# The inverse in copula to go back to original data

Sklar's theorem states that: for any given n-variate distribution function $F$ with uniform margins $F_1,...,F_n$ then there exist copula $C$, such that: $F(x_1,..,x_n) =C(F_1(x_1),...,F_n(x_n))$, then if all margins is continuous, then copula is unique and defined as:

$$C(u_1,...,u_n)=F(F^{-1}(u_1),...,F^{-1}(u_n))\qquad\qquad (1)$$

Hence, since in copula we need to transform all the margins to uniform, then we need to go back to the original data using the inverse as in $(1)$.

However, I think we do not need to have the inverse of copula because based on Sklar's theorem we can write the density of $F$ (in the bivariate case) as follows:

$$f(x_1,x_2)=c(F(_1(x_1),F_2(x_2)) \, f_1 \,f_2$$

where $f_1$ and $f_2$ are the densities of the margins and can be any types of margins. Hence because of that, we do not need to have the inverse of the copula. Is that correct? Any help please?

The conversion to marginal uniformity is achieved by the probability integral transform $U_i=F_{i}(X_i)$. Note that each variable is transformed by its own cdf.
Similarly you transform the uniform margins of the copula back to the original scale via $X_i=F_{i}^{-1}(U_i)$.