MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Hello everyone,

I recently became aware of the existence of the copula concept. So, I have been reading a few things about copulas lately, but I cannot seem to find information on the following question:

  • Let's say that I have marginals for the random variables a,b,c,d.

  • Is it possible to learn a copula from the 4 marginals p(a),p(b),p(c),p(d) that allows me to derive then any marginal I might be interested in?

E.g. once I have obtained the copula c(a,b,c,d), can I calculate the marginal for (a,d)?

Thank you in advance! N.

share|cite|improve this question
up vote 0 down vote accepted

Answer is yes, it's Sklar's theorem.

The copula is merely a form of normalization that makes all your marginals U(0,1). Given a copula and the marginals, you can reconstruct the original distribution, and a fortiori get any marginal you're interested in.

share|cite|improve this answer
Dear Arthur, your prompt answer is much appreciated. Please bear with me, I must sit myself down and have a serious think:) as I seem to have misunderstood some fundamentals... – ngiann Mar 6 '12 at 15:58
@Arthur: I think you should seek clarification from the OP, first. There is no way to "learn" (I assume this means estimate or infer) a copula from the marginal distributions. You can specify a copula and based on that and the assumed marginals, recover the joint distribution of any subset of the random variables, but the only way you could try to infer a copula is if you had a sample (or other collection) of random vectors. – cardinal Mar 6 '12 at 20:12
Thank you both, I suppose one needs sometimes a little push to get unstuck... – ngiann Mar 7 '12 at 10:33
Ah yes, I focused on the last line and missed that bullet point! There are many different distribution that share the same marginals, so marginals aren't enough to recover the original distribution. – Arthur B Mar 7 '12 at 14:25

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.