexcess(x, nbpoint, ...)

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 object inheriting from "zvalueRTDE".
...: arguments to be passed to subsequent methods.

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)\min (f(R_i^X/(n+1)),\omega/(1-\omega) f(R_i^Y/(n+1)) )

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}}\widetilde Z_{\omega,n-m+j,n}/\widetilde Z_{\omega,n-m,n}

where $m$ equals nbpoint, $j=1,\dots, m$ , and c(" $\\widetilde Z_{\\omega,1,n},...,\n$ ", " $\\widetilde Z_{\\omega,n,n}$ ") 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,f(x) = 1/(1-x), f(x) = -1/log(x), f(x) = x,

respectively for unit Pareto (marg="upareto"), unit Frechet (marg="ufrechet") and unit uniform margin (marg="uunif").

Returns

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".

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).

Author(s)

Christophe Dutang

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")

RTDE package Read PDF manual

Maintainer: Christophe Dutang
License: GPL (>= 2)
Last published: 2024-10-16

Useful links

RTDEzvalue function