Skip to main content
SpringerLink
Log in

Estimating population sizes with the Rasch model

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

The Rasch model has been used to estimate the unknown size of a population from multi-list data. It can take both the list effectiveness and individual heterogeneity into account. Estimating the population size is shown to be equivalent to estimating the odds that an individual is unseen. The odds parameter is nonidentifiable. We propose a sequence of estimable lower bounds, including the greatest one, for the odds parameter. We show that a lower bound can be calculated by linear programming. Estimating a lower bound of the odds leads to an estimator for a lower bound of the population size. A simulation experiment is performed and three real examples are studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aaron, D. J., Chang, Y. F., Markovic, N., LaPorte, R. E. (2003). Estimating the lesbian population: a capture-recapture approach. Journal of Epidemiology and Community Health, 57, 207–209.

  • Agresti, A. (1994). Simple capture-recapture models permitting unequal catchability and variable sampling effort. Biometrics, 50, 494–500.

    Article  Google Scholar 

  • Andersen, E. B. (1970). Asymptotic properties of conditional maximum-likelihood estimators. Journal of the Royal Statistical Society, Series B, 32, 283–301.

    MathSciNet  MATH  Google Scholar 

  • Bartolucci, F., Forcina, A. (2001). Analysis of capture-recapture data with a Rasch-type model allowing for conditional dependence and multidimensionality. Biometrics, 57, 714–719.

  • Bezerra, K., Gurgel, R., Ilozue, C., Castaneda, D. (2011). Estimating the number of street children and adolescents in two cities of Brazil using capture-recapture. Journal of Paediatrics and Child Health, 47, 524–529.

  • Chao, A. (1984). Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics, 11, 265–270.

    MathSciNet  Google Scholar 

  • Chao, A. (1987). Estimating the population size for capture-recapture data with unequal catchability. Biometrics, 43, 783–791.

    Article  MathSciNet  MATH  Google Scholar 

  • Chao, A. (1989). Estimating population size for sparse data in capture-recapture experiments. Biometrics, 45, 427–438.

    Article  MathSciNet  MATH  Google Scholar 

  • Chao, A. (2001). An overview of closed capture-recapture models. Journal of Agricultural, Biological, and Environmental Statistics, 6, 138–155.

    Article  Google Scholar 

  • Chao, A., Tsay, P. K., Lin, S.-H., Shau, W.-Y., Chao, D.-Y. (2001). The applications of capture-recapture models to epidemiological data. Statistics in Medicine, 20, 3123–3157.

  • Chao, A., Shen, T. J., Hwang, W. H. (2006). Application of Laplace’s boundary-mode approximations to estimate species and shared species richness. Australian and New Zealand Journal of Statistics, 48, 117–128.

  • Coull, B. A., Agresti, A. (1999). The use of mixed logit models to reflect heterogeneity in capture-recapture studies. Biometrics, 55, 294–301.

  • Holzmann, H., Munk, A., Zucchini, W. (2006). On identifiability in capture-recapture models. Biometrics, 62, 934–936.

  • Huggins, R. (2001). A note on the difficulties associated with the analysis of capture-recapture experiments with heterogeneous capture probabilities. Statistics and Probability Letters, 54, 147–152.

    Article  MathSciNet  MATH  Google Scholar 

  • Lindsay, B. G. (1986). Exponential family mixture models (With least-squares estimators). The Annals of Statistics, 14, 124–137.

    Article  MathSciNet  MATH  Google Scholar 

  • Lindsay, B. G., Clogg, C. C., Grego, J. M. (1991). Semi-parametric estimation in the Rasch model, including a simple latent class model for item analysis. Journal of the American Statistical Association, 86, 96–107.

  • Link, W. A. (2003). Nonidentifiability of population size from capture-recapture data with heterogeneous detection probabilities. Biometrics, 59, 1123–1130.

    Article  MathSciNet  MATH  Google Scholar 

  • Mao, C. X. (2006). Inference of the number of species via geometric lower bounds. Journal of American Statistical Association, 101, 1663–1670.

    Article  MathSciNet  MATH  Google Scholar 

  • Mao, C. X. (2007a). Estimating population sizes for capture-recapture sampling with binomial mixtures. Computational Statistics and Data Analysis, 51, 5211–5219.

    Article  MathSciNet  MATH  Google Scholar 

  • Mao, C. X. (2007b). Testing list effects and diagnosing individual catchabilities in the Rasch model. Statistical Methodology, 4, 416–422.

    Article  MathSciNet  MATH  Google Scholar 

  • Mao, C. X. (2008a). Computing an NPMLE for a mixing distribution in two closed heterogeneous population size models. Biometrical Journal, 50, 983–992.

    Article  MathSciNet  Google Scholar 

  • Mao, C. X. (2008b). On the nonidentifiability of population sizes. Biometrics, 64, 977–979.

    Article  MathSciNet  MATH  Google Scholar 

  • Mao, C. X., Lindsay, B. G. (2007). Estimating the number of classes. Annals of Statistics, 35, 917–930.

  • Mao, C. X., You, N. (2009). On comparison of mixture models for closed population capture-recapture studies. Biometrics, 65, 547–553.

  • Prohaska, T. R., Anderson, L. A., Binstock, R. H. (2012). Public Health for an Aging Society. Baltimore: Johns Hopkins University Press.

  • Rivest, L. (2011). A lower bound model for multiple record systems estimation with heterogeneous catchability. The International Journal of Biostatistics, 7, 1–21.

    Article  MathSciNet  Google Scholar 

  • Rivest, L., Baillargeon, S. (2007). Applications and extensions of Chao’s moment estimator for the size of a closed population. Biometrics, 63, 999–1006.

  • Shukla, P. C. (2005). Street Children and Asphalt Life. Delhi: Isha Books.

    Google Scholar 

Download references

Acknowledgments

We thank the referees, the associated editor and editor for their constructive comments. Our research is supported by the National Natural Science Foundation of China Grant No. 713-731-52.

Author information

Authors and Affiliations

  1. School of Statistics and Management, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, 200433, China

    Chang Xuan Mao, Yitong Yang & Wei Zhuang

  2. School of Public Economics and Administration, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai, 200433, China

    Cuiying Yang

Authors
  1. Chang Xuan Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cuiying Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yitong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Zhuang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang Xuan Mao.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mao, C.X., Yang, C., Yang, Y. et al. Estimating population sizes with the Rasch model. Ann Inst Stat Math 69, 705–716 (2017). https://doi.org/10.1007/s10463-016-0561-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-016-0561-1

Keywords

Search

Navigation