# Multivariate sampling of $F(x_1,...,x_n)$?

Let $$(X_1,...,X_n)\sim F(x_1,...,x_n)$$ (not independent).

How can I sample from this distribution?

In the univariate case, on can use $F^{-1}(u),u\sim U(0,1)$. However, in the multivariate case these uniforms are not independent, such that they cannot be drawn independently.

• I think the rule still applies. Nov 25 '14 at 23:27
• @SeyhmusGüngören No, because then the sampling distribution of dependent and independent variables were the same. Nov 25 '14 at 23:33
• So let $\mathbf{U}$ be multivariate normal and $F^{-1}$ multivariate pseudo inverse of multivariate $F$... Nov 25 '14 at 23:36
• @SeyhmusGüngören No Nov 25 '14 at 23:37

Added: Yes it is possible. So the main idea is to use the Bayes rule. Since with have the joint distribution, we can obtain the conditional marginals. For example for three events $A,B,C$:
$$P(A,B,C)=P(A|B,C)P(B,C)=P(A|B,C)P(B|C)P(C)$$ Now applying the same idea to random variables $X_1,X_2,X_2$, then $P(X_1<x_1,X_2<x_2,X_3<x_3)=...$ in this case we have all marginals but they are conditional. So they are basically all univariate and we can use for each of them the regular way $F^{-1}_i(U)$, $i=1,...$. Then what remains is to combine every sample and obtain the vector valued sample.
• Is there a way to simulate from $F$ without knowing the Copula function? Nov 26 '14 at 1:01