# Is there a common underdispersed discrete distribution with unbounded support for general mean and variance?

I have a mean $$\mu$$ and a variance $$\sigma^2$$ with underdispersion, i.e., $$\sigma^2<\mu$$. Is there a standard discrete distribution with these moments and unbounded-on-the-right support, i.e., support on $$\{0, 1, \dots\}$$?

Bonus points if it is implemented in R.

• I looked at the , but that is only defined if the size parameter $$\frac{\mu}{1-\frac{\sigma^2}{\mu}}$$ is an integer.
• The binomial and binomial compounds like the have bounded support.
• So does the Generalized Poisson distribution (Consul & Jain, 1973) in the case of underdispersion, plus it can only handle underdispersion to a certain degree (note that Consul & Jain require $$|\lambda_2|<1$$ in formula (3.1)). The Generalized Poisson is Joseph Hilbe's main recommendation in this answer of his. His other recommendations might be useful, but he gives no details on them, and searching for the names is not very successful.
• Sampling from under/over-dispersed count data in R is related but does not have a helpful answer.
• Quasi-Poisson models sound like they may be useful (e.g., here), but I haven't been able to find anything helpful outside the context of a regression.
• Hilbe's answer to this question stats.stackexchange.com/questions/67385/… might be helpful. Commented Oct 17, 2018 at 16:27
• Not familiar with applications of this, but how about $\sqrt{X},$ where $X \sim \mathsf{Pois}(\lambda)?$ set.seed(1017); x=rpois(10^5,3); mean(sqrt(x)); var(sqrt(x)) returns $1.62883 > 0.3404474.$ Round or take floor if you need integers. Commented Oct 17, 2018 at 17:33
• @jbowman: Hilbe's main suggestion is the Generalized Poisson, which has bounded support if underdispersed. (I did find his answer when writing this question, upvoted it and looked through his Negative Binomial Regression.) His other suggestions might be helpful if there were just a few pointers to literature. Commented Oct 17, 2018 at 17:34
• The Conway-Maxwell-Poisson distribution has unbounded support and R packages, including one for regression, so may fit your requirements: en.wikipedia.org/wiki/…. When I looked at it some years ago, computation was very slow, but it looks like much faster algorithms have been developed since then. Commented Oct 17, 2018 at 17:50
• There are many possible answers. The most trivial is to add a sufficiently large integer to any lattice variable, thereby increasing $\mu$ without changing $\sigma^2.$ Even restricting distributions to those implemented in R, you could (for instance) reduce the dispersion by using $\lfloor X/k \rfloor$ for $k\gt 1$ with $X$ any non-negative variable (discrete or not). As in almost all such cases, it's likely more constructive to articulate the statistical problem you are trying to solve so that a suitable choice of distribution (family) can be made.
– whuber
Commented Jan 18, 2021 at 20:00

The Conway-Maxwell-Poisson distribution (https://en.wikipedia.org/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution) has unbounded support (on the right) and can model both under- and over-dispersion (relative to the Poisson) seamlessly through the use of a single parameter. The Poisson is a special case. It can't handle any amount of underdispersion, though, and it is relatively computationally intensive. It is a member of the exponential family of distributions.

R packages exist for both estimation and regression:

https://cran.r-project.org/web/packages/CompGLM/CompGLM.pdf

https://cran.r-project.org/web/packages/COMPoissonReg/COMPoissonReg.pdf

https://cran.r-project.org/web/packages/compoisson/compoisson.pdf

although I haven't used them so cannot make any helpful comments about their relative quality / usefulness!

• From my recent experience, Generalized Poisson regression is not well supported in R, CompPoissonReg is well established and good and DGLMExtPois (cran.r-project.org/web/packages/DGLMExtPois/index.html) is useful because it has the mean parametrized CMP, i.e. results are comparable to those from a regular GLM. Commented Jan 11 at 23:06