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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 writtenwhere  is Pickands dependence function, which is a convex function satisfyingObserve that in this case,where  is Kendall'tau, and can be writtenFor instance, ifthen, 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 copula , thenhas distribution functionAnd conversely, Pickands dependence function can be writtenThus, a natural estimator for Pickands function iswhere  is the empirical cumulative distribution function ofThis is the estimator proposed in Capéràa, Fougères  &#38; Genest (1997). Here, we can compute everything here using&#62; library(evd)&#62; X=lossalae&#62; U=cbind(rank(X[,1])/(nrow(X)+1),rank(X[,2])/+ (nrow(X)+1))&#62; Z=log(U[,1])/log(U[,1]*U[,2])&#62; h=function(t) mean(Z&#60;=t)&#62; H=Vectorize(h)&#62; a=function(t){+ f=function(t) (H(t)-t)/(t*(1-t))+ return(exp(integrate(f,lower=0,upper=t,+ subdivisions=10000)$value))+ }&#62; A=Vectorize(a)&#62; u=seq(.01,.99,by=.01)&#62; 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,&#62; A(.5)/2[1] 0.4055346