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.[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 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 is 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 parameters[]
The method of moments estimators are given by[2]
where is the sample mean and is the sample variance.
The maximum likelihood estimator[3] can be found by solving the following equation
where is the observed proportion of zeros.
A closed form solution of this equation is given by[4]
with being the main branch of Lambert's W-function[5] and
- .
Alternatively, the equation can be solved by iteration.[6]
The maximum likelihood estimator for is given by
Related models[]
In 1994, Greene considered the zero-inflated negative binomial (ZINB) model.[7] Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model.[8]
Discrete pseudo compound Poisson model[]
If the count data is such that the probability of zero is larger than the probability of nonzero, namely
then the discrete data obey discrete pseudo compound Poisson distribution.[9]
In fact, let be the probability generating function of . If , then . Then from the Wiener–Lévy theorem,[10] has the probability generating function of the 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[]
- Poisson distribution
- Zero-truncated Poisson distribution
- Compound Poisson distribution
- Sparse approximation
- Hurdle model
Software[]
References[]
- ^ 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.
- ^ Beckett, Sadie; Jee, Joshua; Ncube, Thalepo; Washington, Quintel; Singh, Anshuman; Pal, Nabendu (2014). "Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to model data from natural calamities". Involve. 7 (6): 751–767. doi:10.2140/involve.2014.7.751.
- ^ Johnson, Norman L.; Kotz, Samuel; Kemp, Adrienne W. (1992). Univariate Discrete Distributions (2nd ed.). Wiley. pp. 312–314. ISBN 978-0-471-54897-3.
- ^ Dencks, Stefanie; Piepenbrock, Marion; Schmitz, Georg (2020). "Assessing Vessel Reconstruction in Ultrasound Localization Microscopy by Maximum-Likelihood Estimation of a Zero-Inflated Poisson Model". IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. doi:10.1109/TUFFC.2020.2980063.
- ^ Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W Function". Advances in Computational Mathematics. 5 (1): 329–359. arXiv:1809.07369. doi:10.1007/BF02124750.
- ^ 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. 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.
- Generalized linear models
- Categorical data
- Poisson point processes