AStA Advances in Statistical Analysis

, Volume 93, Issue 1, pp 61–71 | Cite as

CAMCR: Computer-Assisted Mixture model analysis for Capture–Recapture count data

Original Paper


Population size estimation with discrete or nonparametric mixture models is considered, and reliable ways of construction of the nonparametric mixture model estimator are reviewed and set into perspective. Construction of the maximum likelihood estimator of the mixing distribution is done for any number of components up to the global nonparametric maximum likelihood bound using the EM algorithm. In addition, the estimators of Chao and Zelterman are considered with some generalisations of Zelterman’s estimator. All computations are done with CAMCR, a special software developed for population size estimation with mixture models. Several examples and data sets are discussed and the estimators illustrated. Problems using the mixture model-based estimators are highlighted.


CAMCR Capture–recapture Chao’s and Zelterman’s estimator of population size Mixture of truncated Poisson distributions 


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  1. Böhning, D.: Convergence of Simar’s algorithm for finding the maximum likelihood estimate of a compound Poisson process. Ann. Stat. 10, 1006–1008 (1982) MATHCrossRefGoogle Scholar
  2. Böhning, D.: A note on a test for Poisson overdispersion. Biometrika 81, 418–419 (1994) MATHCrossRefMathSciNetGoogle Scholar
  3. Böhning, D.: Computer-Assisted Analysis of Mixtures and Applications. Meta-Analysis, Disease Mapping and Others. Chapman & Hall/CRC, Boca Raton (2000) MATHGoogle Scholar
  4. Böhning, D., Kuhnert, R.: The equivalence of truncated count mixture distributions and mixtures of truncated count distributions. Biometrics 62, 1207–1215 (2006) MATHCrossRefMathSciNetGoogle Scholar
  5. Böhning, D., Suppawattanabodee, B., Kusolvisitkul, W., Viwatwongkasem, C.: Estimating the number of drug users in Bangkok 2001: a capture–recapture approach using repeated entries in one list. Eur. J. Epidemiol. 19, 1075–1083 (2004) CrossRefGoogle Scholar
  6. Bunge, J., Fitzpatrick, M.: Estimating the number of species: A review. J. Am. Stat. Assoc. 88, 364–373 (1993) CrossRefGoogle Scholar
  7. Chao, A.: Estimating the population size for capture–recapture data with unequal catchability. Biometrics 43, 783–791 (1987) MATHCrossRefMathSciNetGoogle Scholar
  8. Chao, A.: Estimating population size for sparse data in capture–recapture experiments. Biometrics 45, 427–438 (1989) MATHCrossRefMathSciNetGoogle Scholar
  9. Dempster, A., Laird, N., Rubin, D.B.: Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). J. R. Stat. Soc. B 39, 1–38 (1977) MATHMathSciNetGoogle Scholar
  10. Horvitz, D.G., Thompson, D.J.: A generalization of sampling without replacement from a finite universe. J. Am. Stat. Assoc. 47, 663–685 (1952) MATHCrossRefMathSciNetGoogle Scholar
  11. Laird, N.: Nonparametric maximum likelihood estimation of a mixing distribution. J. Am. Stat. Assoc. 73, 805–811 (1978) MATHCrossRefMathSciNetGoogle Scholar
  12. Leroux, B.G.: Consistent estimation of a mixing distribution. Ann. Stat. 20, 1350–1360 (1992) MATHCrossRefMathSciNetGoogle Scholar
  13. Lindsay, B.G.: The geometry of mixture likelihoods, Part I: A general theory. Ann. Stat. 11, 783–792 (1983) MATHCrossRefMathSciNetGoogle Scholar
  14. Mao, C.X., Lindsay, B.G.: Diagnostics for the homogeneity of inclusion probabilities in a Bernoulli census. Sankhyā Indian J. Stat. Ser. A 64, 626–639 (2002) MathSciNetGoogle Scholar
  15. Mao, C.X., Lindsay, B.G.: Tests and diagnostics for heterogeneity in the species problem. Comput. Stat. Data Anal. 41, 389–398 (2003) CrossRefMathSciNetGoogle Scholar
  16. McLachlan, G., Krishnan, T.: The EM Algorithm and Extensions. Wiley, New York (1997) MATHGoogle Scholar
  17. McLachlan, G., Peel, D.: Finite Mixture Models. Wiley, New York (2000) MATHGoogle Scholar
  18. Norris, J.L.I., Pollock, K.H.: Nonparametric MLE under two closed capture–recapture models with heterogeneity. Biometrics 52, 639–649 (1996) MATHCrossRefGoogle Scholar
  19. Norris, J.L.I., Pollock, K.H.: Non-parametric MLE for Poisson species abundance models allowing for heterogeneity between species. Environ. Ecol. Stat. 5, 391–402 (1998) CrossRefGoogle Scholar
  20. Oremus, M.: Personal communication (2005) Google Scholar
  21. Schwartz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978) CrossRefGoogle Scholar
  22. Simar, L.: Maximum likelihood estimation of a compound Poisson process. Ann. Stat. 4, 1200–1209 (1976) MATHCrossRefMathSciNetGoogle Scholar
  23. van der Heijden, P.G.M., Bustami, R., Cruy, M., Engbersen, G., van Houwelingen, H.C.: Point and interval estimation of the population size using the truncated Poisson regression model. Stat. Model. Int. J. 3, 305–322 (2003) MATHCrossRefGoogle Scholar
  24. Wilson, R.M., Collins, M.F.: Capture–recapture estimation with samples of size one using frequency data. Biometrika 79, 543–553 (1992) MATHCrossRefGoogle Scholar
  25. Zelterman, D.: Robust estimation in truncated discrete distributions with application to capture–recapture experiments. J. Stat. Plan. Inference 18, 225–237 (1988) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Division for Health of Children and Adolescents, Prevention ConceptsRobert Koch-InstituteBerlinGermany
  2. 2.Quantitative Biology and Applied StatisticsSchool of Biological SciencesReadingUK

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