I'm trying to model some discrete data that's under-dispersed enough that the Poisson distribution doesn't seem to fit. (That is, the variance is significantly less than the mean.)
If the data were over-dispersed, I'd consider the negative binomial distribution, but I don't know of a similarly obvious choice for under-dispersion.
I'm aware of a generalized Poisson distribution that can be over- or under-dispersed, but the under-dispersed case seems to have finite support. I'm looking for something that supports the non-negative (or at least positive) integers; otherwise, the binomial distribution would be a reasonable option.
Sharing the property that a sum of Poisson-distributed variables is also Poisson-distributed would be nice but not necessary. Basically I'm interested in any distributions that have been studied somewhat and have density functions that are easy to work with. (I know, these are fairly vague criteria!)