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

Arguments

Details

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

diPsi() computes (currently only the first two) derivatives of iPsi() (= ψ^(-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)), (u,v) in (0,1]^2 w/o {(1,1)},

where A: [0,1] -> [1/2, 1] is convex and satisfies max(t,1-t) <= A(t) <= 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. (2010). Extreme-value copulas. In Copula theory and its applications, Jaworski, P., Durante, F., Härdle, W. and Rychlik, W., Eds. Springer-Verlag, Lecture Notes in Statistics, 127–146, https://arxiv.org/abs/0911.1015.

See Also

Nonparametric estimators for A() are available, see An.

Examples