As mentioned in the course on copulas, a nice tool to describe dependence it Kendall’s cumulative function. Given a random pair with distribution , define random variable . Then Kendall’s cumulative function is

Genest and Rivest (1993) introduced that function to choose among Archimedean copulas (we’ll get back to this point below).

From a computational point of view, computing such a function can be done as follows,

- for all , compute as the proportion of observation in the lower quadrant, with upper corner , i.e.

- then compute the cumulative distribution function of ‘s.

To visualize the construction of that cumulative distribution function, consider the following animation

Thus, here the code to compute simply that cumulative distribution function is

n=nrow(X)
i=rep(1:n,each=n)
j=rep(1:n,n)
S=((X[i,1]>X[j,1])&(X[i,2]>X[j,2]))
Z=tapply(S,i,sum)/(n-1)

The graph can be obtain either using

plot(ecdf(Z))

or

plot(sort(Z),(1:n)/n,type="s",col="red")

The interesting point is that for an Archimedean copula with generator , then Kendall’s function is simply

If we’re too lazy to do the maths, at least, it is possible to compute those functions numerically. For instance, for Clayton copula,

h=.001
phi=function(t){(t^(-alpha)-1)}
dphi=function(t){(phi(t+h)-phi(t-h))/2/h}
k=function(t){t-phi(t)/dphi(t)}
Kc=Vectorize(k)

Similarly, let us consider Gumbel copula,

phi=function(t){(-log(t))^(theta)}
dphi=function(t){(phi(t+h)-phi(t-h))/2/h}
k=function(t){t-phi(t)/dphi(t)}
Kg=Vectorize(k)

If we plot the empirical Kendall’s function (obtained from the sample), with different theoretical ones, derived from Clayton copulas (on the left, in blue) or Gumbel copula (on the right, in purple), we have the following,

Note that the different curves were obtained when Clayton copula has Kendall’s tau equal to 0, .1, .2, .3, …, .9, 1, and similarly for Gumbel copula (so that Figures can be compared). The following table gives a correspondence, from Kendall’s tau to the underlying parameter of a copula (for different families)

as well as Spearman’s rho,

To conclude, observe that there are two important particular cases that can be identified here: the case of perfect dependent, on the first diagonal when , and the case of independence, the upper green curve, . It should also be mentioned that it is also common to plot not function , but function , defined as ,