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I am trying to understand how copulas help us analyze the dependence structure between random variables. From Sklar's theorem, I know that given any joint distribution, you can extract the associated copula to understand the dependence structure. Similarly, you can construct a joint distribution using any copula and appropriate marginal distributions.

However, I am struggling to grasp how copulas specifically reveal the nature of the dependency between variables. For example, consider the extreme case of the countermonotonic copula,

$C(u, v) = \max(u + v - 1, 0).$

I know this copula represents perfect negative dependence, meaning that if one variable increases, the other decreases in a deterministic way. But my question is: how can I infer or interpret this kind of dependence directly from the copula?

To illustrate, let's imagine $X$ and $Y$ are two random variables such that their joint cumulative distribution function is $F_{X,Y}(x, y)$, and the copula is countermonotonic. This implies that as $X$ increases, $Y$ decreases. However, if we only look at the copula itself (without knowing its name or classification), how can we deduce or interpret the nature of this relationship? Could someone explain this using the properties of the copula or other intuitive reasoning?

I’ve been stuck on this question for three days, and I would greatly appreciate any guidance. Thank you in advance!

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    $\begingroup$ The approach I sketched (for any distribution, not just copulae) at stats.stackexchange.com/a/18200/919 works well for me. (I submitted a paper on this to The American Statistician years ago but it was rejected as being too obvious...: maybe that's a point in its favor as a good, basic way to understand bivariate relationships!) $\endgroup$
    – whuber ♦
    Commented Nov 27 at 15:16
  • $\begingroup$ @whuber thnx very much, Sir. Did you publish your paper somewhere ? can I get access to it ? $\endgroup$
    – Houssem Ajili
    Commented Nov 27 at 15:20
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    $\begingroup$ It remains unpublished. $\endgroup$
    – whuber ♦
    Commented Nov 27 at 16:16
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    $\begingroup$ One of those things that are obvious only after someone actually suggests it... :-( $\endgroup$
    – Dikran Marsupial
    Commented Nov 27 at 17:03
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    $\begingroup$ @whuber I hope your paper will be published soon. Try another journal. Sometimes, people need something too obvious to learn from. $\endgroup$
    – Dr. Statistics
    Commented Dec 7 at 9:46

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Since I can't comment yet I will write this as an answer, which is hopefully acceptable. A copula $C:[0,1]^d \to [0,1]$ is a joint distribution over the unit cube with uniform marginals. Because it's a distribution, you can for example calculate the covariance $\text{Cov}(u,v)$ which will quantify the dependence structure present in this joint distribution. You can also plot this joint distribution. Here is a really good visual illustration of this. This is an example of how copulas can be used to infer correlations of firing rates in neural populations.

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