Abstract
Capture-recapture analysis is applied to estimate population size in ecology, biology, social science, medicine, linguistics and software engineering. The Poisson distribution is one of the simplest models for count data and appropriate for homogeneous populations. On the other hand, it is found to underestimate the counts for overdispersed data. In this study, population size estimation using the mixture of Poisson and Lindley distribution is proposed. It can exhibit overdispersed, equidispersed and underdispersed data. Additionally, it is able to present count data with long tail. As a result of the problem of unobserved individuals, the zero-truncated Poisson Lindley distribution is considered. The parameter of distribution can be estimated using the maximum likelihood estimation. The Horvitz-Thompson estimator based on the zero-truncated Poisson Lindley distribution for modelling the population size is investigated in this study. Point and interval estimation of the target population are presented. The technique of conditioning is used for variance estimation of the population size. Relative bias, relative variance and relative mean square error are used for measuring the accuracy of the estimator. The simulation results show that the Horvitz-Thompson estimator under the zero-truncated Poisson Lindley distribution provides a good fit when compared to the zero-truncated Poisson distribution.
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References
Böhning, D.: A simple variance formula for population size estimators by conditioning. Stat. Methodol. 5(5), 410–423 (2008). https://doi.org/10.1016/j.stamet.2007.10.001
Böhning, D., Suppawattanabodee, B., Kusolvisitkul, W., Viwatwongkasem, C.: Estimating the number of drug users in Bangkok 2001: a capture-recapture approach using repeated entries in one list. Eur. J. Epidemiol. 19(12), 1075–1083 (2004). https://doi.org/10.2307/3583005
Böhning, D.: Ratio plot and ratio regression with applications to social and medical sciences. Stat. Sci. 31(2), 205–218 (2016). https://doi.org/10.1214/16-STS548
Bunge, J., Barger, K.: Parametric models for estimating the number of classes. Biometrical J. 50(6), 971–982 (2008). https://doi.org/10.1002/bimj.200810452
Cruyff, M.J., van der Heijden, P.G.: Point and interval estimation of the population size using a zero-truncated negative binomial regression model. Biometrical J. 50(6), 1035–1050 (2008). https://doi.org/10.1002/bimj.200810455
Edwards, W.R., Eberhardt, L.: Estimating cottontail abundance from livetrapping data. J. Wildlife Manag. 87–96 (1967). https://doi.org/10.2307/3798362
El-Shaarawi, A.H., Zhu, R., Joe, H.: Modelling species abundance using the Poisson-Tweedie family. Environmetrics 22(2), 152–164 (2011). https://doi.org/10.1002/env.1036
Fisher, R.A., Corbet, A.S., Williams, C.B.: The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecol. 42–58 (1943). https://doi.org/10.2307/1411
Ghitany, M., Al-Mutairi, D.K., Nadarajah, S.: Zero-truncated Poisson-Lindley distribution and its application. Math. Comput. Simul. 79(3), 279–287 (2008). https://doi.org/10.1016/j.matcom.2007.11.021
van der Heijden, P.G., Bustami, R., Cruyff, M.J., Engbersen, G., van Houwelingen, H.C.: Point and interval estimation of the population size using the truncated Poisson regression model. Stat. Model. 3(4), 3305–322 (2003)
van der Heijden, P.G., Cruyff, M.J., Böhning, D.: Capture recapture to estimate criminal populations. In: Encyclopedia of Criminology and Criminal Justice pp. 267–276 (2014). https://doi.org/10.2307/41954244
Hidaka, S.: General type token distribution. Briometrika 101(4), 999–1002 (2014). https://doi.org/10.1093/biomet/asu035
Holla, M.: On a Poisson-inverse Gaussian distribution. Metrika 11(1), 115–121 (1967). https://doi.org/10.1007/bf02613581
Horvitz, D.G., Thompson, D.J.: A generalization of sampling without replacement from a finite universe. J. Am. Stat. Assoc. 47(260), 663–685 (1952). https://doi.org/10.1080/01621459.1952.10483446
Mahmoudi, E., Zakerzadeh, H.: Generalized Poisson-Lindley distribution. Commun. Stat.-Theory Methods 39(10), 1785–1798 (2010). https://doi.org/10.1080/03610920902898514
Rocchetti, I., Bunge, J., Böhning, D.: Population size estimation based upon ratios of recapture probabilities. Ann. Appl. Stat. 5(2B), 1512–1533 (2011)
Sankaran, M.: The discrete Poisson-Lindley distribution. Biometrics 145–149 (1970). https://doi.org/10.2307/2529053
Shanker, R., Hagos, F., Sujatha, S., Abrehe, Y.: On zero-truncation of Poisson and Poisson-Lindley distributions and their applications. Biometrics Biostat. Int. J. 2(6), 1–14 (2015)
Zamani, H., Ismail, N., Faroughi, P.: Poisson-weighted exponential univariate version and regression model with applications. J. Math. Stat. 10(2), 148–154 (2014). https://doi.org/10.3844/jmssp.2014.148.154
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This research was supported by Maejo University, Chiang Mai, Thailand.
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Wongprachan, R. (2020). Modelling Population Size Using Horvitz-Thompson Approach Based on the Zero-Truncated Poisson Lindley Distribution. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_18
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