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I do understand the mathematic behind probability density function( PDF) and cumulative distribution function (CDF). My problem starts when I try to understand why copula relies on CDF and not on PDF. I searched and found this:

the probability density function can be hard to work with directly. What I found confused me again.

So,

1- Why copula relies on CDF and not PDf?

2- Why is it hard to work with PDF directly?

3- Why always CDF is used in integral transformation function. That is, why we cannot transform the variables into their uniform distribution using PDF?

Maryam
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  • Sklar's theorem guarantees the existence of a copula C for a joint CDF and marginals CDFs. –  Jul 19 '20 at 10:08
  • @ping Thanks so much for your comment. So, why `cdf` is used instead of `pdf`. Is that because `cdf` is unique? – Maryam Jul 19 '20 at 10:13
  • Sklar's theorem not only guarantees the existence but it also gives a simple way to estimate the copula from the CDF https://en.wikipedia.org/wiki/Copula_(probability_theory)#Sklar's_theorem If all marginal CDFs are continuous, C is unique. –  Jul 19 '20 at 10:17
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    What do you mean by "copula relies on CDF and not on PDF"? Not every distribution may have a pdf but beyond that those two ways of describing a distribution are mathematically equivalent. What description works better depends on the problem you would like to solve. The ubiquitous bivariate scatter plots of copulas are a good example of the pdf-view. – g g Jul 19 '20 at 10:22
  • @ping Thanks again. But why `cdf` is good in this case? what is the problem if we use `pdf` instead? – Maryam Jul 19 '20 at 10:24
  • @gg Thanks for your comment. I mean why copula is a cumulative distribution function? Why we do not work directly with pdf? Why copula joins the `CDF` and not the `PDF`? We first join the `CDF` then, we estimate `PDF` and hence the dependency. Is that because `PDF` is not always known? If yes, how about known `PDF`? – Maryam Jul 19 '20 at 10:28
  • @gg Another point is, why always `CDF` is used in integral transformation function? That is, why we cannot transform the variables into their uniform distribution using `PDF`? – Maryam Jul 19 '20 at 10:30
  • "Copula" is the name given to multivariate distributions with uniform margins. Not more, not less. Given that, you can describe it with whatever tools you like. Think about the pdf of the independent copula. This is not complicated and a good way to understand this particular copula. – g g Jul 19 '20 at 10:31
  • @gg Why always CDF is used in integral transformation function. That is, why we cannot transform the variables into their uniform distribution using PDF? – Maryam Jul 19 '20 at 10:40
  • https://math.stackexchange.com/questions/3473846/why-is-there-a-preference-to-use-the-cumulative-distribution-function-to-charact –  Jul 19 '20 at 10:41

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Sklar's theorem guarantees the existence of a copula $C$, given a joint CDF and marginal CDFs.

The theorem gives an easy method to determine the copula $C$.

Why do we use CDF instead of PDF?
"Every random variable has a CDF. Not every random variable has a PDF"

https://math.stackexchange.com/questions/3473846/why-is-there-a-preference-to-use-the-cumulative-distribution-function-to-charact