An interesting problem - an "ugly" familly of distributions - what can we do? I've recently encountered an interesting problem. Say, we've got a famility of distributions indexed by a parameter $ \theta$. Say it's a family of descrete distributions (you can also tell me what to do when they are continous...). We know they depend on the parameter $\theta$ but the exact relationship is unknown (it's not given by a mathematical formula). We even know the exact probabilities/distributions! But the relationship between distributions and the $\theta$ is not trivial - for instance we know that the distribution "shifts" to the right in its domain along with the increasing $\theta$ and that's it.
My question is: what can we do with such family? What statistical procedure can we involve? How can we calculate approximate probabilities for new $\theta$? Using some kind of regression? What about a family of continous distributions?
Here is the instance I encountered. I ploted 2 initial distributions (black and red) and the last one (green). We see they, on average, shift to the right as the parameter increases. I'm not completly sure I calculated the probabilities correctly, but it doesn't matter here.

For your family of distributions indexed by $\theta$ if you know the exact probabilities over the sample space then I would argue you do know the functional form.  At the very least you could present the probabilities as a piecewise function.  If you are interested in better understanding the family of distributions indexed by $\theta$ this would not require any sampling.  If you are interested in making inference on, say, the mean of a particular distribution based only on a sample you could use non-parametric or semi-parametric methods.
