depfun: Dependence function of Extreme-Value copula

In acopula: Modelling Dependence with Multivariate Archimax (or any User-Defined Continuous) Copulas

Dependence function of Extreme-Value copula

Description

Produce a list containing dependence function of specified EV copula family, its derivatives and parameters bounds. Only Hussler-Reiss family is limited to two dimensions.

ldepPartition3D returns set of 5 dependence functions (see details).

Usage

depfun(name, ...)

dep1(...)
depGalambos(...)
depGumbel(...)
depHuslerReiss(...)
depMax(power = 10, ...)
depTawn(dim = 2, ...)

depCC(depfun = list(dep1(),depGumbel()), 
  dparameters = lapply(depfun,
    function(x) rep(list(NULL),max(1,length(x$parameters)))),
  dim = 2)
depGCC(depfun=list(dep1(),depGumbel()), 
  dparameters = lapply(depfun,
    function(x) rep(list(NULL),max(1,length(x$parameters)))),
  dim = 2, symmetry = FALSE)
    
ldepPartition3D(power = 8)

Arguments

name	character. Code name for Pickands' dependence function, identical with the part after dep.
power	numeric. Parameter of Gumbel family dependence function, which approximates the weakest dependence function in order to bring smoothness.
dim	numeric. Dimension (of copula) of random vector.
depfun	list of dependence function definition lists, also ldepPartition3D can be used.
dparameters	list of dependence function parameters; defaults to list of NULLs which means the parameters are to be estimated.
symmetry	logical. If TRUE, then GCC reduces to standard convex sum and depCC is used.
...	named arguments. Items of the dependence function definition list to be redefined.

Details

Currently implemented families of EV copula dependence functions:

family	dependence function A(t)=	domain	EV.case
1	1		\Pi
Galambos	1 - (\sum_i t_i^{-p})^{-1/p}	[0,10]	1(\Pi),Inf(M)
Gumbel-Hougaard	(\sum_i(t_i^{p}))^{1/p}	[1,Inf]	1(\Pi),Inf(M)
Husler-Reiss	t_1 \Phi(1/p + p \log(t_1/t_2)/2) + \atop + t_2 \Phi(1/p - p \log(t_1/t_2)/2)	[0,Inf]	0(\Pi),Inf(M)
Max	(\sum_i{t_i^{10}})^{1/10}		M
Tawn	1 - \sum_i{p_i t_i} + (\sum_i{(p_i t_i)^{p_0}})^{1/p_0}	[1,Inf]x[0,1]x...	{1,0,...}(W),{Inf,1,...}(M)

Since \sum_i t_i=1 a dependence function accepts argument vector with the last element omitted.

Value

parameters	numeric vector to be used whenever parameters of depfun are not supplied to procedure that use it, or as starting values in estimation
dep	function of two arguments; the first is depfun argument, the another is depfun parameters
dep.der	function; depfun first derivative
dep.der2	function; depfun second derivative
kendall,spearman	list. Correlation coefficient as function of copula parameter (coef), its inverse (icoef) and range (bounds). Available only for 1-parameter families.
lower,upper	numeric; parameters boundary
id	character; identification of depfun family
combpars,rescalepars	function; extract the combination parameters from the set of provided parameters and rescale them if not fulfilling inner conditions of the (general) convex combination

Author(s)

Tomas Bacigal

References

Bacigál, T., Mesiar, R.: 3-dimensional Archimax copulas and their fitting to real data. In: COMPSTAT 2012, 20th International conference on computational statistics. Limassol,Cyprus,27.-31.8.2012. The International Statistical Institute, 81–88 (2012).

Gudendorf, G., Segers, J. (2010): Extreme-value copulas. In Copula Theory and Its Applications. Springer Berlin Heidelberg, 127-145.

Insightful Corp.: EVANESCE Implementation in S-PLUS FinMetrics Module (2002). https://faculty.washington.edu/ezivot/book/QuanCopula.pdf Cited 6th July 2013.

See Also

pCopula, generator, copula

Examples

## the following gives the same definition list
depGumbel()
depfun("Gumbel")

## any list item can be modified upon function call
depGumbel(parameters=2.2,upper=10)

## general convex combination of 5 basic depfuns that arise from 
## partitioning method for 3 dimensions; it results in 
## (3x5)-parametric Pickand's dependence function definition list
depGCC(depfun=ldepPartition3D(), dim = 3)

acopula documentation built on Sept. 11, 2023, 1:08 a.m.