http://www.stat.ubc.ca/lib/FCKuserfiles/file/huacopula.pdf is the best so far. But I feel all the texts seem to skip some essential clarification. We often have some knowledge about the marginal distributions of 2 rvars. We often have calibrated models for each. But how do we model the dependency? If we have either a copula or a joint CDF, then we can derive the other. I there are 2 distinct contexts — A) known CDF -> copula, or B) propose copula -> CDF
–Context A: known joint CDF
I feel this is not a practical context but an academic context, but students need to build this theoretical foundation.
Given 2 marginal distro F1 and F2 and the joint distro (let’s call it F(u1,u2) ) between them, we can directly produce the true copula. Denoted CF(u1, u2) on P72, True copula := the copula to reproduce the joint CDF. This true copula C contains all information on the dependence structure between U1 and U2.
http://www.stat.ncsu.edu/people/bloomfield/courses/st810j/slides/copula.pdf P9 points that if the joint CDF is known (lucky!) then we can easily find the “true” copula that’s specific to that input distro.
In contrast to Context B, the true copula for a given joint distro is constructed using the input distros.
— Context A2:
Assume the joint distribution between 2 random variables X1 and X2 is, hmm ….. stable, then there exists a definite, concrete albeit formless CDF function H(x1, x2). If the marginal CDFs are continuous, then the true copula is unique by Sklar’s theorem.
–Context B: unknown joint CDF — “model the copula i.e. dependency, and thereby the CDF between 2 observable rvars”
This is the more common situation in practice. Given 2 marginal distro F1 and F2 without the joint distro and without the dependency structure, we can propose several candidate copula distributions. Each candidate copula would produce a joint CDF. I think often we have some calibrated parametric formula for the marginal distros, but we don’t know the joint distro, so we “guess” the dependency using these candidate copulas.
* A Clayton copula (a type of Archimedean copula) is one of those proposed copulas. The generic Clayton copula can apply to a lot of “input distros”
* the independence copula
* the comonotonicity copula
* the countermonotonicity copula
* Gaussian copula
In contrast to Context A, these “generic” copulas are defined without reference to the input distros. All of these copulas are agnostic of the input random variables or input distributions. They apply to a lot of different input distros. I don’t think they match the “true” copula though. Each proposed copula describes a unique dependency structure.
Perhaps this is similar — we have calibrated models of the SPX smile curve at short tenor and long tenor. What’s the term structure of vol? We propose various models of the term structure, and we examine their quality. We improve on the proposed models but we can never say “Look this is the true term structure”. I would say there may not exist a stable term structure.
A copula is a joint distro, a CDF of 2 (or more) random variables. Not a density function. As such, C(u1, u2) := Pr(U1<u1, U2<u2). It looks (and is) a function, often parameterized.