Why are there no one-inflated count data models? I am working on zero-inflated count data models using the pscl package. I am just wondering why there is no development of models for one-inflated count data models! Also why there is no development of bimodal, say zero-and-2-inflated, count data models! Once I generated one-inflated Poisson data and found that neither the glm with family=poisson model nor the negative binomial (glm.nb) model was good enough to fit the data well. If any one can shed some light on my thought, eccentric though it might be, it would be very helpful for me.
 A: The R package VGAM has function vglm which can be used to fit all sorts of Poisson-esque models. You can use it to specify a one-inflated model, so something like vglm(Y~X,family=oipospoisson(),data=data). See here for more details. 
A: A one-inflated Poisson model for a count $Y_i$ is
$$\begin{align}\Pr(Y_i = 1) &= \pi_i +(1-\pi_i)\cdot\mu_i\mathrm{e}^{-\mu_i}\\
\Pr(Y_i = y_i) &= (1-\pi_i)\cdot\frac{\mu_i^{y_i}\mathrm{e}^{-\mu_i}}{y_i!} \qquad \text{when } y_i\neq 1
\end{align}$$
where the Poisson mean $\mu_i$ & Bernoulli probability $\pi_i$ are related to the predictors through appropriate link functions. You can define a similar model to inflate probabilities for any values you choose.
Still, zero has a special (& once controversial) place among the counting numbers—in a sense representing the absence of anything to count. And it's the "nothing" vs "something" distinction, rather than the "one" vs "any other count" distinction that tends to be relevant across a wide range of phenomena we like to model: there's one process that gives a nought, one, two, ... count & another that gives no count at all.
