# Approximation of copulas

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.

Copulas are typically numerically tractable in moderate dimensions, so there is no need for approximations. Even in high dimensions there are tractable copulas. Taylor approximations for copulas is most likely a bad idea, although it may be interesting to look at their expansion (if they have a nice expression, which I doubt). There exist other types of approximations, which are mentioned in the references below.

Nelsen's book is a great introduction to copulas, but it is becoming a bit outdated. I would rather suggest looking at more recent literature on this topic such as:

Principles of Copula Theory by Fabrizio Durante and Carlo Sempi.

Dependence Modeling with Copulas by Harry Joe.

Copula Theory and Its Applications - Proceedings.

• Thank you for the suggestions! Skimmed through Durante and Sempi as a parallel to Nelsen, but will definitely take a second look. For the 2nd and 3rd -- will see if I can get my hands on them.
– runr
Oct 31 '18 at 9:58
• @Nutle Indeed, the first one is more introductory but it has a chapter on approximations that might be of interest. B-OK might help you get your hands on them ;). Oct 31 '18 at 10:02