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.

• I wrote a grant proposal once to work on software for this, but it got turned down. – Peter Flom Jul 3 '14 at 9:34
• I expect there's little to say than there's less call for it so people don't mention it as much, but it's not like it wouldn't ever have been done; it may not have been discussed much. The ideas aren't really introducing much that's substantively different from the common 0-case. – Glen_b Aug 25 '16 at 4:03

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.