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
http://freakonometrics.blog.free.fr/public/perso6/CFG5.gif

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

Observe that in this case,
http://freakonometrics.blog.free.fr/public/perso6/CFG12.gif

where http://freakonometrics.blog.free.fr/public/perso6/CFG14.gif is Kendall'tau, and can be written
http://freakonometrics.blog.free.fr/public/perso6/CFG13.gif

For instance, if
http://freakonometrics.blog.free.fr/public/perso6/CFG15.gif

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 http://freakonometrics.blog.free.fr/public/perso6/CFG6.gif has copula http://latex.codecogs.com/gif.latex?C, then
http://freakonometrics.blog.free.fr/public/perso6/CFG3.gif

has distribution function
http://freakonometrics.blog.free.fr/public/perso6/CFG2.gif

And conversely, Pickands dependence function can be written
 
http://freakonometrics.blog.free.fr/public/perso6/CFG7.gif

Thus, a natural estimator for Pickands function is
http://freakonometrics.blog.free.fr/public/perso6/CFG9.gif

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

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