# Do copula describe multivariate distributions more accurately than their moments?

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?

## 1 Answer

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.

• Even though the copula allows the joint to be theoretically reconstructed exactly, how easy is it to do this in practice as number of variables in the joint distribution become ever larger? Do non-parametric (empirical) estimators fail compared to parametric copulas? Commented Sep 10, 2020 at 13:43