, Volume 81, Issue 5, pp 549–568 | Cite as

Likelihood ratio confidence interval for the abundance under binomial detectability models

  • Yang Liu
  • Yukun Liu
  • Yan Fan
  • Han Geng


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.


Abundance Binomial detectability models Capture-recapture models Confidence interval Distance sampling models 



We are grateful to the editor and two anonymous referees for their insightful and constructive comments which led to an improved presentation of this article. The research was supported by National Natural Science Foundation of China (Grant Nos. 11501354, 11771144, 11371142, and 11501208), Program of Shanghai Subject Chief Scientist (14XD1401600) and the 111 Project (B14019).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Alho JM (1990) Logistic regression in capture–recapture models. Biometrics 46:623–635MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barnard J, Emam K, Zubrow D (2003) Using capture–recapture models for the reinspection decision. Softw Qual Prof 5:11–20Google Scholar
  3. Borchers DL, Zucchini W, Fewster RM (1998) Mark-recapture models for line transect surveys. Biometrics 54:1207–1220CrossRefzbMATHGoogle Scholar
  4. Borchers DL, Buckland ST, Zucchini W (2002) Estimating animal abundance: closed population. Springer, LondonCrossRefzbMATHGoogle Scholar
  5. Borchers DL, Stevenson BC, Kidney D, Thomas L, Marques TA (2015) A unifying model for capture-recapture and distance sampling surveys of wildlife populations. J Am Stat Assoc 110:195–204MathSciNetCrossRefzbMATHGoogle Scholar
  6. Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L (2001) Introduction to distance sampling. Oxford University Press, OxfordzbMATHGoogle Scholar
  7. Chao A (1987) Estimating the population size for capture–recapture data with unequal catchability. Biometrics 43:783–791MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chao A, Chu W, Hsu CH (2000) Capture–recapture when time and behavioral response affect capture probabilities. Biometrics 56:427–433CrossRefzbMATHGoogle Scholar
  9. Chao A, Tsay PK, Lin SH, Shau WY, Chao DY (2001) The applications of capture–recapture models to epidemiological data. Stat Med 20:3123–3157CrossRefGoogle Scholar
  10. Chen SX, Lloyd CJ (2002) Estimation of population size from biased samples using non-parametric binary regression. Stat. Sin. 12:505–518MathSciNetzbMATHGoogle Scholar
  11. Cormack RM (1992) Interval estimation for mark-recapture studies of closed population. Biometrics 48:567–576MathSciNetCrossRefGoogle Scholar
  12. Evans MA, Bonett DG (1994) Bias reduction for multiple recapture estimators of closed population size. Biometrics 50:388–395CrossRefGoogle Scholar
  13. Evans MA, Kim H, O’Bren TE (1996) An application of profile-likelihood based confidence interval to capture–recapture estimators. J Agric Biol Environ Stat 1(1):131–140MathSciNetCrossRefGoogle Scholar
  14. Fancy SG, Snetsinger TJ, Jacobi JD (1997) Translocation of the Palila, an endangered Hawaiian honeycreeper. Pac Conserv Biol 3:39–46CrossRefGoogle Scholar
  15. Fewster RM, Jupp PE (2009) Inference on population size in binomial detectability models. Biometrika 96:805–820MathSciNetCrossRefzbMATHGoogle Scholar
  16. Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction, and estimation. J Am Stat Assoc 102:359–378MathSciNetCrossRefzbMATHGoogle Scholar
  17. Heinze D, Broome L, Mansergh I (2004) A review of the ecology and conservation of the mountain pygmy-possum Burramys parvus. In: Goldingay RL, Jackson SM (eds) The biology of Australian Possums and Gliders. Baulkham Hills, Surrey Beatty & Sons, pp 254–267Google Scholar
  18. Hjort NL, Pollard D (2011) Asymptotics for minimisers of convex processes. arXiv:1107.3806v1
  19. Hogan H (2000) Accuracy and coverage evaluation 2000: decomposition of dual system estimate components. U.S. Census Bureau, WashingtonGoogle Scholar
  20. Huggins RM (1989) On the statistical analysis of capture experiments. Biometrika 76:133–140MathSciNetCrossRefzbMATHGoogle Scholar
  21. Huggins R, Hwang WH (2007) Non-parametric estimation of population size from capture–recapture data when the capture probability depends on a covariate. J R Stat Soc Ser C 56:429–443MathSciNetCrossRefGoogle Scholar
  22. Liu Y, Li P, Qin J (2017) Maximum empirical likelihood estimation for abundance in a closed population from capture–recapture data. Biometrika 104(3):527–543MathSciNetGoogle Scholar
  23. Marques FFC, Buckland ST (2004) Covariate models for the detection function. In: Buckland ST, Anderson DR, Burnham KP, Laake JL, Borchers DL, Thomas L (eds) Advanced distance sampling: estimating abundance of biological populations. Oxford University Press, OxfordGoogle Scholar
  24. Marques TA, Thomas L, Fancy SG, Buckland ST (2007) Improving estimates of bird density using multiple-covariate distance sampling. Auk 124(4):1229–1243CrossRefGoogle Scholar
  25. Otis DL, Burnham KP, White GC, Anderson DR (1978) Statistical inference from capture data on closed animal populations. Wildl Monogr 62:1–135zbMATHGoogle Scholar
  26. Pollock KH (2000) Capture–recapture models. J Am Stat Assoc 95:293–296CrossRefGoogle Scholar
  27. Qin J, Lawless J (1994) Empirical likelihood and general estimating equations. Ann Stat 22(1):300–325MathSciNetCrossRefzbMATHGoogle Scholar
  28. Sanathanan L (1972) Estimating the size of a multinomial population. Ann Math Stat 43:142–152MathSciNetCrossRefzbMATHGoogle Scholar
  29. Serfling RJ (1980) Approximation theorem of mathematical statistics. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  30. Stoklosa J, Hwang WH, Wu SH, Huggins R (2011) Heterogeneous capture–recapture models with covariates: a partial likelihood approach for closed populations. Biometrics 67:1659–1665MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of StatisticsEast China Normal UniversityShanghaiChina
  2. 2.School of Statistics and InformationShanghai University of International Business and EconomicsShanghaiChina

Personalised recommendations