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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.

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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.

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  • $\begingroup$ 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... $\endgroup$
    – ngiann
    Commented Mar 6, 2012 at 15:58
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    $\begingroup$ @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. $\endgroup$
    – cardinal
    Commented Mar 6, 2012 at 20:12
  • $\begingroup$ Thank you both, I suppose one needs sometimes a little push to get unstuck... $\endgroup$
    – ngiann
    Commented Mar 7, 2012 at 10:33
  • $\begingroup$ 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. $\endgroup$
    – Arthur B
    Commented Mar 7, 2012 at 14:25

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