# Sampling from copula given a particular value of marginal

I am working in the bivariate case with copulas as follows: I have two marginal Gamma distributions $f_1=Ga(\alpha_1, 1)$ and $f_2=Ga(\alpha_2, 1)$ that are bound by a copula Frank copula $C$, with parameter 10. Assume now, that I observe a value from $f_1=5$.

The question: How can I obtain a value for $f_2$, assuming that $f_1, f_2$ admit the copula $C$?

Initially I though it was a matter of conditinal probability with copulas as in this question, but I am not convinced about my approach. I tried with cCopula, but I am not sure I am on the right path. Any code sample would be highly appreciated.

Your question is somewhat related as both are based on conditional probabilities and thus partial derivatives of copulas. However, the sampling cannot easily be obtained from the single density values, but from the conditional distribution function $F(y|x) = \frac{\partial}{\partial F_1(x)} C(F_1(x),F_2(y))$. For ease of notation, let us only look at the copula part. The strategy I would suggest is to use the general sampling scheme from a copula $C(u,v)$ (going back to the univariate probability integral transform) where one takes two independent random samples $\bf u$ and $\bf y$ of the same length from a uniform distribution $U(0,1)$, then: $${\bf v} = \left(\frac{\partial}{\partial u|_{u=\bf u}} C\right)^{-1}({\bf y})$$ In your case, you do not want to obtain a sample for $u$ and $v$ following $C$, but just a sample of $v$, given a particular value of $u$. Hence, you do not sample a vector $\bf u$, but just fix it at your desired value and sample a vector $\bf y$ that is used in the inversion of the partial derivative of the copula as in the general case above.

An approach in R could look like:

library(copula)
library(VineCopula)
library(spcopula)

cop <- tawnT1Copula(c(3, 0.7))

# easy bivariate sampling
plot(rCopula(500, cop), asp=1)

# step-wise
u <- runif(500)
y <- runif(500)

plot(u, invdduCopula(u, cop, y), asp=1)

# conditional on u=0.3
u <- rep(0.3, 500)
y <- runif(500)
hist(invdduCopula(u, cop, y), main="Sample of v | u = 0.3")

# conditional on u=0.7
u <- rep(0.7, 500)
y <- runif(500)

hist(invdduCopula(u, cop, y), main="Sample of v | u = 0.7")´´