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

Histogram of a zero-inflated Poisson distribution

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 (ZIP) 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 distribution is described as follows:

$\Pr(Y=0)=\pi +(1-\pi )e^{-\lambda }$

$\Pr(Y=y_{i})=(1-\pi ){\frac {\lambda ^{y_{i}}e^{-\lambda }}{y_{i}!}},\qquad y_{i}=1,2,3,...$

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

The mean is $(1-\pi )\lambda$ and the variance is $\lambda (1-\pi )(1+\pi \lambda )$.

Estimators of ZIP parameters

The method of moments estimators are given by

${\hat {\lambda }}_{mo}={\frac {s^{2}+m^{2}}{m}}-1,$

${\hat {\pi }}_{mo}={\frac {s^{2}-m}{s^{2}+m^{2}-m}},$

where $m$ is the sample mean and $s^{2}$ is the sample variance.

The maximum likelihood estimator can be found by solving the following equation

$m(1-e^{-{\hat {\lambda }}_{ml}})={\hat {\lambda }}_{ml}\left(1-{\frac {n_{0}}{n}}\right).$

where ${\frac {n_{0}}{n}}$ is the observed proportion of zeros.

A closed form solution of this equation is given by

${\hat {\lambda }}_{ml}=W_{0}(-se^{-s})+s$

with $W_{0}$ being the main branch of Lambert's W-function and

$s={\frac {m}{1-{\frac {n_{0}}{n}}}}$.

Alternatively, the equation can be solved by iteration.

The maximum likelihood estimator for $\pi$ is given by

${\hat {\pi }}_{ml}=1-{\frac {m}{{\hat {\lambda }}_{ml}}}.$

Related models

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

Discrete pseudo compound Poisson model

If the count data $Y$ is such that the probability of zero is larger than the probability of nonzero, namely

$\Pr(Y=0)>0.5$

then the discrete data $Y$ obey discrete pseudo compound Poisson distribution.

In fact, let $G(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}$ be the probability generating function of $y_{i}$. If $p_{0}=\Pr(Y=0)>0.5$, then $|G(z)|\geqslant p_{0}-\sum \limits _{i=1}^{\infty }p_{i}=2p_{0}-1>0$. Then from the Wiener–Lévy theorem, $G(z)$ has the probability generating function of the discrete pseudo compound Poisson distribution.

We say that the discrete random variable $Y$ satisfying probability generating function characterization

$G_{Y}(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}=\exp \left(\sum _{k=1}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)$

has a discrete pseudo compound Poisson distribution with parameters

$(\lambda _{1},\lambda _{2},\ldots )=(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left(\sum _{k=1}^{\infty }\alpha _{k}=1,\sum \limits _{k=1}^{\infty }|\alpha _{k}|<\infty ,\alpha _{k}\in \mathbb {R} ,\lambda >0\right).$

When all the $\alpha _{k}$ are non-negative, it is the discrete compound Poisson distribution (non-Poisson case) with overdispersion property.

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See also

• Poisson distribution
• Zero-truncated Poisson distribution
• Compound Poisson distribution
• Sparse approximation
• Hurdle model

Software

• pscl and brms R packages