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
http://freakonometrics.hypotheses.org/files/2016/05/CFG5.gif
where https://latex.codecogs.com/gif.latex?A(\cdot) is Pickands dependence function, which is a convex function satisfying
http://freakonometrics.hypotheses.org/files/2016/05/CFG11.gif
Observe that in this case,
http://freakonometrics.hypotheses.org/files/2016/05/CFG12.gifwhere https://latex.codecogs.com/gif.latex?\tau is Kendall’tau, and can be written
http://freakonometrics.hypotheses.org/files/2016/05/CFG13.gifFor instance, if
http://freakonometrics.hypotheses.org/files/2016/05/CFG15.gifthen, 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 https://latex.codecogs.com/gif.latex?(U,V) has copula https://latex.codecogs.com/gif.latex?C, then
http://freakonometrics.hypotheses.org/files/2016/05/CFG3.gifhas distribution function
http://freakonometrics.hypotheses.org/files/2016/05/CFG2.gifAnd conversely, Pickands dependence function can be written
http://freakonometrics.hypotheses.org/files/2016/05/CFG7.gif
Thus, a natural estimator for Pickands function is
http://freakonometrics.hypotheses.org/files/2016/05/CFG9.gif
where https://latex.codecogs.com/gif.latex?\widehat{H}_n is the empirical cumulative distribution function of
http://freakonometrics.hypotheses.org/files/2016/05/cfg1.gifThis 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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