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 binomial-distribution, 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 beta-binomial 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.
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. $\endgroup$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. $\endgroup$