# Underdispersed Poisson-like discrete probability distribution

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!)

• Have you heard of the Conway-Maxwell-Poisson distribution (not computationally super nice)? Apr 19, 2021 at 9:06
• @Memming I have. I hadn't really considered it for this because it's cumbersome to work with, but now I'm rethinking that. I'll have to play with it and see how it does on my data. Apr 21, 2021 at 13:21
• Do you have a response to the answer below? Nov 23, 2021 at 15:41
• @iosif-pinelis while that family could certainly fit my data, I'm not sure how well it meets the criteria of "have been studied somewhat" or "have density functions that are easy to work with". So far the CMP distribution seems like the best choice, although the lack of a closed form means it isn't quite as easy to work with as I'd hoped. Nov 23, 2021 at 19:24

Let $$Z:=X+Y$$, where $$X$$ and $$Y$$ are independent random variables, $$X$$ has the binomial distribution with parameters $$n$$ and $$p\in(0,1)$$, and $$Y$$ has the Poisson distribution with parameter $$\lambda$$. Then the support of the distribution of $$X$$ is the set of all nonnegative integers, and the ratio $$\frac{EZ}{Var\,Z}=\frac{np+\lambda}{np(1-p)+\lambda}$$ can be made however large by choosing $$p$$ to be close to $$1$$ and at the same time choosing $$\lambda$$ to be much smaller than $$n$$.
Moreover, all the moments of $$Z$$, as well its moment generating function, are easy to express, and the family of the distributions of $$Z$$, depending on the three parameters -- $$n,p,\lambda$$ -- seems rich and flexible enough for modeling.