As mentioned in the course last week Venter (2003) suggested nice functions to illustrate tail dependence (see also some slides used in Berlin a few years ago).

- Joe (1990)’s lambda

Joe (1990) suggested a (*strong*)* tail dependence index*. For lower tails, for instance, consider

i.e

**Upper and lower strong tail (empirical) dependence functions**

The idea is to plot the function above, in order to visualize limiting behavior. Define

for the lower tail, and

while

Now, one can easily derive empirical conterparts of those function, i.e.

and

Thus, for upper tail, on the right, we have the following graph

and for the lower tail, on the left, we have

For the code, consider some real data, like the loss-ALAE dataset.

> library(evd) > X=lossalae

The idea is to plot, on the left, the lower tail concentration function, and on the right, the upper tail function.

> U=rank(X[,1])/(nrow(X)+1) > V=rank(X[,2])/(nrow(X)+1) > Lemp=function(z) sum((U<=z)&(V<=z))/sum(U<=z) > Remp=function(z) sum((U>=1-z)&(V>=1-z))/sum(U>=1-z) > u=seq(.001,.5,by=.001) > L=Vectorize(Lemp)(u) > R=Vectorize(Remp)(rev(u)) > plot(c(u,u+.5-u[1]),c(L,R),type="l",ylim=0:1, + xlab="LOWER TAIL UPPER TAIL") > abline(v=.5,col="grey")

Now, we can compare this graph, with what should be obtained for some parametric copulas that have the same Kendall’s tau (e.g.). For instance, if we consider a Gaussian copula,

> tau=cor(lossalae,method="kendall")[1,2] > library(copula) > paramgauss=sin(tau*pi/2) > copgauss=normalCopula(paramgauss) > Lgaussian=function(z) pCopula(c(z,z),copgauss)/z > Rgaussian=function(z) (1-2*z+pCopula(c(z,z),copgauss))/(1-z) > u=seq(.001,.5,by=.001) > Lgs=Vectorize(Lgaussian)(u) > Rgs=Vectorize(Rgaussian)(1-rev(u)) > lines(c(u,u+.5-u[1]),c(Lgs,Rgs),col="red")

or Gumbel’s copula,

> paramgumbel=1/(1-tau) > copgumbel=gumbelCopula(paramgumbel, dim = 2) > Lgumbel=function(z) pCopula(c(z,z),copgumbel)/z > Rgumbel=function(z) (1-2*z+pCopula(c(z,z),copgumbel))/(1-z) > u=seq(.001,.5,by=.001) > Lgl=Vectorize(Lgumbel)(u) > Rgl=Vectorize(Rgumbel)(1-rev(u)) > lines(c(u,u+.5-u[1]),c(Lgl,Rgl),col="blue")

That’s nice (isn’t it?), but since we do not have any confidence interval, it is still hard to conclude (even if it looks like Gumbel copula has a much better fit than the Gaussian one). A strategy can be to generate samples from those copulas, and to visualize what we had. With a Gaussian copula, the graph looks like

> u=seq(.0025,.5,by=.0025); nu=length(u) > nsimul=500 > MGS=matrix(NA,nsimul,2*nu) > for(s in 1:nsimul){ + Xs=rCopula(nrow(X),copgauss) + Us=rank(Xs[,1])/(nrow(Xs)+1) + Vs=rank(Xs[,2])/(nrow(Xs)+1) + Lemp=function(z) sum((Us<=z)&(Vs<=z))/sum(Us<=z) + Remp=function(z) sum((Us>=1-z)&(Vs>=1-z))/sum(Us>=1-z) + MGS[s,1:nu]=Vectorize(Lemp)(u) + MGS[s,(nu+1):(2*nu)]=Vectorize(Remp)(rev(u)) + lines(c(u,u+.5-u[1]),MGS[s,],col="red") + }

(including – pointwise – 90% confidence bands)

> Q95=function(x) quantile(x,.95) > V95=apply(MGS,2,Q95) > lines(c(u,u+.5-u[1]),V95,col="red",lwd=2) > Q05=function(x) quantile(x,.05) > V05=apply(MGS,2,Q05) > lines(c(u,u+.5-u[1]),V05,col="red",lwd=2)

while it is

with Gumbel copula. Isn’t it a nice (graphical) tool ?

But as mentioned in the course, the statistical convergence can be slow. Extremely slow. So assessing if the underlying copula has tail dependence, or not, it now that simple. Especially if the copula exhibits tail independence. Like the Gaussian copula. Consider a sample of size 1,000. This is what we obtain if we generate random scenarios,

or we look at the left tail (with a log-scale)

Now, consider a 10,000 sample,

or with a log-scale

We can even consider a 100,000 sample,

or with a log-scale

On those graphs, it is rather difficult to conclude if the limit is 0, or some strictly positive value (again, it is a classical statistical problem when the value of interest is at the border of the support of the parameter). So, a natural idea is to consider a weaker tail dependence index. Unless you have something like 100,000 observations…

Hi Arthur,

If you could find time off your busy schedule, do please help to fix the links!

In any case, I would like you to know that your materials have been very helpful and would like to thank you for sharing them 🙂

Sincerely,

EH

Thank you for your post. Noticed that not everything is showing up in the above post. Instead it shows the weblink. Could you please fix.

yes, there were problems when I transfered the blog ! I will try to fix it asap.

Hi Arthur, I have been immensly helped by your blogs, thanks a lot. I was very interested in the 3 posts on Copula and tail dependency. Since you are busy to fix the missing links , is it possible to get them by email.(pradiptaparhi@rediffmail.com). Would highly appreciate your help.

Best.

Prad

Would it be possible to get the pictures? Thank you for any reply!

I am big trouble on my blog… I removed the previous one, but a lot of pictures were not transfered… I will try during the Xmas holiday to fix all that… sorry for the inconvenience !

Thanks a lot for very useful course. Would it be possible to get the pictures?