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Some General Comparative Points on Chao's and Zelterman's Estimators of the Population Size

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  • DANKMAR BÖHNING

Abstract

. Two simple and frequently used capture–recapture estimates of the population size are compared: Chao's lower‐bound estimate and Zelterman's estimate allowing for contaminated distributions. In the Poisson case it is shown that if there are only counts of ones and twos, the estimator of Zelterman is always bounded above by Chao's estimator. If counts larger than two exist, the estimator of Zelterman is becoming larger than that of Chao's, if only the ratio of the frequencies of counts of twos and ones is small enough. A similar analysis is provided for the binomial case. For a two‐component mixture of Poisson distributions the asymptotic bias of both estimators is derived and it is shown that the Zelterman estimator can experience large overestimation bias. A modified Zelterman estimator is suggested and also the bias‐corrected version of Chao's estimator is considered. All four estimators are compared in a simulation study.

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  • Dankmar Böhning, 2010. "Some General Comparative Points on Chao's and Zelterman's Estimators of the Population Size," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 221-236, June.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:2:p:221-236
    DOI: 10.1111/j.1467-9469.2009.00676.x
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    1. William A. Link, 2003. "Nonidentifiability of Population Size from Capture-Recapture Data with Heterogeneous Detection Probabilities," Biometrics, The International Biometric Society, vol. 59(4), pages 1123-1130, December.
    2. Shirley Pledger, 2000. "Unified Maximum Likelihood Estimates for Closed Capture–Recapture Models Using Mixtures," Biometrics, The International Biometric Society, vol. 56(2), pages 434-442, June.
    3. Hajo Holzmann & Axel Munk & Walter Zucchini, 2006. "On Identifiability in Capture–Recapture Models," Biometrics, The International Biometric Society, vol. 62(3), pages 934-936, September.
    4. John M. Roberts & Devon D. Brewer, 2006. "Estimating the prevalence of male clients of prostitute women in Vancouver with a simple capture–recapture method," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 169(4), pages 745-756, October.
    5. Shirley Pledger, 2005. "The Performance of Mixture Models in Heterogeneous Closed Population Capture–Recapture," Biometrics, The International Biometric Society, vol. 61(3), pages 868-873, September.
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