zvalueRTDE {RTDE}

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 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 R 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)

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").

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".

Author(s)

Christophe Dutang

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")