1
$\begingroup$

In the univariate case, one can easily sample a distribution via random numbers $u\sim[0,1]$ and plugging into $F^{-1}(u)$.

I have a bivariate distribution constructed via Sklar's theorem on Copulas:

$$F(x,y)=C(F(x),F(y))$$

The distribution is not in closed form.

How do I sample from this bivariate distribution $F(x,y)$?

$\endgroup$
0

1 Answer 1

2
$\begingroup$

We would need more information on how you estimate/construct your copula $C$ to give any detailed advise.

In general, sampling from a copula can be achieved following a similar idea as the univariate case, but involving the conditional marginal distribution $F(v|u)$. To sample from the copula, one would take two independent uniform samples $\bf x$ and $\bf y$ of desired length $n$. For each $i \in \{1, \dots, n\}$, set $u_i=x_i$ and seek $v_i$ such that: $$y_i=F(v_i|u_i)=\frac{\partial C (u_i,v_i)}{\partial u_i}.$$

The pairs $(u_i,v_i)$ are now a sample from the copula. Applying the inverse of the marginals $F_X$ and $F_Y$ gives a sample of the bivariate distribution $F_{XY}(x,y)=C(F_X(x), F_Y(y))$. See as well the thread on this question.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.