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In copula model, some researchers, identify it as a multivariate distribution function, while other present it as a cumulative distribution function. I believe multivariate differs of cumulative. But it seems researcher used them as exchangeable definition. Is that correct?

My other point is, copula is known to be with standard uniform margins, so why author used the following form of definition:

Copula is a multivariate distribution function with standard uniform margins U(0,1) on [0,1].

So, my point is, why they say U(0,1) on [0,1]. What is the difference between U(0,1) and on [0,1]?

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The two terms are not synonyms.

Copulas are necessarily multivariate.

The conventional definition of a copula function is of $C$, a cdf but someone might occasionally refer to the density $c$ as a copula since it contains the same information. I'd be more likely to say the density of the copula but not everyone does, particularly in informal discussion.

So $C$ will be a multivariate cumulative distribution function.

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  • $\begingroup$ Thanks a lot. What about 'U(0,1)' on [0,1]? What is the difference between U(0,1) and [0,1]? $\endgroup$
    – Alice
    Commented Jun 21, 2022 at 9:51
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    $\begingroup$ Sorry, missed the bit at the end of your question. It's more a mathematics notation question rather than a stats question; specifically it's about open vs closed intervals. see en.wikipedia.org/wiki/Interval_(mathematics)#Terminology . It's $0<u<1$ vs $0\leq u\leq 1$. For a continuous random variable the practical difference is not of consequence (it's essentially the difference between 'strictly between 0 and 1, always' -- and '... almost always', the difference is an 'event of probability 0'. $\endgroup$
    – Glen_b
    Commented Jun 21, 2022 at 21:58
  • $\begingroup$ Thanks. So, even though the margins must be standard uniform, it must be 'strictly' between 0 and 1, otherwise is not a copula data. Is that correct? $\endgroup$
    – Alice
    Commented Jun 23, 2022 at 11:08
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    $\begingroup$ I think you could quite reasonably choose to use open or closed intervals in different situations without doing much violence to anything. Open intervals are more typical but I don't know that it's an absolute requirement. If you explicitly define a copula to be on $(0,1)^d$ then that's how you define it (and there may indeed be some advantages to avoiding the boundaries), but I don't know that you absolutely have to define it that way. $\endgroup$
    – Glen_b
    Commented Jun 24, 2022 at 3:26
  • $\begingroup$ Great. Thanks a lot. I really appreciate your help. $\endgroup$
    – Alice
    Commented Jun 24, 2022 at 5:56

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