# Zero-inflated model

In statistics, a zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e. a distribution that allows for frequent zero-valued observations.

The zero-inflated Poisson model concerns a random event containing excess zero-count data in unit time.[1] For example, the number of claims to an insurance company by any given covered person is almost always zero, otherwise substantial losses would cause the insurance company to go bankrupt. The zero-inflated Poisson (ZIP) model employs two components that correspond to two zero generating processes. The first process is governed by a binary distribution that generates structural zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The two model components are described as follows:

$\Pr (y_j = 0) = \pi + (1 - \pi) e^{-\lambda}$
$\Pr (y_j = h_i) = (1 - \pi) \frac{\lambda^{h_i} e^{-\lambda}} {h_i!},\qquad h_i \ge 1$

where the outcome variable $y_j$ has any non-negative integer value, $\lambda_i$ is the expected Poisson count for the $i$th individual; $\pi$ is the probability of extra zeros.

## Related properties

In 1994, Greene considered the zero-inflated negative binomial (ZINB) model.[2] Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model.[3]