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Inference on population size in binomial detectability models

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  • R. M. Fewster
  • P. E. Jupp

Abstract

Many models for biological populations, including simple mark-recapture models and distance sampling models, involve a binomially distributed number, n, of observations x 1 , …, x n on members of a population of size N. Two popular estimators of (N, θ), where θ is a vector parameter, are the maximum likelihood estimator and the conditional maximum likelihood estimator based on the conditional distribution of x 1 , …, x n given n. We derive the large-N asymptotic distributions of and , and give formulae for the biases of and . We show that the difference is, remarkably, of order 1 and we give a simple formula for the leading part of this difference. Simulations indicate that in many cases this formula is very accurate and that confidence intervals based on the asymptotic distribution have excellent coverage. An extension to product-binomial models is given. Copyright 2009, Oxford University Press.

Suggested Citation

  • R. M. Fewster & P. E. Jupp, 2009. "Inference on population size in binomial detectability models," Biometrika, Biometrika Trust, vol. 96(4), pages 805-820.
    • Handle: RePEc:oup:biomet:v:96:y:2009:i:4:p:805-820
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    File URL: http://hdl.handle.net/10.1093/biomet/asp051
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    Citations

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    Cited by:

    1. Yang Liu & Yukun Liu & Yan Fan & Han Geng, 2018. "Likelihood ratio confidence interval for the abundance under binomial detectability models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(5), pages 549-568, July.
    2. Shirley Pledger & Kenneth H. Pollock & James L. Norris, 2010. "Open Capture–Recapture Models with Heterogeneity: II. Jolly–Seber Model," Biometrics, The International Biometric Society, vol. 66(3), pages 883-890, September.
    3. Alessio Farcomeni, 2015. "Latent class recapture models with flexible behavioural response," Statistica, Department of Statistics, University of Bologna, vol. 75(1), pages 5-17.
    4. Yauck, Mamadou & Rivest, Louis-Paul, 2019. "On the estimation of population sizes in capture–recapture experiments," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 512-524.
    5. Fewster, R.M. & Jupp, P.E., 2013. "Information on parameters of interest decreases under transformations," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 34-39.
    6. R. T. R. Vale & R. M. Fewster & E. L. Carroll & N. J. Patenaude, 2014. "Maximum likelihood estimation for model M t,α for capture–recapture data with misidentification," Biometrics, The International Biometric Society, vol. 70(4), pages 962-971, December.
    7. Wen-Han Hwang & Richard Huggins, 2016. "Estimating Abundance from Presence–Absence Maps via a Paired Negative-Binomial Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(2), pages 573-586, June.
    8. Matthew R. Schofield & Richard J. Barker & William A. Link & Heloise Pavanato, 2023. "Estimating population size: The importance of model and estimator choice," Biometrics, The International Biometric Society, vol. 79(4), pages 3803-3817, December.
    9. Hannah Worthington & Rachel S. McCrea & Ruth King & Richard A. Griffiths, 2019. "Estimation of Population Size When Capture Probability Depends on Individual States," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(1), pages 154-172, March.
    10. R. M. Fewster, 2011. "Variance Estimation for Systematic Designs in Spatial Surveys," Biometrics, The International Biometric Society, vol. 67(4), pages 1518-1531, December.

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