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On the post A formal definition of a “measure of association” @kjetil b halvorsen commented the following:

A copula could be seen as a characterization of association, so maybe a "measure of association" is a functional of the bivariate distribution (,) that only depends on the copula (,).

From Wikipedia:

[A map] $C: [0,1]^d \mapsto [0,1]$ is a $d$-dimensional copula if $C$ is a joint distribution of ad $d$-dimensional random vector on the unit cube $[0,1]^d$ with uniform marginals.

With Sklar's theorem I can see how this is a useful mathematical tool for studying distributions in general, but it hasn't hit me yet why the copula is a characterization of association. At this moment it seems more like a multivariate CDF with normalized support, and it isn't clear to me that a functional of a CDF which depends on the copula would necessarily be a measure of association.

Just as mutual independence for a collection of random variables $\{X_j\}_{j=1}^{n}$ can be given by the satisfying of the equality $F_{X_1, \cdots, X_n}(x_1,\cdots,x_n) = \prod_{j}^{n} F_{x_j}(x_j)$, do we simply rephrase this in terms of the copulas?

$$C_{X_1, \cdots, X_n}(x_1,\cdots,x_n) = \prod_{j}^{n} C_{x_j}(x_j)$$

How can a copula be seen as a characterization of association?

Footnote: There are some related questions that are interesting but do not address my question.

Galen
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    I posted two questions last year that are possibly of interest: 1 2. – Dave Jul 19 '21 at 13:18
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    Oops, I posted those earlier this year. Anyway, the key point is that, if we can rectify what I question in those two posts (what I call "borked" about the relationship between mutual information, copula, and copula entropy calculation), then the copula has an intimate relationship with the mutual information. – Dave Jul 19 '21 at 13:35
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    I hate to say it, but isn't this a trivial observation? The copula factors out the univariate marginal distributions, which obviously have nothing to do with any concept of "association." Thus, any measure of association perforce is some functional of the copula. – whuber Jul 19 '21 at 13:39
  • @whuber It may seem trivial prima facie, but I've asked in case I missed something counter-intuitive to me. – Galen Jul 19 '21 at 13:45
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    @whuber: It is not that trivial, the Pearson correlation is not a functional of the copula ... – kjetil b halvorsen Jul 19 '21 at 22:52
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    @Kjetil That's an interesting observation. Maybe it says something important about Pearson correlation! – whuber Jul 20 '21 at 12:49
  • If Pearson's R is a measure of association, and Pearson's R is not a functional of the copula, then not all measures of association are a functional of a copula. – Galen Jul 20 '21 at 13:47
  • Copula can measure the dependency through its dependency parameters. – Alice Jul 22 '21 at 13:22

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