I have two Skellam-distributed random variables, $X_1$ and $X_2$, with rate parameters $\mu_{1,+}, \mu_{1,-}$ and $\mu_{2,+}, \mu_{2,-}$ respectively. (For my case, it just so happens that their distributions are symmetric, i.e. $\mu_{1,+} = \mu_{1,-}$ and $\mu_{2,+} = \mu_{2,-}$.) Now I want to figure out how to describe/model/estimate the joint distribution of these processes. Note, I can simulate time series of $X_1$ and $X_2$ through a highly-complex nonlinear model.
There is literature here that discusses an extension to the Common Shock Model that is often applied to RVs with Poisson marginals. I wonder if somehow the Common Shock Model can be applied to Skellam RVs? Of if there is another way to approach this problem?
(Can a moderator please add the tags "skellam-distribution" and possibly "copula" or "common-shock-model"?)