RTDE (version 0.2-0)

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, …)

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.

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".

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 $${\tilde{Z}}_{\omega ,i}=min(f\left(\frac{R_i^X}{n+1}\right),\frac{\omega }{1-\omega }f\left(\frac{R_i^Y}{n+1}\right))$$ where $f(x)$ is the margin transformation and $i=1,...,n$; (2) for output="relexcess", the $Z_j$ variable $$\frac{{\tilde{Z}}_{\omega ,n-m+j,n}}{{\tilde{Z}}_{\omega ,n-m,n}}$$ where $m$ equals nbpoint, $j=1,\dots ,m$, and ${\tilde{Z}}_{\omega ,1,n},...,{\tilde{Z}}_{\omega ,n,n}$ are the order statistics of ${\tilde{Z}}_{\omega ,1},...,{\tilde{Z}}_{\omega ,n}$. 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

# NOT RUN {
#####
# (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")

		
# }