satdad (version 1.1)

copArchimaxMevlog: cop-ell-psi-psiinv- functions for Archimax Mevlog models.

Description

Copula function, stable tail dependence function, psi function, psi inverse function for Archimax Mevlog models.

Usage

copArchimaxMevlog(x, ds,  dist = "exp", dist.param = 1)
ellArchimaxMevlog(x, ds)
psiArchimaxMevlog(t, dist = "exp", dist.param = 1)
psiinvArchimaxMevlog(t, dist = "exp", dist.param = 1)

Value

When the underlying distribution dist is

• "exp" ; For a positive \(\lambda\) given by dist.param, \(\psi(t)=\frac{\lambda}{t+\lambda}\) and \(\psi^{-1}(t)=\lambda \frac{1-t}{t}\).

• "gamma" ; For positive scale \(\sigma\) and shape \(a\) given by dist.param, \(\psi(t)=\frac{1}{(t+\sigma)^a}\) and \(\psi^{-1}(t)=\frac{t^{-1/a}-1}{\sigma}\).

• "ext" ; \(\psi(t)=\exp(-t)\) and \(\psi^{-1}(t)=-\ln(t)\).

copArchimaxMevlog returns the copula function \(C(x_1,...,x_d) = \psi(\ell(\psi^{-1}(x_1),...,\psi^{-1}(x_d)))\).

ellArchimaxMevlog returns the stable tail dependence function \(\ell(x_1,...,x_d)\).

psiArchimaxMevlog returns the psi function \(\psi(t)\).

psiinvArchimaxMevlog returns the psi inverse function \(\psi^{-1}(t)\).

Arguments

x

A vector of size d or a (N.x times d) matrix.

ds

An object of class ds.

dist

The underlying distribution. A character string among "exp" (the default value), "gamma" and "ext".

dist.param

The parameter associated with the choice dist. If dist is "exp", then dist.param is a postive real, the parameter of an exponential distribution. The default value is 1. If dist is "gamma", then dist.param is a vector that concatenates the shape and scale parameters (in this order) of a gamma distribution.

t

A non negative scalar or vector.

Author

Cécile Mercadier ([e-mail])

Details

The tail dependence structure is set by a ds object. See Section Value in gen.ds.

Turning to Archimax structures, we follow Charpentier et al. (2014). Their algorithm (4.1 of p. 124) has been applied in rArchimaxMevlog to generate observations sampled from the copula

\(C(x_1,...,x_d) = \psi(\ell(\psi^{-1}(x_1),...,\psi^{-1}(x_d)))\)

when \(\ell\) is here the stable tail dependence function of a Mevlog model. In this package, the stdf function \(\ell\) is completely characterized by the ds object. See ellMevlog.

References

Charpentier, A., Fougères, A.-L., Genest, C. and Nešlehová, J.G. (2014) Multivariate Archimax copulas. Journal of Multivariate Analysis, 126, 118--136.

See Also

rArchimaxMevlog, gen.ds, ellMevlog

Examples

## Fix a 7-dimensional tail dependence structure
ds7 <- gen.ds(d = 7)

## Fix the parameters for the underlying distribution
(lambda <- runif(1, 0.01, 5))
(shape <- runif(1, 0.01, 5))
(scale <- runif(1, 0.01, 5))

## Fix x and t
x <- c(0.8, 0.9, 0.5, 0.8, 0.4, 0.9, 0.9)
t <- 2

## Evaluate the functions under the underlying exponential construction
copArchimaxMevlog(x = x, ds = ds7, dist = "exp", dist.param = lambda)
ellArchimaxMevlog(x = x, ds = ds7)
psiArchimaxMevlog(t = t, dist = "exp", dist.param = lambda)
psiinvArchimaxMevlog(t = t, dist = "exp", dist.param = lambda)

## Evaluate the functions under the underlying gamma construction
copArchimaxMevlog(x = x, ds = ds7, dist = "gamma", dist.param = c(shape, scale))
ellArchimaxMevlog(x = x, ds = ds7)
psiArchimaxMevlog(t = t, dist = "gamma", dist.param = c(shape, scale))
psiinvArchimaxMevlog(t = t, dist = "gamma", dist.param = c(shape, scale))