Yauck, Mamadou; Rivest, Louis-Paul On the estimation of population sizes in capture-recapture experiments. (English) Zbl 1422.62203 J. Multivariate Anal. 173, 512-524 (2019). Summary: This work considers a nested mark-recapture experiment with two levels of sampling: within each primary sampling period of an open population model, there are secondary capture occasions to estimate the size of the population at that primary period. This scheme is known as Pollock’s robust design. Two sources of information are then available to estimate the population size for a primary period: the within and the between primary period data. This work proves that the population size estimators derived from these two sources are asymptotically independent for a large class of closed population models. In this context, the robust design maximum likelihood estimator of population size is shown to be asymptotically equivalent to a weighted sum of the estimators for the open population Jolly-Seber model [G. M. Jolly, Biometrika 52, 225–247 (1965; Zbl 0141.36601); G. A. F. Seber [ibid. 52, 249–259 (1965; Zbl 0141.36602)] and for the closed population model. This article shows that the weighted estimator is more efficient than the moment estimator of Kendall et al. (1995). A closed form expression for the efficiency associated with this estimator is given and evaluated in a Monte Carlo study and in a numerical example about the estimation of the size of dolphin populations discussed by N. L. Santostasi et al. [“A robust design capture-recapture analysis of abundance, survival and temporary emigration of three Odontocete species in the Gulf of Corinth, Greece”, PLOS ONE 11, No. 12, Article ID e0166650, (2016; doi:10.1371/journal.pone.0166650)]. . MSC: 62H17 Contingency tables 62P12 Applications of statistics to environmental and related topics Keywords:asymptotics; heterogeneity; Jolly-Seber model; mark-recapture study; multinomial distribution; Poisson regression; robust design Citations:Zbl 0141.36601; Zbl 0141.36602 Software:MARK; Program MARK; CRAN PDFBibTeX XMLCite \textit{M. Yauck} and \textit{L.-P. Rivest}, J. Multivariate Anal. 173, 512--524 (2019; Zbl 1422.62203) Full Text: DOI References: [1] Baillargeon, S.; Rivest, L.-P., The rcapture package: loglinear models for capture-recapture in r, J. Stat. Softw., 19, 5 (2007), , Rcapture CRAN URL: [2] Berrow, S.; O’Brien, J.; Groth, L.; Foley, A.; Voigt, K., Abundance estimate of bottlenose dolphins (tursiops truncatus) in the lower river shannon candidate special area of conservation, ireland, Aquat. Mamm., 38, 2, 136-144 (2012) [3] E.G. Cooch, G.C. White, Program MARK: A Gentle Introduction, New York, 18th edition, 2018.; E.G. Cooch, G.C. White, Program MARK: A Gentle Introduction, New York, 18th edition, 2018. [4] Cormack, R. M., Loglinear models for capture-recapture, Biometrics, 45, 395-413 (1989) · Zbl 0707.62244 [5] Cormack, R. M., Interval estimation for mark-recapture studies of closed populations (ack: v49 p315; ref: 91statmed v10 p717-721), Biometrics, 48, 2, 567-576 (1992) [6] Darroch, J. N.; Fienberg, S. E.; Glonek, G. F.V.; Junker, B. W., A three-sample multiple-recapture approach to census population estimation with heterogeneous catchability, J. Amer. Statist. Assoc., 88, 1137-1148 (1993) [7] Farcomeni, A., A general class of recapture models based on the conditional capture probabilities, Biometrics, 72, 116-124 (2016) · Zbl 1393.62061 [8] Fewster, R.; Jupp, P., Inference on population size in binomial detectability models, Biometrika, 96, 4, 805-820 (2009) · Zbl 1179.62033 [9] Jolly, G. M., Explicit estimates from capture-recapture data with both death and immigration-stochastic model, Biometrika, 52, 225-247 (1965) · Zbl 0141.36601 [10] Kendall, W. L.; Pollock, K. H.; Brownie, C., A likelihood-based approach to capture-recapture estimation of demographic parameters under the robust design, Biometrics, 51, 293-308 (1995) · Zbl 0826.62096 [12] Nichols, J. D.; Hines, J. E.; Lebreton, J. D.; Pradel, R., Estimation of contributions to population growth: a reverse-time capture – recapture approach, Ecology, 81, 3362-3376 (2000) [14] Pradel, R., Utilization of capture-mark-recapture for the study of recruitment and population growth rate, Biometrics, 52, 703-709 (1996) · Zbl 0875.62538 [15] Rivest, L.-P.; Baillargeon, S., Applications and extensions of chao’s moment estimator for the size of a closed population, Biometrics, 63, 999-1006 (2007) · Zbl 1274.62862 [16] Rivest, L.-P.; Daigle, G., Loglinear models for the robust design in mark-recapture experiments, Biometrics, 60, 1, 100-107 (2004) · Zbl 1130.62349 [17] Rivest, L.-P.; Lévesque, T., Improved log-linear model estimators of abundance in capture-recapture experiments, Canad. J. Statist., 29, 4, 555-572 (2001) · Zbl 0994.62104 [18] Sandland, R. L.; Cormack, R. M., Statistical inference for poisson and multinomial models for capture-recapture experiments, Biometrika, 71, 1, 27-33 (1984) · Zbl 0537.62092 [19] Santostasi, N. L.; Bonizzoni, S.; Bearzi, G.; Eddy, L.; Gimenez, O., A robust design capture-recapture analysis of abundance, survival and temporary emigration of three odontocete species in the gulf of corinth, greece, PLoS ONE, 11, 1-21 (2016) [20] Schwarz, C. J.; Arnason, A., A general methodology for the analysis of capture-recapture experiments in open populations, Biometrics, 860-873 (1996) · Zbl 0875.62540 [21] Seber, G. A.F., A note on the multiple-recapture census, Biometrika, 52, 249-259 (1965) · Zbl 0141.36602 [24] White, G.; Burnham, K. P., Program mark: survival estimation from populations of marked animals, Bird Study, 46, Supplement, 120-138 (1999) [25] Yang, H.-C.; Chao, A., Modeling animals’ behavioral response by markov chain models for capture-recapture experiments, Biometrics, 61, 1010-1017 (2005) · Zbl 1087.62136 [26] Yauck, M.; Rivest, L.-P.; Rothman, G., CaPture-recapture methods for data on the activation of applications on mobile phones, J. Amer. Statist. Assoc., 114, 525, 105-114 (2019) · Zbl 1462.62742 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.