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

The first zero-inflated model is zero-inflated Poisson model. The zero-inflated Poisson model concerns a random event containing excess zero-count data in unit time.[1] 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 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.

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

Estimators of ZIP

The method of moments estimators are given by

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

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

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

The maximum likelihood estimator[2] can be found by solving the following equation

$\bar{x}(1- e^{-\hat{\lambda}_{ml}}) = \hat{\lambda}_{ml} \left( 1 - \frac{n_0}{n} \right).$

Where $\bar{x}$ is the sample mean, and $\frac{n_0}{n}$ is the observed proportion of zeros.

This can be solved by iteration,[3] and the maximum likelihood estimator for $\pi$ is given by

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

Related models

1994, Greene considered the zero-inflated negative binomial (ZINB) model.[4] Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model.[5]

Discrete pseudo compound Poisson model

If the count data with the feature that the probability of zero is larger than the probability of nonzero, namely

$\Pr (y_i = 0) > 0.5$

then the discrete data $y_i$ obey discrete pseudo compound Poisson distribution.[6]

In fact, let ${G_i(z)}= \sum\limits_{n = 0}^\infty P(y_i = n)z^n$ be the probability generating function of $y_i$. If $p_0=\Pr (y_i = 0) > 0.5$, then $\left| {G_i(z)} \right| \geqslant {p_0} - \sum\limits_{i = 1}^\infty {{p_i}} = 2{p_0}-1 > 0$. Then from Wiener-Lévy theorem,[7] we show that ${G(z)}$ have the probability generating function of discrete pseudo compound Poisson distribution.

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

$G_Y(z) = \sum\limits_{n = 0}^\infty P(Y = n)z^n = \exp\left(\sum\limits_{k = 1}^\infty \alpha_k \lambda (z^k - 1)\right), \quad (|z| \le 1)$

has a discrete pseudo compound Poisson distribution with parameters $(\alpha_1 \lambda,\alpha_2 \lambda, \ldots ) \in \mathbb{R}^\infty \left( {\sum\limits_{k = 1}^\infty {{\alpha _k}} = 1,\sum\limits_{k = 1}^\infty {\left| {{\alpha _k}} \right|} < \infty ,{\alpha _k} \in {\rm{R}},\lambda > 0} \right)$.

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