An alternative to describe tail dependence can be found in the Ledford & Tawn (1996) for instance. The intuition behind can be found in Fischer & Klein (2007)). Assume that and have the same distribution. Now, if we assume that those variables are (strictly) independent,
So assume that there is a such that
Then =2 can be interpreted as independence while =1 means strong (perfect) positive dependence. Thus, consider the following transformation to get a parameter in [0,1], with a strength of dependence increasing with the index, e.g.
In order to derive a tail dependence index, assume that there exists a limit to
which will be interpreted as a (weak) tail dependence index. Thus define concentration functions
for the lower tail (on the left) and
> library(evd); > data(lossalae) > X=lossalae > U=rank(X[,1])/(nrow(X)+1) > V=rank(X[,2])/(nrow(X)+1 > fL2emp=function(z) 2*log(mean(U<=z))/ + log(mean((U<=z)&(V<=z)))-1 > fR2emp=function(z) 2*log(mean(U>=1-z))/ + log(mean((U>=1-z)&(V>=1-z)))-1 > u=seq(.001,.5,by=.001) > L=Vectorize(fL2emp)(u) > R=Vectorize(fR2emp)(rev(u)) > plot(c(u,u+.5-u),c(L,R),type="l",ylim=0:1, + xlab="LOWER TAIL UPPER TAIL") > abline(v=.5,col="grey")
and again, it is possible to plot those empirical functions against some parametric ones, e.g. the one obtained from a Gaussian copula (with the same Kendall’s tau)
> tau=cor(lossalae,method="kendall")[1,2] > library(copula) > paramgauss=sin(tau*pi/2) > copgauss=normalCopula(paramgauss) > Lgaussian=function(z) 2*log(z)/log(pCopula(c(z,z), + copgauss))-1 > Rgaussian=function(z) 2*log(1-z)/log(1-2*z+ + pCopula(c(z,z),copgauss))-1 > u=seq(.001,.5,by=.001) > Lgs=Vectorize(Lgaussian)(u) > Rgs=Vectorize(Rgaussian)(1-rev(u)) > lines(c(u,u+.5-u),c(Lgs,Rgs),col="red")
or Gumbel copula,
> paramgumbel=1/(1-tau) > copgumbel=gumbelCopula(paramgumbel, dim = 2) > Lgumbel=function(z) 2*log(z)/log(pCopula(c(z,z), + copgumbel))-1 > Rgumbel=function(z) 2*log(1-z)/log(1-2*z+ + pCopula(c(z,z),copgumbel))-1 > Lgl=Vectorize(Lgumbel)(u) > Rgl=Vectorize(Rgumbel)(1-rev(u)) > lines(c(u,u+.5-u),c(Lgl,Rgl),col="blue")
Again, one should look more carefully at confidence bands, but is looks like Gumbel copula provides a good fit here.