Consider our loss-ALAE dataset, and - as in Frees & Valdez (1998) - let us fit a parametric model, in order to price a reinsurance treaty. The dataset is the following,
> library(evd)
> data(lossalae)
> Z=lossalae
> X=Z;Y=Z
...

Following the course of this afternoon, I will just upload some codes to make interactive 3d plots, in R.
> library(rgl)
> library(evd);
> data(lossalae)
> U=rank(lossalae+rnorm(nrow(lossalae),
+ mean=0,sd=.001))/(nrow(lossalae)+1)
...

Today, we will go further on the inference of copula functions. Some codes (and references) can be found on a previous post, on nonparametric estimators of copula densities (among other related things). Consider (as before) the loss-ALAE data...

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 ...

An alternative to describe tail dependence can be found in the Ledford & Tawn (1996) for instance. The intuition behind can be found in Fischer & Klein (2007)). Assume that and have the same distribution. Now, if we assume that those vari...

As mentioned in the course last week Venter (2003) suggested nice functions to illustrate tail dependence (see also some slides used in Berlin a few years ago).
Joe (1990)'s lambda
Joe (1990) suggested a (strong) tail dependence index. For lower t...

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) intr...

Following the course, in order to define assocation measures (from Kruskal (1958)) or concordance measures (from Scarsini (1984)), define a concordance function as follows: let be a random pair with copula , and with copula . Then define
the so-...

First, let us recall a standard result from linear algebra: "real symmetric matrices are diagonalizable by orthogonal matrices". Thus, any variance-covariance matrix can be written
since a variance-covariance matrix is also definite positive. In ...

In the course, still introducing some concept of dependent distributions, we will talk about the Dirichlet distribution (which is a distribution over the simplex of ). Let denote the Gamma distribution with density (on )
Let denote independent...