copula (version 0.999-14)

rF01FrankJoe: Sample Univariate Distributions Involved in Nested Frank and Joe Copulas

Description

rF01Frank: Generate a vector of random variates $V01 ~ F01$ with Laplace-Stieltjes transform $${\psi }_{01}(t;V_0)=(\frac{1-(1-\exp (-t)(1-e^{-{\theta }_1}){)}^{{\theta }_0/{\theta }_1}}{1-e^{-{\theta }_0}}{)}^{V_0}.$$ for the given realizations $V0$ of Frank's $F0$ and the parameters $theta0, theta1 in (0,Inf)$ such that $theta0 <= theta1$.="" 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 $F01$. If $ rej < V0*theta_0*(1-e^{-theta0})^(V0-1)$ a rejection from $F01$ 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 $V01 ~ F01$ with Laplace-Stieltjes transform $${\psi }_{01}(t;V_0)=(1-(1-\exp (-t){)}^{\alpha }{)}^{V_0}.$$ for the given realizations $V0$ of Joe's $F0$ and the parameter $alpha in (0,1]$. This distribution appears on sampling nested Joe copulas. Here, $alpha = theta0/theta1$, where $theta0, theta1 in [1,Inf)$ such that $theta0 <= theta1$.="" the="" parameter="" approx denotes the largest number of summands in the sum-representation of $V01$ 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 $V01$.

Usage

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

Arguments

V0

a vector of random variates from $F0$.

theta0, theta1, alpha

parameters $theta0, theta1$ 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)