Copulas and tail dependence, part 3

[This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

We have seen extreme value copulas in the section where we did consider general families of copulas. In the bivariate case, an extreme value can be written

https://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG5.gif?w=578

where http://latex.codecogs.com/gif.latex?A(\cdot) is Pickands dependence function, which is a convex function satisfying
https://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG11.gif?w=578

Observe that in this case,
https://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG12.gif?w=578

where https://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG14.gif?w=578 is Kendall’tau, and can be written
https://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG13.gif?w=578

For instance, if
https://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG15.gif?w=578

then, we obtain Gumbel copula. This is what we’ve seen in the section where we introduced this family.
Now, let us talk about (nonparametric) inference, and more precisely the estimation of the dependence function. The starting point of the most standard estimator is to observe that if https://i1.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG6.gif?w=578 has copula http://latex.codecogs.com/gif.latex?C, then
https://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG3.gif?w=578

has distribution function
https://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG2.gif?w=578

And conversely, Pickands dependence function can be written
 
https://i1.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG7.gif?w=578

Thus, a natural estimator for Pickands function is
https://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG9.gif?w=578

where http://latex.codecogs.com/gif.latex?\widehat{H}_n is the empirical cumulative distribution function of
https://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/cfg1.gif?w=578

This is the estimator proposed in Capéràa, Fougères  & Genest (1997). Here, we can compute everything here using

> library(evd)
> X=lossalae
> U=cbind(rank(X[,1])/(nrow(X)+1),rank(X[,2])/(nrow(X)+1))
> Z=log(U[,1])/log(U[,1]*U[,2])
> h=function(t) mean(Z<=t)
> H=Vectorize(h)
> a=function(t){
+ f=function(t) (H(t)-t)/(t*(1-t))
+ return(exp(integrate(f,lower=0,upper=t,
+ subdivisions=10000)$value))
+ }
> A=Vectorize(a)
> u=seq(.01,.99,by=.01)
> plot(c(0,u,1),c(1,A(u),1),type="l",col="red",
+ ylim=c(.5,1))

Even integrate to get an estimator of Pickands’ dependence function. Note that an interesting point is that the upper tail dependence index can be visualized on the graph, above,


> A(.5)/2
[1] 0.4055346

To leave a comment for the author, please follow the link and comment on their blog: Freakonometrics - Tag - R-english.

R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job.
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)