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Likelihood ratio confidence interval for the abundance under binomial detectability models

Author

Listed:
  • Yang Liu

    (East China Normal University)

  • Yukun Liu

    (East China Normal University)

  • Yan Fan

    (Shanghai University of International Business and Economics)

  • Han Geng

    (East China Normal University)

Abstract

Binomial detectability models are often used to estimate the size or abundance of a finite population in biology, epidemiology, demography and reliability. Special cases include incompletely observed multinomial models, capture–recapture models, and distance sampling models. The most commonly-used confidence interval for the abundance is the Wald-type confidence interval, which is based on the asymptotic normality of a reasonable point estimator of the abundance. However, the Wald-type confidence interval may have poor coverage accuracy and its lower limit may be less than the number of observations. In this paper, we rigorously establish that the likelihood ratio test statistic for the abundance under the binomial detectability models follows the chisquare limiting distribution with one degree of freedom. This provides a solid theoretical justification for the use of the proposed likelihood ratio confidence interval. Our simulations indicate that in comparison to the Wald-type confidence interval, the likelihood ratio confidence interval not only has more accurate coverage rate, but also exhibits more stable performance in a variety of binomial detectability models. The proposed interval is further illustrated through analyzing three real data-sets.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metrik:v:81:y:2018:i:5:d:10.1007_s00184-018-0655-2
    DOI: 10.1007/s00184-018-0655-2
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    References listed on IDEAS

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    1. Jakub Stoklosa & Wen-Han Hwang & Sheng-Hai Wu & Richard Huggins, 2011. "Heterogeneous Capture–Recapture Models with Covariates: A Partial Likelihood Approach for Closed Populations," Biometrics, The International Biometric Society, vol. 67(4), pages 1659-1665, December.
    2. D. L. Borchers & B. C. Stevenson & D. Kidney & L. Thomas & T. A. Marques, 2015. "A Unifying Model for Capture-Recapture and Distance Sampling Surveys of Wildlife Populations," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 195-204, March.
    3. Yukun Liu & Pengfei Li & Jing Qin, 2017. "Maximum empirical likelihood estimation for abundance in a closed population from capture-recapture data," Biometrika, Biometrika Trust, vol. 104(3), pages 527-543.
    4. Richard Huggins & Wen‐Han Hwang, 2007. "Non‐parametric estimation of population size from capture–recapture data when the capture probability depends on a covariate," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 56(4), pages 429-443, August.
    5. Anne Chao & Wenten Chu & Chiu-Hsieh Hsu, 2000. "Capture–Recapture When Time and Behavioral Response Affect Capture Probabilities," Biometrics, The International Biometric Society, vol. 56(2), pages 427-433, June.
    6. R. M. Fewster & P. E. Jupp, 2009. "Inference on population size in binomial detectability models," Biometrika, Biometrika Trust, vol. 96(4), pages 805-820.
    7. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
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