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 (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:

:$\Pr (Y = 0) = \pi + (1 - \pi) e^$
:$\Pr (Y = y_i) = (1 - \pi) \frac ,\qquad y_i = 1,2,3,...$

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

: $\hat_ = \frac - 1,$

: $\hat_ = \frac,$

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

: $m(1- e^) = \hat_ \left( 1 - \frac \right).$

where $\frac$ is the observed proportion of zeros.

A closed form solution of this equation is given by

:$\hat_ = W_(-s e^)+s$

with $W_0$ being the main branch of Lambert's W-function and

:$s = \frac$.

Alternatively, the equation can be solved by iteration.

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

: $\hat_ = 1 - \frac.$

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 $G(z) = \sum\limits_^\infty P(Y = n)z^n$ be the probability generating function of $y_i$. If $p_0=\Pr (Y = 0) > 0.5$, then $, G(z), \geqslant p_0 - \sum\limits_^\infty p_i = 2p_0-1 > 0$. 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

:$G_Y(z) = \sum\limits_^\infty P(Y = n)z^n = \exp\left(\sum_^\infty \alpha_k \lambda (z^k - 1)\right), \quad (, z, \le 1)$

has a discrete pseudo compound Poisson distribution with parameters

: $(\lambda_1, \lambda_2, \ldots ) = (\alpha_1 \lambda,\alpha_2 \lambda, \ldots ) \in \mathbb^\infty \left( \sum_^\infty \alpha _k = 1, \sum\limits_^\infty , \alpha_k, < \infty, \alpha_k \in \mathbb,\lambda > 0 \right).$

When all the $\alpha_k$ are non-negative, it is the discrete compound Poisson distribution (non-Poisson case) with overdispersion property.

See also

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

Software

pscl
an
brms
R packages

References

{{least squares and regression analysis

Generalized linear models
Categorical data
Poisson point processes