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 \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. (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 A()
are available, see
An
.
Examples
## List the available methods (and their definitions):
showMethods("A")
showMethods("iPsi", incl=TRUE)