Copulas and tail dependence, part 3

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\cdot) is Pickands dependence function, which is a convex function satisfying
Observe that in this case,\tau is Kendall’tau, and can be written instance, if, 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,V) has copula, then distribution function conversely, Pickands dependence function can be written
Thus, a natural estimator for Pickands function is
where\widehat{H}_n is the empirical cumulative distribution function of 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

One thought on “Copulas and tail dependence, part 3”

  1. 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?

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