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The copula is a very interesting tool to describe the dependence structure. However, I read that if the margins are continuous then copula is unique. However, if margins are discrete then copula is not unique. I cannot understand this two points. That is, I do not understand why if margins are continuous then copula is unique, otherwise then copula is not unique?

Could someone help me, please? I hope I can have an example to make it so clear to me.

  • Intuitively (hence commenting and not answering) I would guess it is likely to do with the fact that continuous margins can take on infinite number of values so it is very unlikely to get multiple identical copulas and discrete margins mean there is a limited number of values possible so overlap is likely. If this is indeed a rough approximation of what is going on though it would suggest that as the dimensionality of the copula increases then the probability of equivalent copulas decreases as the number of combinations of values grows. – ReneBt Apr 17 '18 at 11:22
  • @ReneBt I read that the reason is that discrete margins have jump. However, I am really not sure about this part. –  Apr 17 '18 at 12:05
  • Copulas are almost never unique. Where did you read about the dependence of their uniqueness on the continuity of the marginals? – Matt F. Jul 27 '22 at 18:11

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