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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

    as
    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. repec:bla:biomet:v:71:y:2015:i:4:p:1042-1049 is not listed on IDEAS
    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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