generator: Generator Functions for Archimedean and Extreme-Value Copulas

Description

Methods to evaluate the generator function, the inverse generator function, and derivatives of the inverse of the generator function for Archimedean copulas. For extreme-value copulas, the Pickands dependence function plays the role of a generator function.

Usage

psi(copula, s)iPsi(copula, u, ...)
diPsi(copula, u, degree=1, log=FALSE, ...)
A(copula, w)
dAdu(copula, w)

Arguments

copula

an object of class "copula".

u, s, w

numerical vector at which these functions are to be evaluated.

...

further arguments for specific families.

degree

the degree of the derivative (defaults to 1).

log

logical indicating if the log of the absolute derivative is desired. Note that the derivatives of $psi$ alternate in sign.

Details

psi() and iPsi() are, respectively, the generator function $\psi()$ and its inverse $\psi^{(-1)}$ for an Archimedean copula, see pnacopula for definition and more details.

diPsi() computes (currently only the first two) derivatives of iPsi() ($= \psi^{(-1)}$).

A(), the Pickands dependence function, can be seen as the generator function of an extreme-value copula. For instance, in the bivariate case, we have the following result (see, e.g., Gudendorf and Segers 2009):

A bivariate copula $C$ is an extreme-value copula if and only if $$C(u,v) = (uv)^{A(\log(v) / \log(uv))}, \qquad (u,v) \in (0,1]^2 \setminus {(1,1)},$$ where $A: [0, 1] \to [1/2, 1]$ is convex and satisfies $\max(t, 1-t) \le A(t) \le 1$ for all $t \in [0, 1]$. In the $d$-variate case, there is a similar characterization, except that this time, the Pickands dependence function $A$ is defined on the $d$-dimensional unit simplex.

dAdu() returns a data.frame containing the 1st and 2nd derivative of A().

References

Gudendorf, G. and Segers, J. (2009) {Nonparametric estimation of an extreme-value copula in arbitrary dimensions}, {ArXiv}, http://arxiv.org/abs/0911.1015.

Examples

Run this code

## List the available methods (and their definitions):
showMethods("A")
showMethods("iPsi", incl=TRUE)