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.

Zero-inflated Poisson

One well-known zero-inflated model is Diane Lambert's zero-inflated Poisson model, which concerns a random event containing excess zero-count data in unit time. For example, the number of insurance claims within a population for a certain type of risk would be zero-inflated by those people who have not taken out insurance against the risk and thus are unable to claim. The zero-inflated Poisson model mixes two zero generating processes. The first process generates zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The mixture is described as follows:
where the outcome variable has any non-negative integer value, is the expected Poisson count for the th individual; is the probability of extra zeros.
The mean is and the variance is.

Estimators of ZIP Parameters

The method of moments estimators are given by
where is the sample mean and is the sample variance.
The maximum likelihood estimator can be found by solving the following equation
where is the observed proportion of zeros.
A closed form solution of this equation is given by
with being the main branch of Lambert's W-function and
Alternatively, the equation can be solved by iteration.
The maximum likelihood estimator for is given by

Related models

1994, Greene considered the zero-inflated negative binomial model. Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial model.

Discrete pseudo compound Poisson model

If the count data is such that the probability of zero is larger than the probability of nonzero, namely
then the discrete data obey discrete pseudo compound Poisson distribution.
In fact, let be the probability generating function of. If, then. Then from the Wiener–Lévy theorem, has the probability generating function of the discrete pseudo compound Poisson distribution.
We say that the discrete random variable satisfying probability generating function characterization
has a discrete pseudo compound Poisson distribution with parameters
When all the are non-negative, it is the discrete compound Poisson distribution with overdispersion property.