×

Capture – recapture estimation by means of empirical Bayesian smoothing with an application to the geographical distribution of hidden scrapie in Great Britain. (English) Zbl 1225.62154

Summary: This paper discusses population size estimation on the basis of a frequency distribution of zero-truncated counts and is motivated by a study on the geographical distribution of hidden scrapies in Great Britain. Aggregation of scrapie cases is considered at the county level and results in sparse zero-truncated count distributions which make the application of conventional capture – recapture procedures for estimating the hidden part of the scrapie-affected populations difficult. We suggest a smoothed generalization of Zelterman’s estimator of population size which overcomes the overestimation bias of the conventional Zelterman estimator and instead produces a lower bound, which is typically larger than Chao’s lower bound estimator. The estimator uses an empirical Bayes approach with various choices for the prior distribution including a parametric choice of the gamma distribution as well as various non-parametric distributions. A simulation study investigates the performance of the new estimators, and also in comparison with conventional estimators. The empirical Bayes estimator with a non-parametric mixture model as prior performs well and the boundary problem of the conventional non-parametric discrete mixture model estimator leading to spurious population size is avoided. In the application to hidden scrapies in Great Britain the new estimators lead to maps of scrapie of observed – hidden ratios as well as completeness of the current surveillance system.

MSC:

62P12 Applications of statistics to environmental and related topics
62C12 Empirical decision procedures; empirical Bayes procedures
65C60 Computational problems in statistics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Böhning, The EM algorithm with gradient function update for discrete mixtures with known (fixed) number of components, Statist. Comput. 13 pp 257– (2003) · doi:10.1023/A:1024222817645
[2] Böhning, A simple variance formula for population size estimators by conditioning, Statist. Methodol. 5 pp 410– (2008) · Zbl 1248.62006 · doi:10.1016/j.stamet.2007.10.001
[3] Böhning, Estimating the hidden number of scrapie affected holdings in Great Britain using a simple, truncated count model allowing for heterogeneity, J. Agric. Biol. Environ. Statist. 13 pp 1– (2008) · Zbl 1306.62248 · doi:10.1198/108571108X277904
[4] Böhning, The equivalence of truncated count mixture distributions and mixtures of truncated count distributions, Biometrics 62 pp 1207– (2006) · Zbl 1114.62114 · doi:10.1111/j.1541-0420.2006.00565.x
[5] Bunge, Estimating the number of species: a review, J. Am. Statist. Ass. 88 pp 364– (1993) · doi:10.2307/2290733
[6] Carlin, Bayes and Empirical Bayes Methods for Data Analysis (1997) · Zbl 0871.62012
[7] Chao, Estimating the population size for capture-recapture data with unequal catchability, Biometrics 43 pp 783– (1987) · Zbl 0715.62286 · doi:10.2307/2531532
[8] Chao, Estimating population size for sparse data in capture-recapture experiments, Biometrics 45 pp 427– (1989) · Zbl 0715.62285 · doi:10.2307/2531487
[9] Del Rio Vilas, Analysis of data from the passive surveillance of scrapie in Great Britain between 1993 and 2002, Veter. Rec. 159 pp 799– (2006)
[10] Del Rio Vilas, A case study of capture-recapture methodology using scrapie surveillance data in Great Britain, Prev. Veter. Med. 67 pp 303– (2005) · doi:10.1016/j.prevetmed.2004.12.003
[11] Dorazio, Mixture models for estimating the size of a closed population when capture rates vary among individuals, Biometrics 59 pp 351– (2005) · Zbl 1210.62226 · doi:10.1111/1541-0420.00042
[12] Gale, Good-Turing frequency estimation without tears, J. Quant. Ling. 2 pp 217– (1995) · Zbl 05431514 · doi:10.1080/09296179508590051
[13] Good, The population frequencies of species and the estimation of population parameters, Biometrika 40 pp 237– (1953) · Zbl 0051.37103 · doi:10.1093/biomet/40.3-4.237
[14] Hampel, Robust Statistics-the Approach based on Influence Functions (1986) · Zbl 0593.62027
[15] van der Heijden, Estimating the size of a criminal population from police records using the truncated Poisson regression model, Statist. Neerland. 57 pp 1– (2003) · Zbl 04575593 · doi:10.1111/1467-9574.00232
[16] Hoinville, Descriptive epidemiology of scrapie in Great Britain: results of a postal survey, Veter. Rec. 146 pp 455– (2000) · doi:10.1136/vr.146.16.455
[17] Holzmann, On identifiability in capture-recapture models, Biometrics 62 pp 934– (2006) · doi:10.1111/j.1541-0420.2006.00637_1.x
[18] Lindsay, The geometry of mixture likelihoods, part I: a general theory, Ann. Statist. 11 pp 783– (1983) · Zbl 0534.62002 · doi:10.1214/aos/1176346245
[19] Link, Nonidentifiability of population size from capture-recapture data with heterogeneous detection probabilities, Biometrics 59 pp 1123– (2003) · Zbl 1274.62821 · doi:10.1111/j.0006-341X.2003.00129.x
[20] Link, Response to a paper by Holzmann, Munk and Zucchini, Biometrics 62 pp 936– (2006) · doi:10.1111/j.1541-0420.2006.00637_2.x
[21] McKendrick, Application of mathematics to medical problems, Proc. Edinb. Math. Soc. 44 pp 98– (1926) · JFM 52.0542.04 · doi:10.1017/S0013091500034428
[22] Mosley, An epidemiological assessment of cholera control programs in rural East Pakistan, Int. J. Epidem. 1 pp 5– (1972) · doi:10.1093/ije/1.1.5-a
[23] Pledger, The performance of mixture models in heterogeneous closed population capture-recapture, Biometrics 61 pp 868– (2005) · doi:10.1111/j.1541-020X.2005.00411_1.x
[24] Robbins, Proc. 3rd Berkeley Symp. Mathematical Statistics and Probability pp 157– (1955)
[25] Waller, Applied Spatial Statistics for Public Health Data (2004) · Zbl 1057.62106 · doi:10.1002/0471662682
[26] Wang, A penalized nonparametric maximum likelihood approach to species richness estimation, J. Am. Statist. Ass. 100 pp 942– (2005) · Zbl 1117.62439 · doi:10.1198/016214504000002005
[27] Wang, An exponential partial prior for improving nonparametric maximum likelihood estimation in mixture models, Statist. Methodol. 5 pp 30– (2008) · Zbl 1248.62049 · doi:10.1016/j.stamet.2007.03.004
[28] Wilson, Capture-recapture estimation with samples of size one using frequency data, Biometrika 79 pp 543– (1992) · Zbl 0775.62315 · doi:10.1093/biomet/79.3.543
[29] Zelterman, Robust estimation in truncated discrete distributions with applications to capture-recapture experiments, J. Statist. Planng Inf. 18 pp 225– (1988) · Zbl 0642.62021 · doi:10.1016/0378-3758(88)90007-9
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.