Advertisement

A Varying Coefficients Model For Estimating Finite Population Totals: A Hierarchical Bayesian Approach

  • Ciro Velasco-CruzEmail author
  • Luis Fernando Contreras-Cruz
  • Eric P. Smith
  • José E. Rodríguez
Article
  • 163 Downloads

Abstract

In some finite sampling situations, there is a primary variable that is sampled, and there are measurements on covariates for the entire population. A Bayesian hierarchical model for estimating totals for finite populations is proposed. A nonparametric linear model is assumed to explain the relationship between the dependent variable of interest and covariates. The regression coefficients in the linear model are allowed to vary as a function of a subset of covariates nonparametrically based on B-splines. The generality of this approach makes it robust and applicable to data collected using a variety of sampling techniques, provided the sample is representative of the finite population. A simulation study is carried out to evaluate the performance of the proposed model for the estimation of the population total. Results indicate accurate estimation of population totals using the approach. The modeling approach is used to estimate the total production of avocado for a large group of groves in Mexico.

Keywords

Bayesian hierarchical model Population total Varying coefficient model Auxiliary information Nonparametric regression model 

References

  1. Basu, D., 2011. Selected Works of Debabrata Basu. Springer New York, New York, NY, Ch. An Essay on the Logical Foundations of Survey Sampling, Part One, Ed. By Anirban DasGupta, pp. 167–206.Google Scholar
  2. Binder, D. A., 1982. Non-parametric Bayesian models for samples from finite populations. Journal of the Royal Statistical Society. Series B(Methodological) 44, 388–393.Google Scholar
  3. Cleveland, W. S., Grosse, E., Shyu, M. J., 1992. Local regression models. In: Chambers J. M. , Hastie T. J. , eds. Statistical Models in S. Pacific Grove, CA: Wadsworth, Ch. 8, pp. 309–376.Google Scholar
  4. Eilers, P. H. C., Marx, B. D., 1996. Flexible smoothing with b-splines and penalties (with discussion). Statistical Science 11, 89–121.Google Scholar
  5. Ericson, W. A., 1969. Subjective Bayesian models in sampling finite populations. Journal of the Royal Statistical Society. Series B(Methodological) 31, 195–234.Google Scholar
  6. —— 1988. Bayesian inference in finite populations. Handbook of Statistics 6, 213–246.Google Scholar
  7. Eubank, R. L., 1999. Nonparametric regression and spline smoothing. New York: Marcel Dekker.Google Scholar
  8. Fan, J., Zhang, W., 2008. Statistical methods with varying coefficient models. Statistics and its Interface 1, 179–195.Google Scholar
  9. Gelfand, A. E., 2003. Spatial modeling with spatially varying coefficient processes. Journal of the American Statistical Association 98 (462), 387–396.Google Scholar
  10. Gelfand, A. E., Ghosh, S. K., 1998. Model choice: A minimum posterior predictive loss approach. Biometrika 85 (1), 1–11.Google Scholar
  11. Gelman, A., 2006. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 1 (3), 515–533.Google Scholar
  12. Ghosh, M., Meeden, G., 1997. Bayesian methods for finite population sampling. London: Chapman & Hall.Google Scholar
  13. Gregoire, T. G., 1998. Design-based and model-based inference in survey sampling: appreciating the difference. Can. J. For. Res. 28, 1429–1447.Google Scholar
  14. Hastie, T., Tibshirani, R., 1993. Varying-coefficient models. Journal of the Royal Statistical Society. Series B(Methodological) 55, 757–796.Google Scholar
  15. Hastie, T., Tibshirani, R., Friedman, J., 2009. The elements of statistical learning: data mining, inference, and prediction., 2nd Edition. Springer.Google Scholar
  16. Hogan, J. W., Tchernis, R., 2004. Bayesian factor analysis for spatially correlated data, with application to summarizing area-level material deprivation from census data. Journal of the American Statistical Association (Applications and Case Studies) 99 (466), 314–324.Google Scholar
  17. Hoover, D. R., Rice, J. A., Wu, C. O., Yang, L. P., 1998. Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85, 809–822.Google Scholar
  18. Lambert, P. C., Sutton, A. J., Burton, P. R., Abrams, K. R., Jones, D. R., 2005. How vague is vague? A simulation study of the impact of the use of vague prior distributions in MCMC using WinBUGS. Statistics in Medicine 24, 2401–2428.Google Scholar
  19. Little, R. J., 2004. To model or not to model? Competing modes of inference for finite population sampling. American Statistical Association 99, 546–556.Google Scholar
  20. Novák, P., Kosina, V., 2012. Using the superpopulation model for imputation and variance computation in survey sampling. Statistika 49 (1), 56–69.Google Scholar
  21. Plummer, M., Best, N., Cowles, K., Vines, K., Sarkar, D., Bates, D., Almond, R., Magnusson, A., 2015. Output Analysis and Diagnostics for MCMC. R Foundation for Statistical Computing, version 0.18-1 Edition.Google Scholar
  22. Ruppert, D., Wand, M., Carroll, R. J., 2003. Semiparametric regression, 1st Edition. Cambridge University Press, Cambridge UK.Google Scholar
  23. Ruppert, D., Wand, M. P., Carroll, R. J., 2009. Semiparametric regression during 2003-2007. Electronic Journal of Statistics 3, 1193–1256.Google Scholar
  24. Sarndal, C. E., Swensson, B., Wretman, J., 1992. Model assisted survey sampling, 1st Edition. Springer Verlag, New York.Google Scholar
  25. Scott, A. J., 1977. Large-sample posterior distributions for finite populations. The Annals of Mathematical Statistics 42, 1113–1117.Google Scholar

Copyright information

© International Biometric Society 2016

Authors and Affiliations

  • Ciro Velasco-Cruz
    • 1
    Email author
  • Luis Fernando Contreras-Cruz
    • 2
  • Eric P. Smith
    • 3
  • José E. Rodríguez
    • 4
  1. 1.Statistics ProgramColegio de PostgraduadosMexicoMexico
  2. 2.Departamento de FitotécniaUniversidad Autónoma ChapingoTexcocoMexico
  3. 3.Department of StatisticsVirginia TechBlacksburgUSA
  4. 4.Universidad de GuanajuatoGuanajuatoMexico

Personalised recommendations