rF01FrankJoe {copula}

Sample Univariate Distributions Involved in Nested Frank and Joe Copulas

Description

rF01Frank: Generate a vector of random variates V_{01}\sim F_{01} with Laplace-Stieltjes transform

\psi_{01}(t;V_0)= \Bigl(\frac{1-(1-\exp(-t)(1-e^{-\theta_1}))^{\theta_0/\theta_1}}{% 1-e^{-\theta_0}}\Bigr)^{V_0}.

for the given realizations V_0 of Frank's F_0 and the parameters \theta_0,\theta_1\in(0,\infty) such that \theta_0\le\theta_1. This distribution appears on sampling nested Frank copulas. The parameter rej is used to determine the cut-off point of two algorithms that are involved in sampling F_{01}. If \code{rej} < V_0\theta_0(1-e^{-\theta_0})^{V_0-1} a rejection from F_{01} of Joe is applied (see rF01Joe; the meaning of the parameter approx is explained below), otherwise a sum is sampled with a logarithmic envelope for each summand.

rF01Joe: Generate a vector of random variates V_{01}\sim F_{01} with Laplace-Stieltjes transform

\psi_{01}(t;V_0)=(1-(1-\exp(-t))^\alpha)^{V_0}.

for the given realizations V_0 of Joe's F_0 and the parameter \alpha\in(0,1]. This distribution appears on sampling nested Joe copulas. Here, \alpha=\theta_0/\theta_1, where \theta_0,\theta_1\in[1,\infty) such that \theta_0\le\theta_1. The parameter approx denotes the largest number of summands in the sum-representation of V_{01} before the asymptotic

V_{01}=V_0^{1/\alpha}S(\alpha,1,\cos^{1/\alpha}(\alpha\pi/2), \mathbf{1}_{\{\alpha=1\}};1)

is used to sample V_{01}.

Usage

rF01Frank(V0, theta0, theta1, rej, approx)
rF01Joe(V0, alpha, approx)

Arguments

V0

a vector of random variates from F_0.

theta0, theta1, alpha

parameters \theta_0,\theta_1 and \alpha as described above.

rej

parameter value as described above.

approx

parameter value as described above.

Value

A vector of positive integers of length n containing the generated random variates.

References

Hofert, M. (2011). Efficiently sampling nested Archimedean copulas. Computational Statistics & Data Analysis 55, 57–70.

See Also

rFFrank, rFJoe, rSibuya, and rnacopula.

rnacopula

Examples

## Sample n random variates V0 ~ F0 for Frank and Joe with parameter
## chosen such that Kendall's tau equals 0.2 and plot histogram
n <- 1000
theta0.F <- copFrank@iTau(0.2)
V0.F <- copFrank@V0(n,theta0.F)
hist(log(V0.F), prob=TRUE); lines(density(log(V0.F)), col=2, lwd=2)
theta0.J <- copJoe@iTau(0.2)
V0.J <- copJoe@V0(n,theta0.J)
hist(log(V0.J), prob=TRUE); lines(density(log(V0.J)), col=2, lwd=2)

## Sample corresponding V01 ~ F01 for Frank and Joe and plot histogram
## copFrank@V01 calls rF01Frank(V0, theta0, theta1, rej=1, approx=10000)
## copJoe@V01 calls rF01Joe(V0, alpha, approx=10000)
theta1.F <- copFrank@iTau(0.5)
V01.F <- copFrank@V01(V0.F,theta0.F,theta1.F)
hist(log(V01.F), prob=TRUE); lines(density(log(V01.F)), col=2, lwd=2)
theta1.J <- copJoe@iTau(0.5)
V01.J <- copJoe@V01(V0.J,theta0.J,theta1.J)
hist(log(V01.J), prob=TRUE); lines(density(log(V01.J)), col=2, lwd=2)