copula (version 0.999-2)

dnacopula: Density Evaluation for (Nested) Archimedean Copulas

Description

Evaluates the density of an Archimedean copula.

Usage

dnacopula(x, u, log=FALSE, ...)

Arguments

x

an object of class "outer_nacopula".

u

argument of the copula x. Note that u can be a matrix in which case the density is computed for each row of the matrix and the vector of values is returned.

log

logical indicating if the log of the density should be returned.

...

optional arguments passed to the copula's dacopula function (slot), such as n.MC (non-negative integer) for possible Monte Carlo evaluation (see dacopula in acopula

Value

• A numeric vector containing the values of the density of the Archimedean copula at u.

Details

If it exists, the density of an Archimedean copula $C$ with generator $\psi$ at $\bm{u}\in(0,1)^d$ is given by $$c(\text{bm}u)={\psi }^{(d)}({\psi }^{-1}(u_1)+\cdots +{\psi }^{-1}(u_d))\prod _{j=1}^d({\psi }^{-1}(u_j){)}^{\prime }=\frac{{\psi }^{(d)}({\psi }^{-1}(u_1)+\cdots +{\psi }^{-1}(u_d))}{\prod _{j=1}^d{\psi }^{\prime }({\psi }^{-1}(u_j))}.$$

References

Hofert, M., Mächler{Maechler}, M., and McNeil, A. J. (2011a), Estimators for Archimedean copulas in high dimensions: A comparison, submitted.

Hofert, M., Mächler{Maechler}, M., and McNeil, A. J. (2011b), Likelihood inference for Archimedean copulas, submitted.

See Also

For more details about the derivatives of an Archimedean generator, see, for example, absdPsi in class acopula.

Examples

## Construct a twenty-dimensional Gumbel copula with parameter chosen
## such that Kendall's tau of the bivariate margins is 0.25.
theta <- copJoe@iTau(.25)
C20 <- onacopula("J", C(theta, 1:20))

## Evaluate the copula density at the point u = (0.5,...,0.5)
u <- rep(0.5, 20)
dnacopula(C20, u)

## the same with Monte Carlo based on 10000 simulated "frailties"
dnacopula(C20, u, n.MC = 10000)

## Evaluate the exact log-density at several points
u <- matrix(runif(100), ncol=20)
dnacopula(C20, u, log = TRUE)