## Electronic Journal of Statistics

### Identifiability and inferential issues in capture-recapture experiments with heterogeneous detection probabilities

#### Abstract

We focus on a capture-recapture model in which capture probabilities arise from an unspecified distribution $F$. We show that model parameters are identifiable based on the unconditional likelihood. This is not true with the conditional likelihood. We also clarify that consistency and asymptotic equivalence of maximum likelihood estimators based on conditional and unconditional likelihood do not hold. We show that estimates of the undetected fraction of population based on the unconditional likelihood converge to the so-called estimable sharpest lower bound and we derive a new asymptotic equivalence result. We finally provide theoretical and simulation arguments in favor of the use of the unconditional likelihood rather than the conditional likelihood especially when one is willing to infer on the sharpest lower bound.

#### Article information

Source
Electron. J. Statist., Volume 6 (2012), 2602-2626.

Dates
First available in Project Euclid: 11 January 2013

https://projecteuclid.org/euclid.ejs/1357913090

Digital Object Identifier
doi:10.1214/12-EJS758

Mathematical Reviews number (MathSciNet)
MR3020278

Zentralblatt MATH identifier
1302.62099

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62F12: Asymptotic properties of estimators

#### Citation

Farcomeni, Alessio; Tardella, Luca. Identifiability and inferential issues in capture-recapture experiments with heterogeneous detection probabilities. Electron. J. Statist. 6 (2012), 2602--2626. doi:10.1214/12-EJS758. https://projecteuclid.org/euclid.ejs/1357913090

#### References

• [1] A. P. Basu (1983). Icing the tails to limit theorems, lecture notes in economics and mathematical systems, 192. In: Samuel Kotz, Norman L. Johnson, and Campbell B. Read, eds., Encyclopedia of statistical sciences. Vol. 4, A Wiley-Interscience Publication, ix+657. John Wiley & Sons Inc., New York.
• [2] Sanjib Basu and Nader Ebrahimi (2001). Bayesian capture-recapture methods for error detection and estimation of population size: heterogeneity and dependence., Biometrika, 88, 1, 269–279.
• [3] K. P. Burnham and W. S. Overton (1978). Estimation of the size of a closed population when capture probabilities vary among animals (corr: V68 p345)., Biometrika, 65, 625–634.
• [4] Anne Chao (1989). Estimating population size for sparse data in capture-recapture experiments., Biometrics, 45, 427–438.
• [5] A. DasGupta and Herman Rubin (2005). Estimation of binomial parameters when both n, p are unknown., Journal of Statistical Planning and Inference, 130, 1–2, 391–404.
• [6] Holger Dette and William J. Studden (1997)., The theory of canonical moments with applications in statistics, probability, and analysis. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication.
• [7] Robert M. Dorazio and J. Andrew Royle (2005). Rejoinder to “the performance of mixture models in heterogeneous closed population capture-recapture”., Biometrics, 61, 3, 874–876.
• [8] A. Farcomeni and L. Tardella (2010). Reference Bayesian methods for alternative recapture models with heterogeneity., TEST, 19, 187–208.
• [9] Hajo Holzmann and Axel Munk (2008). On the nonidentifiability of population sizes (rejoinder)., Biometrics, 64, 3, 977–979.
• [10] Hajo Holzmann, Axel Munk, and Walter Zucchini (2006). On identifiability in capture-recapture models., Biometrics, 62, 3, 934–936.
• [11] R. Huggins (2001). A note on the difficulties associated with the analysis of capture-recapture experiments with heterogeneous capture probabilities., Statistics & Probability Letters, 54, 147–152.
• [12] Richard M. Huggins and Paul S. F. Yip (2001). A note on nonparametric inference for capture-recapture experiments with heterogeneous capture probabilities., Statistica Sinica, 11, 3, 843–853.
• [13] Wen-Han Hwang and Richard Huggins (2005). An examination of the effect of heterogeneity on the estimation of population size using capture-recapture data., Biometrika, 92, 1, 229–233.
• [14] Joseph B. Kadane (1975). The role of identification in Bayesian theory. In: Stephen E. Fienberg and Arnold Zellner, eds., Studies in Bayesian econometrics and statistics in honor of Leonard J. Savage, 175–191. Elsevier/North-Holland [Elsevier Science Publishing Co., New York; North-Holland Publishing Co., Amsterdam].
• [15] Bruce G. Lindsay (1995)., Mixture Models: Theory, Geometry, and Applications. Institute of Mathematical Statistics.
• [16] William A. Link (2003). Nonidentifiability of population size from capture-recapture data with heterogeneous detection probabilities., Biometrics, 59, 4, 1123–1130.
• [17] C. X. Mao and N. You (2009). On comparison of mixture models for closed population capture-recapture studies., Biometrics, 65, 547–553.
• [18] Chang Xuan Mao (2007). Estimating population sizes for capture-recapture sampling with binomial mixtures., Comput. Stat. Data Anal., 51, 11, 5211–5219.
• [19] Chang Xuan Mao (2008). On the nonidentifiability of population sizes., Biometrics, 64, 3, 977–979.
• [20] James L. III Norris and Kenneth H. Pollock (1996). Nonparametric MLE under two closed capture-recapture models with heterogeneity., Biometrics, 52, 639–649.
• [21] D. L. Otis, K. P. Burnham, G. C. White, and D. R. Anderson (1978)., Statistical Inference From Capture Data on Closed Animal Populations. Wildlife Monographs.
• [22] Carlos D. Paulino and Carlos A. Pereira de Barganca (1994). On identifiability of parametric statistical models., Journal of the Italian Statistical Society, 1, 3, 125–151.
• [23] Shirley Pledger (2005). The performance of mixture models in heterogeneous closed population capture-recapture., Biometrics, 61, 3, 868–876.
• [24] C. R. Rao (1965)., Linear Statistical Inference. Wiley, New York.
• [25] John E. Rolph (1968). Bayesian estimation of mixing distributions., The Annals of Mathematical Statistics, 39, 1289–1302.
• [26] Lalitha Sanathanan (1972). Estimating the size of a multinomial population., The Annals of Mathematical Statistics, 43, 142–152.
• [27] L. Tardella and A. Farcomeni (2009). Identifiability of population size from capture-recapture data with heterogeneous detection probabilities., Tech. Rep. 2, Department of Statistics, Sapienza - University of Rome.
• [28] Luca Tardella (2002). A new Bayesian method for nonparametric capture-recapture models in presence of heterogeneity., Biometrika, 89, 4, 807–817.
• [29] Ji-Ping Wang and Bruce G. Lindsay (2008). An exponential partial prior for improving nonparametric maximum likelihood estimation in mixture models., Stat. Methodol., 5, 1, 30–45.
• [30] O. Yoshida, J. G. Leite, and H. Bolfarine (1999). Stochastic monotonicity properties of Bayes estimation of the population size for capture-recapture data., Statistics and Probability Letters, 42, 257–266.