Do copula describe multivariate distributions more accurately than their moments?

Question

Moments approach:

A common way to characterize and describe the density (pdf) of a random variable is to only look at the mean and standard deviation ($\mu_1$, $\sigma_1$) of its pdf. This mentality is also extended to bivariate data where we rely on ($\mu_1$, $\sigma_1$) and ($\mu_2$, $\sigma_2$) to help us describe their joint empirical distribution by switching to vector notation: ($\boldsymbol\mu, \boldsymbol\Sigma$), where the vector of means and covariance matrix are now used.

Copula approach:

Can it be said that estimating the joint probability of two variables, $X$ and $Y$, with an empirical copula, is a more intricate way of describing their joint pdf beyond just their first two moments? Can it also be said that empirical copula are a more accurate method of estimating pdf's than higher-moment approaches?

Answer 1

Well, a copula is just a different way to present the joint distribution. From the copula and the marginals the joint distribution can be reconstructed exactly (by using Sklar's theorem.)

Moments is a different story, you cannot in general recontsruct the joint distribution exactly from any finite set of moments, sometimes you can get good approximations. So I don't really see the point of your comparison, the uses of moments and copulas are quite distinct.