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