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,






Thus, a natural estimator for Pickands function is

where

> 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 April 19, 2025 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?