Zero-Inflated, Zero-Altered and Positive Discrete Distributions

  • Thomas W. Yee
Part of the Springer Series in Statistics book series (SSS)


This chapter looks at positive (0-truncated), zero-inflated and zero-altered (hurdle) distributions, with the focus on discrete distributions. Zero-deflated distributions are also mentioned. Specific examples include the zero-inflated Poisson and positive-binomial distributions. Another example concerns closed-population capture–recapture estimation, which is described in relatively more detail as an application of a positive-Bernoulli distribution. Reduced-rank variants of some of the above models are also considered.


Wing Length Sampling Occasion Family Function Parent Distribution Deer Mouse 
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  1. Amstrup, S. C., T. L. McDonald, and B. F. J. Manly 2005. Handbook of Capture–Recapture Analysis. Princeton: Princeton University Press.Google Scholar
  2. Baillargeon, S. and L.-P. Rivest 2007. Rcapture: Loglinear models for capture–recapture in R. Journal of Statistical Software 19(5):1–31.CrossRefGoogle Scholar
  3. Burnham, K. P. and D. R. Anderson 2002. Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach (Second ed.). New York: Springer.Google Scholar
  4. Cameron, A. C. and P. K. Trivedi 2013. Regression Analysis of Count Data (Second ed.). Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  5. Hilbe, J. M. 2011. Negative Binomial Regression (Second ed.). Cambridge, UK; New York, USA: Cambridge University Press.Google Scholar
  6. Horvitz, D. G. and D. J. Thompson 1952. A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47(260):663–685.zbMATHMathSciNetCrossRefGoogle Scholar
  7. Huggins, R. and W.-H. Hwang 2011. A review of the use of conditional likelihood in capture–recapture experiments. International Statistical Review 79(3):385–400.zbMATHCrossRefGoogle Scholar
  8. Huggins, R. M. 1989. On the statistical analysis of capture experiments. Biometrika 76(1):133–140.zbMATHMathSciNetCrossRefGoogle Scholar
  9. Huggins, R. M. 1991. Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics 47(2):725–732.CrossRefGoogle Scholar
  10. Hwang, W.-H. and R. Huggins 2011. A semiparametric model for a functional behavioural response to capture in capture–recapture experiments. Australian & New Zealand Journal of Statistics 53(4):403–421.MathSciNetCrossRefGoogle Scholar
  11. Johnson, N. L., A. W. Kemp, and S. Kotz 2005. Univariate Discrete Distributions (Third ed.). Hoboken, NJ, USA: John Wiley & Sons.zbMATHCrossRefGoogle Scholar
  12. Kleiber, C. and A. Zeileis 2008. Applied Econometrics with R. New York, USA: Springer.zbMATHCrossRefGoogle Scholar
  13. Liu, H. and K. S. Chan 2010. Introducing COZIGAM: An R package for unconstrained and constrained zero-inflated generalized additive model analysis. Journal of Statistical Software 35(11):1–26.CrossRefGoogle Scholar
  14. McCrea, R. S. and B. J. T. Morgan 2015. Analysis of Capture–Recapture Data. Boca Raton, FL, USA: Chapman & Hall/CRC.Google Scholar
  15. Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson 1978. Statistical inference from capture data on closed animal populations. Wildlife Monographs 62:3–135.Google Scholar
  16. Tutz, G. 2012. Regression for Categorical Data. Cambridge: Cambridge University Press.Google Scholar
  17. Vuong, Q. H. 1989. Likelihood ratio tests for model selection and nonnested hypotheses. Econometrica 57(2):307–333.zbMATHMathSciNetCrossRefGoogle Scholar
  18. Webb, M. H., S. Wotherspoon, D. Stojanovic, R. Heinsohn, R. Cunningham, P. Bell, and A. Terauds 2014. Location matters: Using spatially explicit occupancy models to predict the distribution of the highly mobile, endangered swift parrot. Biological Conservation 176:99–108.CrossRefGoogle Scholar
  19. Welsh, A. H., R. B. Cunningham, C. F. Donnelly, and D. B. Lindenmayer 1996. Modelling the abundances of rare species: statistical models for counts with extra zeros. Ecological Modelling 88(1–3):297–308.CrossRefGoogle Scholar
  20. Welsh, A. H., D. B. Lindenmayer, and C. F. Donnelly 2013. Fitting and interpreting occupancy models. PLOS One 8(1):1–21.CrossRefGoogle Scholar
  21. Williams, B. K., J. D. Nichols, and M. J. Conroy 2002. Analysis and Management of Animal Populations. London: Academic Press.Google Scholar
  22. Winkelmann, R. 2008. Econometric Analysis of Count Data (5th ed.). Berlin: Springer.Google Scholar
  23. Winkelmann, R. and S. Boes 2006. Analysis of Microdata. Berlin: Springer.zbMATHGoogle Scholar
  24. Yang, H.-C. and A. Chao 2005. Modeling animals’ behavioral response by Markov chain models for capture–recapture experiments. Biometrics 61(4):1010–1017.zbMATHMathSciNetCrossRefGoogle Scholar
  25. Yee, T. W. 2014. Reduced-rank vector generalized linear models with two linear predictors. Computational Statistics & Data Analysis 71:889–902.MathSciNetCrossRefGoogle Scholar
  26. Yee, T. W., J. Stoklosa, and R. M. Huggins 2015. The VGAM package for capture–recapture data using the conditional likelihood. Journal of Statistical Software 65(5):1–33.CrossRefGoogle Scholar

Copyright information

© Thomas Yee 2015

Authors and Affiliations

  • Thomas W. Yee
    • 1
  1. 1.Department of StatisticsUniversity of AucklandAucklandNew Zealand

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