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Population size estimation and heterogeneity in capture–recapture data: a linear regression estimator based on the Conway–Maxwell–Poisson distribution

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Abstract

The purpose of the study is to estimate the population size under a truncated count model that accounts for heterogeneity. The proposed estimator is based on the Conway–Maxwell–Poisson distribution. The benefit of using the Conway–Maxwell–Poisson distribution is that it includes the Bernoulli, the Geometric and the Poisson distributions as special cases and, furthermore, allows for heterogeneity. Parameter estimates can be obtained by exploiting the ratios of successive frequency counts in a weighted linear regression framework. The results of the comparisons with Turing’s, the maximum likelihood Poisson, Zelterman’s and Chao’s estimators reveal that our proposal can be beneficially used. Furthermore, our proposal outperforms its competitors under all heterogeneous settings. The empirical examples consider the homogeneous case and several heterogeneous cases, each with its own features, and provide interesting insights on the behavior of the estimators.

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References

  • Alunni-Fegatelli D, Tardella L (2013) Improved inference on capture recapture models with behavioural effects. Stat Methods Appl 22:45–66

    Article  MathSciNet  MATH  Google Scholar 

  • Baksh MF, Böhning D, Lerdsuwansri R (2011) An extension of an over-dispersion test for count data. Comput Stat Data Anal 55:466–474

    Article  MathSciNet  MATH  Google Scholar 

  • Bartolucci F, Forcina A (2006) A class of latent marginal models for capturerecapture data with continuous covariates. J Am Stat Assoc 101:786–794

    Article  MathSciNet  MATH  Google Scholar 

  • Blumenthal JA, Williams RB, Kong Y, Schanberg SM, Thompson LW (1978) Type A behavior pattern and coronary atherosclerosis. Circulation 58:634–639

    Article  Google Scholar 

  • Böhning D, Dietz E, Kuhnert R, Schön D (2005) Mixture models for capture–recapture count data. Stat Methods Appl 14:29–43

    Article  MathSciNet  MATH  Google Scholar 

  • Böhning D, Schön D (2005) Nonparametric maximum likelihood estimation of population size based on the counting distribution. J R Stat Soc Ser C 54:721–737

    Article  MathSciNet  MATH  Google Scholar 

  • Böhning D (2008) A simple variance formula for population size estimators by conditioning. Stat Methodol 5:410–423

    Article  MathSciNet  MATH  Google Scholar 

  • Böhning D, Baksh MF, Lerdsuwansri R, Gallagher J (2013) Use of the ratio plot in capturerecapture estimation. J Comput Gr Stat 22:135–155

    Article  MathSciNet  Google Scholar 

  • Borchers DL, Buckland ST (2002) Estimating animal abundance: closed populations. Springer, New York

    Book  MATH  Google Scholar 

  • Bunge J, Barger K (2008) Parametric models for estimating the number of classes. Biom J 50:971–982

    Article  MathSciNet  Google Scholar 

  • Chao A (1987) Estimating the population size for capture–recapture data with unequal catchability. Biometrics 43:783–791

    Article  MathSciNet  MATH  Google Scholar 

  • Chao A (1989) Estimating population size for sparse data in capture–recapture experiments. Biometrics 45:427–438

    Article  MathSciNet  MATH  Google Scholar 

  • Chiu CH, Wang YT, Walther BA, Chao A (2014) An improved non-parametric lower bound of species richness via a modified good-turing frequency formula. Biometrics 70:671–682

    Article  MathSciNet  MATH  Google Scholar 

  • Dorazio RM, Royle AJ (2003) Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics 59:351–364

    Article  MathSciNet  MATH  Google Scholar 

  • Farcomeni A (2011) Recapture models under equality constraint. Biometrika 98:237–242

    Article  MathSciNet  MATH  Google Scholar 

  • Farcomeni A, Scacciatelli D (2013) Heterogeneity and behavioural response in continuous time capture–recapture, with application to street cannabis use in Italy. Ann Appl Stat 7:2293–2314

    Article  MathSciNet  MATH  Google Scholar 

  • Gerritse S, van der Heijden PGM, Bakker B (2015) Sensitivity of population size estimation for violating parametric assumptions in loglinear models. J Off Stat 31:357–379

    Google Scholar 

  • Guikema SD, Coffelt JP (2008) A flexible count data regression model for risk analysis. Risk Anal 28:213–223

    Article  Google Scholar 

  • Kuhnert R, Böhning D (2009) CAMCR: computer-assisted mixture model analysis for CaptureRecapture count data. AStA Adv Stat Anal 93:61–71

    Article  MathSciNet  MATH  Google Scholar 

  • Lanumteang K (2011) Estimating of size of a target population using capture–recapture methods based upon multi-source and continuous time experiments. Ph.D. thesis, University of Reading

  • Lanumteang K, Böhning D (2011) An extension of Chao’s estimator of population size based on the first three capture frequency counts. Comput Stat Data Anal 55:2302–2311

    Article  MathSciNet  MATH  Google Scholar 

  • Lerdsuwansri R (2012) Generlisation of the Lincoln-Petersen approach to non-binary source variable. Ph.D. thesis, University of Reading

  • Lindsay BG, Roeder K (1987) A unified treatment of integer parameter models. J Am Stat Assoc 82:758–764

    Article  MathSciNet  MATH  Google Scholar 

  • Link WA (2003) Nonidentifiability of population size from capture-recapture data with heterogeneous detection probabilities. Biometrics 59:1123–1130

    Article  MathSciNet  MATH  Google Scholar 

  • Mao CX, Lindsay BG (2003) Tests and diagnostics for heterogeneity in the species problem. Comput Stat Data Anal 41:389–398

    Article  MathSciNet  MATH  Google Scholar 

  • McCrea RS, Morgan BJT (2014) Analysis of capture–recapture data. CRC Press, Boca Raton

    MATH  Google Scholar 

  • McKendrick AG (1926) Application of mathematics to medical problems. Proc Edinb Math Soc 44:98–130

    Article  Google Scholar 

  • Meurant G (1992) A review on the inverse of symmetric tridiagonal and block matrices. SIAM J Matrix Anal Appl 13:707–728

    Article  MathSciNet  MATH  Google Scholar 

  • Morgan BJT, Ridout MS (2008) A new mixture model for capture heterogeneity. J R Stat Soc Ser C 57:433–446

    Article  MathSciNet  MATH  Google Scholar 

  • Niwitpong SA, Böhning D, van der Heijden PG, Holling H (2013) Capturerecapture estimation based upon the geometric distribution allowing for heterogeneity. Metrika 76:495–519

    Article  MathSciNet  MATH  Google Scholar 

  • Pledger S (2005) The performance of mixture models in heterogeneous closed population capturerecapture. Biometrics 61:868–873

    Article  MathSciNet  Google Scholar 

  • Rocchetti I, Bunge J, Böhning D (2011) Population size estimation based upon ratios of recapture probabilities. Ann Appl Stat 5:1512–1533

    Article  MathSciNet  MATH  Google Scholar 

  • Rocchetti I, Alfó M, Böhning D (2014) A regression estimator for mixed binomial capture–recapture data. J Stat Plan Inference 145:165–178

    Article  MathSciNet  MATH  Google Scholar 

  • Scollnik DP (1997) Inference concerning the size of the zero class from an incomplete Poisson sample. Commun Stat Theory Methods 26:221–236

    Article  MathSciNet  MATH  Google Scholar 

  • Shmueli G, Minka TP, Kadane JB, Borle S, Boatwright P (2005) A useful distribution for fitting discrete data: revival of the ConwayMaxwellPoisson distribution. J R Stat Soc Ser C 54:127–142

    Article  MathSciNet  MATH  Google Scholar 

  • van Der Heijden PG, Bustami R, Cruyff MJ, Engbersen G, Van Houwelingen HC (2003) Point and interval estimation of the population size using the truncated Poisson regression model. Stat Model 3:305–322

    Article  MathSciNet  MATH  Google Scholar 

  • Viwatwongkasem C, Kunhert R, Sativipawee P (2008) A Comparison of population size estimators under the truncated count model with and without allowance for contaminations. Biom J 50:1006–1021

    Article  MathSciNet  Google Scholar 

  • Zelterman D (1988) Robust estimation in truncated discrete distributions with application to capture-recapture experiments. J Stat Plan Inference 18:225–237

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Authors would like to thank the reviewers for the very helpful comments which lead to considerable improvement of the paper as well as to interesting and promising modifications of the proposed estimator. Authors are grateful to the Ministry of Science and Technology, the Royal Thai Government for providing Ph.D. funding for the first author.

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Authors and Affiliations

  1. School of Mathematics and Southampton Statistical Sciences Research Institute, University of Southampton, Southampton, UK

    Orasa Anan & Dankmar Böhning

  2. Dipartimento di Scienze Economiche, Politiche e delle Lingue Moderne, Libera Universitá Maria Ss. Assunta, Via Pompeo Magno 22, 001921, Roma, Italy

    Antonello Maruotti

  3. Centre for Innovation and Leadership in Health Sciences, University of Southampton, Southampton, UK

    Antonello Maruotti

Authors
  1. Orasa Anan
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  2. Dankmar Böhning
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  3. Antonello Maruotti
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Correspondence to Antonello Maruotti.

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Anan, O., Böhning, D. & Maruotti, A. Population size estimation and heterogeneity in capture–recapture data: a linear regression estimator based on the Conway–Maxwell–Poisson distribution. Stat Methods Appl 26, 49–79 (2017). https://doi.org/10.1007/s10260-016-0358-7

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  • DOI: https://doi.org/10.1007/s10260-016-0358-7

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