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.
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    $\begingroup$ Hilbe's answer to this question stats.stackexchange.com/questions/67385/… might be helpful. $\endgroup$ – jbowman Oct 17 '18 at 16:27
  • $\begingroup$ 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. $\endgroup$ – BruceET Oct 17 '18 at 17:33
  • $\begingroup$ @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. $\endgroup$ – Stephan Kolassa Oct 17 '18 at 17:34
  • $\begingroup$ @BruceET: thanks, but that presupposes a very specific relationship between the mean and the variance, and I'd like for something that works for general moments. $\endgroup$ – Stephan Kolassa Oct 17 '18 at 17:35
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    $\begingroup$ 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. $\endgroup$ – jbowman Oct 17 '18 at 17:50

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:




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


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