Copulas and tail dependence, part 3

September 18, 2012
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(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)

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://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG5.gif?w=456

where http://latex.codecogs.com/gif.latex?A(\cdot) is Pickands dependence function, which is a convex function satisfying
http://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG11.gif?w=456

Observe that in this case,
http://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG12.gif?w=456

where http://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG14.gif?w=456 is Kendall'tau, and can be written
http://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG13.gif?w=456

For instance, if
http://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG15.gif?w=456

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://i1.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG6.gif?w=456 has copula http://latex.codecogs.com/gif.latex?C, then
http://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG3.gif?w=456

has distribution function
http://i2.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG2.gif?w=456

And conversely, Pickands dependence function can be written
 
http://i1.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG7.gif?w=456

Thus, a natural estimator for Pickands function is
http://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/CFG9.gif?w=456

where http://latex.codecogs.com/gif.latex?\widehat{H}_n is the empirical cumulative distribution function of
http://i0.wp.com/freakonometrics.blog.free.fr/public/perso6/cfg1.gif?w=456

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

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