Zero-inflated model

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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[edit]

The first zero-inflated model is Diane Lambert's zero-inflated Poisson model, which 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:

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[edit]

The method of moments estimators are given by

where is the sample mean and is the sample variance.

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

where is the sample mean, and is the observed proportion of zeros.

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

Related models[edit]

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[edit]

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

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

In fact, let be the probability generating function of . If , then . Then from Wiener–Lévy theorem,[7] we show that have the probability generating function of 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 (non-Poisson case) with overdispersion property.

See also[edit]


  1. ^ Lambert, Diane (1992). "Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing". Technometrics. 34 (1): 1–14. doi:10.2307/1269547. JSTOR 1269547.
  2. ^ Johnson, Norman L.; Kotz, Samuel; Kemp, Adrienne W. (1992). Univariate Discrete Distributions (2nd ed.). Wiley. pp. 312–314. ISBN 0-471-54897-9.
  3. ^ Böhning, Dankmar; Dietz, Ekkehart; Schlattmann, Peter; Mendonca, Lisette; Kirchner, Ursula (1999). "The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology". Journal of the Royal Statistical Society, Series A. Wiley Online Library. 162 (2): 195–209. doi:10.1111/1467-985x.00130.
  4. ^ Greene, William H. (1994). "Some Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models". Working Paper EC-94-10: Department of Economics, New York University. SSRN 1293115.
  5. ^ Hall, Daniel B. (2000). "Zero-Inflated Poisson and Binomial Regression with Random Effects: A Case Study". Biometrics. 56 (4): 1030–1039. doi:10.1111/j.0006-341X.2000.01030.x.
  6. ^ Huiming, Zhang; Yunxiao Liu; Bo Li (2014). "Notes on discrete compound Poisson model with applications to risk theory". Insurance: Mathematics and Economics. 59: 325–336. doi:10.1016/j.insmatheco.2014.09.012.
  7. ^ Zygmund, A. (2002). Trigonometric series. Cambridge: Cambridge University Press. p. 245.