zvalueRTDE {RTDE} | R Documentation |
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 |
marg |
a character string for the empirical margin transformation:
either |
x , object |
an R object inheriting from |
... |
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")