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I want to transform two known marginal distributions into a joint distribution by using copula.

As I understand commonly used copulas cannot make sure that in the joint distribution one variable is always larger than the other. I wonder if there is any that can help. Thanks a lot.

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1 Answer 1

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If $\mu$ and $\nu$ are the two marginal distributions and if there exists a coupling $(X,Y)$ of $\mu$ and $\nu$ such that $\Pr(X \geqslant Y) = 1$, then this implies that $\mu$ stochastically dominates $\nu$.

So this stochastic domination is a necessary condition for the existence of your copula. Now, if this condition holds true, you can take the copula $C(u, v) = \min(u,v)$, which is the bivariate cumulative distribution function of a random pair $(U, U)$ where $U \sim \mathcal{U}(0,1)$.

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