The problem sounds quite easy, and I still think it is. I somehow have the feeling that I just went too far and just miss the easiest solution now. The numerical solution I came up with is just not satisfying for all cases.
A priori I know the values of $\mathcal{P}[X<k]$, $\mathcal{P}[X=k]$ and $\mathcal{P}[X>k]$ for some fixed $k \in \mathbb{N}$ for a $\mathit{Skellam}_{\lambda,\mu}$-distributed ($\Rightarrow$ discrete) random variable $X$. What I need to estimate are the parameters $\lambda$ and $\mu$. However, since the probability density function of the Skellam distribution contains modified Bessel functions, I refrained from calculating derivatives thus far (even if there exist recurrence relations for the mod. Bessel functions).
As a workaround, I applied a BOBYQA optimizer in order to find a pair of parameters that give good estimates. However, in several cases this approach is highly inaccurate since the solution of the optimizer depends on the bounds I use as inputs (I applied a bounded optimizer since $\lambda, \mu >0$).
