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. 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
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 sample mean, and is the observed proportion of zeros.
This can be solved by iteration, and the maximum likelihood estimator for is given by
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 with the feature that the probability of zero is larger than the probability of nonzero, namely
In fact, let be the probability generating function of . If , then . Then from Wiener–Lévy theorem, 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
- Poisson distribution
- Zero-truncated Poisson distribution
- Compound Poisson distribution
- Sparse approximation
- 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.
- Johnson, Norman L.; Kotz, Samuel; Kemp, Adrienne W. (1992). Univariate Discrete Distributions (2nd ed.). Wiley. pp. 312–314. ISBN 0-471-54897-9.
- 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.
- 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.
- 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.
- 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.
- Zygmund, A. (2002). Trigonometric series. Cambridge: Cambridge University Press. p. 245.