Suppose we have some count data, and we want to use a model that allows for "overdispersion" or "underdispersion" in the data (i.e., higher or lower variance than the Poisson distribution). Let $X$ be our count variable and let $\phi = \mathbb{E}(X)/\mathbb{S}(X)$ denote the coefficient of variation. Overdispersed data is often modelled using the negative-binomial distribution and both over or underdispersed data can be modelled using the generalised Poisson distribution. However, both of these distributions still have a restricted range for the allowable coefficient of variation. Specifically, for a given mean $\mu$, the allowable values of the coefficient of variation under each distribution (allowing for corner values) are:

$$\begin{aligned} &\text{Poisson} & & & \phi &= 1, \\[10pt] &\text{Negative Binomial} & & & \phi &\geqslant 1, \\[6pt] &\text{Generalised Poisson} & & & \phi &\geqslant \max \big( \tfrac{1}{4}, \tfrac{(4-\mu)^2}{4} \big), \\[6pt] \end{aligned}$$

The generalised Poisson distribution is quite a good generalisation of the Poisson, and can be used for count data, but it still has a restricted value for the coefficient of variation that does not allow you to model count data that is highly underdispersed. If the dispersion is extremely low then even the generalised Poisson would not model it well.

Question: Is there any distribution generalising the Poisson distribution, that can reasonably be used for modelling count data, and which has an unrestricted range for the coefficient of variation (i.e., it allows $\phi \geqslant 0$)?


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