I have been spending dozens of hours trying to understand what copula is and how it works, but still I am not able to get my head straight.

I am reading the wiki page https://en.wikipedia.org/wiki/Copula_(probability_theory)#Mathematical_definition

math definition of copula in wikipedia

Question #1:

how do I get C(u1,u2...ud) ?

Question #2:

After I get C, to use C, is below formula correct ?

CDF of (X1,X2,X3...) = Marginal distribution of X1 * Marginal distribution of X2 ... * C ?

  • $\begingroup$ What do you mean by 'how do I get C'? Defining what a copula is does not pin down $C$. C can be any CDF whatsoever according to what we choose (that is the point of defining a copula), as long as we can recover the marginals from it correctly. $\endgroup$
    – Arshdeep
    Jun 27 at 17:07
  • $\begingroup$ @Arshdeep by "how do I get C", I meant, since now I have random variables (u1,u2,...ud), what is the joint CDF of (u1,u2,...ud) ? By "as long as we can recover the marginals from it correctly", do you mean the copula can be chosen arbitrarily, instead of being determined by the original (x1,x2,...,xd ) or (u1,u2...ud ) ? $\endgroup$
    – ROSS XIE
    Jun 28 at 1:34
  • $\begingroup$ The joint distribution is what you have to pin down in some way, it is obviously not pinned down by the marginal distributions. Yes, you can choose the copula function arbitrarily as long as it is an admissible CDF. It is just a function of n variables which also is a CDF. A notable way of choosing this joint distribution is the gaussian copula, whose advantage is that specifying just one parameter specifies the joint distribution (copula). $\endgroup$
    – Arshdeep
    Jun 28 at 5:27
  • $\begingroup$ @ROSS XIE Copula is just a function that imposes dependence on the marginal distributions. If you have two marginal distributions and draw a random sample from those two marginals then there is no dependence between them (the value of x from first marginal won't tell you anything about value y from the other marginal). Therefore you impose dependence, the function C can be anything as long as it is proper CDF. For example it can be multivariate normal CDF with marginals that are lognormal (or any other marginal distribution). For your Q2: No, there should be c - density, C is for CDF $\endgroup$
    – emot
    Jun 28 at 19:11
  • $\begingroup$ @Arshdeep If a copula is just a CDF of n variables and can be arbitrarily chosen, what is the usage of (u1,u2,....ud ) ? $\endgroup$
    – ROSS XIE
    Jun 29 at 14:36

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