How to interpret this zero-one-inflated model? I came across a paper here that has developed the following model:
\begin{align}
Pr(X=0)&=1-\alpha\\
Pr(X=1)&=\alpha\beta\\
Pr(X=x)&=\alpha(1-\beta)f(x;\lambda)\quad; x\geq2,
\end{align}
where $f(x;\theta)$ is the displaced log distribution.
The authors explained the development of the model with assumptions:

*

*Only those families are considered in which at least one birth prior to the survey has occurred.

*At the survey point, a family either has experienced a child loss or not. Let $\alpha$ and $(1-\alpha)$ be the respective proportions.

*Out of $\alpha$ proportion of families, Let $\beta$ be the proportion of families in which only one child death has occurred.

*Remaining $\alpha(1-\beta)$ proportion of families, experiencing multiple child deaths, follows a displaced log distribution with parameter  according to the number of child deaths.

My question is if I have a model like this:
\begin{align}
Pr(X=0)&=(1-\alpha)(1-\beta)\\
Pr(X=1)&=(1-\alpha)\beta\\
Pr(X=x)&=\alpha f(x;\lambda)\quad; x\geq2,
\end{align}
What are reasonable assumptions that I can use so that the model like this makes sense? I think I need two events to explain this model. Possible assumptions:

*

*Let Event A happening with probability $\beta$ and not happening with probability $1-\beta$.

*Out of $\beta$ proportion, Event B happening with probability $1-\alpha$ and not happening with probability $\alpha$.

*The remaining $\alpha$ proportion, experiencing multiple events, follows a distribution with pmf $f(x;\lambda)$.

Therefore, Event X not happening when Event A not happen but Event B happens. Event X happening once when Event A not happening but Event B happening. Event X happening more than once for the remaining proportion and follows a distribution.
Is there any real life example/motivation to this model?

This is the same model --- all you've done is to transform the parameters so that they now refer to different conditional probabilities in the model.  The model constrains the outcomes $x = 2,3,4,...$ to the specified distributional form, but it is otherwise uncontrained.  As to when this model might be applicable, the linked paper applies it as a count model for the number of child deaths in families.  With sufficient amounts of data (may we all hope that this doesn't exist), it would be possible to examine the tails of the empirical count distribution to see if the stipulated model fits the data well.
