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
where
Observe that in this case,
where 
For instance, if
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 
has distribution function
And conversely, Pickands dependence function can be written
Thus, a natural estimator for Pickands function is
where
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
OpenEdition suggests that you cite this post as follows:
Arthur Charpentier (September 18, 2012). Copulas and tail dependence, part 3. Freakonometrics. Retrieved December 3, 2024 from https://doi.org/10.58079/oumk
this is a fantastic introduction to tail dependence and copulae – but it seams that most of the equations have been lost during blog migration – is there any possibility to fix that?