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I understand that the joint density of two random variables $f(x,y)$ can be decomposed as the product of its marginals and a copula: $f(x,y) = g(x)k(y) \times c(G(x),K(y))$. Alternatively this may be written using CDFs as $F(x,y) = C(G(x),K(y))$.

Is it possible to directly estimate the joint density $f(x,y)$ or the joint distribution $F(x,y)$ without estimating the copula? What are the benefits/costs of obtaining the joint density by first estimating the marginals and copula instead?

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  • $\begingroup$ For instance suppose I observe datapoints $x_i,y_i$ for $i=1,\cdots,n$. Suppose $F(x,y)$ is the multivariate Gaussian distribution. Instead of fitting the associated copula for $F$, could I not directly estimate the parameters of the multivariate Gaussian distribution which maximize the likelihood? $\endgroup$
    – lasoon
    Commented Jul 25, 2022 at 12:42

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It is possible to estimate the joint distribution without using copulas. If you know joint distribution, $f(x,y; \Theta) $ up to an unknown parameter vector, $\Theta$, you shouldn't use a copula.

Copulas are useful when you know there is dependence between random variables, but you don't know a joint distribution that can describe this dependence. Say, $g(x) = Gamma(\alpha, \beta)$ and $k(y) = Lognormal(\mu, \sigma)$. There is no closed-form expression for $f(x,y)$. In this case, you can use a copula to estimate the joint distribution from the marginals.

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  • $\begingroup$ Thanks this is very helpful. Based on your response, does it follow that in general copula-based approaches are more flexible in permitting the marginals to be drawn from different distributions? $\endgroup$
    – lasoon
    Commented Jul 25, 2022 at 13:36
  • $\begingroup$ Moreover why should copulas not be used if you know the joint distribution $f(x,y;\Theta)$ up to an unknown parameter vector, $\Theta$? If you know both the joint and marginals, shouldn't you in principle be able to back out the joint density exactly by using the copula? Or is some information lost in the process of using copulas which prevents you from backing out the joint dist? $\endgroup$
    – lasoon
    Commented Jul 25, 2022 at 13:39
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    $\begingroup$ You have to choose a copula function, $C$, to get the joint distribution. You need to specify $C$ correctly to recover the joint distribution. I think there would be less error if you estimate the joint distribution directly instead of modeling through a copula, but I don't know of any theory which would confirm this. $\endgroup$
    – Eli
    Commented Jul 25, 2022 at 13:46
  • $\begingroup$ How about the case when you know neither the joint distribution nor the copula? A choice will need to be made for either $C$ or $F$ in this case. My intuition is that the copula-based approach might be less restrictive/sensitive to assumptions since you can estimate the marginals non-parametrically; whereas you don't have this flexibility if you assume $F$. $\endgroup$
    – lasoon
    Commented Jul 25, 2022 at 15:06
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    $\begingroup$ It then means imposing a specific probability model, which is the same as with copulas. $\endgroup$
    – Xi'an
    Commented Jul 25, 2022 at 15:23

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