R: Generator Functions for Archimedean and Extreme-Value Copulas

generator {copula} | R Documentation

## 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 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

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

[Package copula version 1.0-1 Index]