I am trying to create a joint distribution that has a specific copula (e.g. Clayton) and whose marginals come from a mixture distribution (e.g. the mixture of two Gaussian distributions). My idea was to first sample from the copula and then transform the marginals using the inverse of the marginal CDFs.
Since I am working with mixed marginals, I don't have an analytic expression of the inverse CDF. My first workaround was to try this method:
Say I have a mixture of two marginals, marginal 1 has inverse CDF $P_1$ and marginal 2 has inverse CDF $P_2$. I also have copula samples $C$. For each of the two dimensions of the copula samples I follow this approach:
- Draw samples from a uniform distribution
- Use the uniform samples to decide which inverse CDF ($P_1$ or $P_2$) to use for transforming the copula sample dimension.
However, the resulting joint distribution does not seem to have the same copula structure.
Is there a way to have an analytical expression of the inverse of the mixed CDF? I am now calculating it by numerically inverting the CDF, but this takes quite long and I would prefer an analytical expression. If there is no analytical expression, is there another workaround that preserves the copula structure of the joint distribution?