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I am trying to simulate the following structure:

$C_{123}(C_{12}(u_1, u_2; \theta_1), u_3; \theta_2)$

I am able to simulate the inner $C_{12}$, I do simply use the method of the conditional copulas. Starting with $u_1$, $u_2$ $\sim U(0,1)$

$u_1 = u_1$

$u_2 = \frac{dC(u_1,v)}{du}$

by inversion I succeed to find $v$. My question is about the outer $C_{123}$: can I simply use as the new $u_1$ the probabilities coming from $C_{12}(u_1,v)$, and then going through the very same procedure? I have read lots of papers but never found a clear explanation.

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You need to use the inverse of the h function if you would like to simulate data from vine copula. It is really complicated.

Have you look at "Pair-copula construction for multiple dependency structures."

They have a nice explanation of the simulation process from vine copula.

Hope this help.

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