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A modification of Chao’s lower bound estimator in the case of one-inflation

Author

Listed:
  • Dankmar Böhning

    (University of Southampton)

  • Panicha Kaskasamkul

    (Naresuan University)

  • Peter G. M. Heijden

    (University of Southampton
    University of Utrecht)

Abstract

For zero-truncated count data, as they typically arise in capture-recapture modelling, the nonparametric lower bound estimator of Chao is a frequently used estimator of population size. It is a simple, nonparametric estimator involving only counts of one and counts of two. The estimator is asymptotically unbiased if the count distribution is a member of the power series family and is providing a lower bound estimator if the distribution is a mixture of a member of the power series family. However, if there is one-inflation Chao’s estimator can severely overestimate as we show here. This is also illustrated by routinely collected country-wide data on family violence in the Netherlands. A new lower bound estimator is developed which involves only counts of twos and threes, thus avoiding the overestimation caused by one-inflation. We show that the new estimator is asymptotically unbiased for a power series distribution with and without one-inflation and provides a lower bound estimator under a mixture of power series distributions with and without one-inflation. For all estimators bias-adjusted versions are developed that reduce the bias considerably when the sample size is small. A simulation study compares the modified Chao estimator with the conventional estimator as well as with an estimator suggested by Chiu and Chao more recently.

Suggested Citation

  • Dankmar Böhning & Panicha Kaskasamkul & Peter G. M. Heijden, 2019. "A modification of Chao’s lower bound estimator in the case of one-inflation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(3), pages 361-384, April.
    • Handle: RePEc:spr:metrik:v:82:y:2019:i:3:d:10.1007_s00184-018-0689-5
    DOI: 10.1007/s00184-018-0689-5
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    References listed on IDEAS

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    1. Chun-Huo Chiu & Yi-Ting Wang & Bruno A. Walther & Anne Chao, 2014. "An improved nonparametric lower bound of species richness via a modified good–turing frequency formula," Biometrics, The International Biometric Society, vol. 70(3), pages 671-682, September.
    2. Pedro Puig & Célestin C. Kokonendji, 2018. "Non†parametric Estimation of the Number of Zeros in Truncated Count Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 45(2), pages 347-365, June.
    3. Amy Willis & John Bunge, 2015. "Estimating diversity via frequency ratios," Biometrics, The International Biometric Society, vol. 71(4), pages 1042-1049, December.
    4. Mao, Chang Xuan, 2006. "Inference on the Number of Species Through Geometric Lower Bounds," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1663-1670, December.
    5. 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.
    6. Wang, Ji-Ping Z. & Lindsay, Bruce G., 2005. "A Penalized Nonparametric Maximum Likelihood Approach to Species Richness Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 942-959, September.
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    Cited by:

    1. Dankmar Böhning & Rattana Lerdsuwansri & Patarawan Sangnawakij, 2023. "Modeling COVID‐19 contact‐tracing using the ratio regression capture–recapture approach," Biometrics, The International Biometric Society, vol. 79(4), pages 3818-3830, December.

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