# RTDEzvalue: The Z-value random variable In RTDE: Robust Tail Dependence Estimation

## Description

Compute the Z-value variable from a bivariate dataset.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 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_{ω,i} variable

\min ( f(R_i^X/(n+1)), ω/(1-ω) 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

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

where m equals nbpoint, j=1,…, m, and \widetilde Z_{ω,1,n},..., \widetilde Z_{ω,n,n} are the order statistics of \widetilde Z_{ω,1},...,\widetilde Z_{ω,n}. The margin transformation is

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").

## 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 fitRTDE for the fitting process and dataRTDE for the data-handling process.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ##### # (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")