I'm studying copulas, finished the Introduction to Copulas by Nelsen. I'm interested in the latest/best known/etc approaches for approximating any Copula, or some families of copulas, so would be grateful for literature/article recommendation on this topic.
To make this more clear, I'm looking for ways of approximating (theoretically, not just through empirical smoothers) a family of copula with controlled roughness. Not sure if I'm expressing this clearly, so please comment if I need to expand on this. But I'm thinking about, for example, an analogy might be with various Taylor approximations for continuous functions, and how we can control the roughness of approximation by adding additional power terms. Intuitively, for me it seems something like this may be possible for copulas, i.e., "power 1" approximation would be very rough, but the same for some family of copulas. "power 2" would get more accurate, but differ between some members of the family, and etc.
Would be grateful for literature suggestions if something like this is already described theoretically or implemented in software.