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 |
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 |
Details
psi()
and iPsi()
are, respectively, the generator
function and its inverse
for
an Archimedean copula, see
pnacopula
for definition and
more details.
diPsi()
computes (currently only the first two) derivatives of
iPsi()
().
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 is an extreme-value copula if and only if
where is convex and satisfies
for all
.
In the -variate case, there is a similar characterization,
except that this time, the Pickands dependence function
is
defined on the
-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, doi:10.1007/978-3-642-12465-5; preprint at https://arxiv.org/abs/0911.1015.
See Also
Nonparametric estimators for are available, see
An
.
Examples
## List the available methods (and their definitions):
showMethods("A")
showMethods("iPsi", incl=TRUE)