| Peer-Reviewed

On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total

Received: 17 February 2017     Accepted: 1 March 2017     Published: 21 March 2017
Views:       Downloads:
Abstract

The precision of an estimator is at times discussed regarding the variance. Usually, the exact value of the variance is unknown. The discussion relies on unknown populace quantities. When a researcher obtains the survey data, an estimate of the variance can, therefore, be calculated. When survey results are presented, it is good practice to provide variance estimates for the estimator used in the study. The estimator of the variance can further be used to construct confidence interval, assuming that the sampling distribution of estimator is approximately normal. This study proposes estimation of standard error and confidence interval for a nonparametric regression estimator for a finite population using bootstrapping method. The idea behind bootstrapping is to carry out computations on the collected data. Computation activity assists in estimating the disparity of statistics that are themselves computed from the same data. The variance of the Nadaraya-Watson estimator is derived, based on bootstrap procedure. This operation has led to the derivation of confidence interval associated with Nadaraya-Watson estimator of the population total. A simulation study has been carried out. The overall conclusion is that the confidence interval associated with Nadaraya-Watson estimator is tighter than all the other estimators (Horvitz-Thompson estimator, Local linear estimator, and Ratio estimator).

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 2)
DOI 10.11648/j.ajtas.20170602.17
Page(s) 117-122
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Bootstrap, Nonparametric Regression Model, Confidence Interval, Finite Population Total

References
[1] Altman, N., & Léger, C. (1995). Bandwidth selection for kernel distribution function estimation. Journal of Statistical Planning and Inference, 46 (2), 195–214.
[2] Binder, D. A. (1983). On the variances of asymptotically normal estimators from complex surveys. International Statistical Review/Revue Internationale de Statistique, 279–292.
[3] Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 353–360.
[4] Breidt, F., Claeskens, G., & Opsomer, J. (2005). Model-assisted estimation for complex surveys using penalised splines. Biometrika, 92 (4), 831–846.
[5] Breidt, F. J., & Opsomer, J. D. (2000). Local polynomial regression estimators in survey sampling. Annals of Statistics, 1026–1053.
[6] Chambers, R. L., & Dunstan, R. (1986). Estimating distribution functions from survey data. Biometrika, 73 (3), 597–604.
[7] Davison, A. C., & Hinkley, D. V. (1997). Bootstrap methods and their application (Vol. 1). Cambridge university press.
[8] Deshpande, J. V., Frey, J., & Ozturk, O. (2006). Nonparametric ranked-set sampling confidence intervals for quantiles of a finite population. Environmental and Ecological Statistics, 13 (1), 25–40.
[9] DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical science, 189–212.
[10] Dorfman, A. H. (1992). Nonparametric regression for estimating totals in finite populations. In Proceedings of the section on survey research methods (pp. 622–625).
[11] Efron, B. (1979). Bootstrap methods: Another look at.
[12] Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American statistical Association, 82 (397), 171–185.
[13] Green, P. J., & Silverman, B. W. (1993). Nonparametric regression and generalized linear models: a roughness penalty approach. CRC Press.
[14] Hall, P., Sheather, S. J., Jones, M., & Marron, J. S. (1991). On optimal data-based bandwidth selection in kernel density estimation. Biometrika, 263–269.
[15] Hardle, W. (1990). Applied nonparametric regression. Cambridge, UK.
[16] Kish, L., & Frankel, M. R. (1974). Inference from complex samples. Journal of the Royal Statistical Society. Series B (Methodological), 1–37.
[17] Nadaraya, E. A. (1964). On estimating regression. Theory of Probability & Its Applications, 9 (1), 141–142.
[18] Royall, R. M. (1986). Model robust confidence intervals using maximum likelihood estimators. International Statistical Review/Revue Internationale de Statistique, 221–226.
[19] Royall, R. M., & Cumberland, W. G. (1978). Variance estimation in finite population sampling. Journal of the American Statistical Association, 73 (362), 351–358.
[20] Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 65–78.
[21] Sarda, P. (1993). Smoothing parameter selection for smooth distribution functions. Journal of Statistical Planning and Inference, 35 (1), 65–75.
[22] Särndal, C.-E., Swensson, B., & Wretman, J. H. (1989). The weighted residual technique for estimating the variance of the general regression estimator of the finite population total. Biometrika, 76 (3), 527–537.
[23] Watson, G. S. (1964). Smooth regression analysis. Sankhy-a: The Indian Journal of Statistics, Series A, 359–372.
[24] Woodroofe. (1970). Bandwidth selection: classical or plug-in? Annals of Statistics, 415-438.
[25] Zheng, H., & Little, J. (2005). Inference for the population total from probability-proportionalto-size samples based on predictions from a penalized spline nonparametric model. Journal of Official Statistics, 21 (1), 1.
[26] Zheng, H., & Little, R. (2003). Penalized spline nonparametric mixed models for inference about a finite population mean from two-stage samples.
[27] Dorfman, A. H. Nonparametric Regression and the Two Sample Problem October 2009.
Cite This Article
  • APA Style

    Nicholas Makumi, Romanus Odhiambo, George Otieno Orwa, Stellamaris Adhiambo. (2017). On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total. American Journal of Theoretical and Applied Statistics, 6(2), 117-122. https://doi.org/10.11648/j.ajtas.20170602.17

    Copy | Download

    ACS Style

    Nicholas Makumi; Romanus Odhiambo; George Otieno Orwa; Stellamaris Adhiambo. On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total. Am. J. Theor. Appl. Stat. 2017, 6(2), 117-122. doi: 10.11648/j.ajtas.20170602.17

    Copy | Download

    AMA Style

    Nicholas Makumi, Romanus Odhiambo, George Otieno Orwa, Stellamaris Adhiambo. On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total. Am J Theor Appl Stat. 2017;6(2):117-122. doi: 10.11648/j.ajtas.20170602.17

    Copy | Download

  • @article{10.11648/j.ajtas.20170602.17,
      author = {Nicholas Makumi and Romanus Odhiambo and George Otieno Orwa and Stellamaris Adhiambo},
      title = {On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {6},
      number = {2},
      pages = {117-122},
      doi = {10.11648/j.ajtas.20170602.17},
      url = {https://doi.org/10.11648/j.ajtas.20170602.17},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170602.17},
      abstract = {The precision of an estimator is at times discussed regarding the variance. Usually, the exact value of the variance is unknown. The discussion relies on unknown populace quantities. When a researcher obtains the survey data, an estimate of the variance can, therefore, be calculated. When survey results are presented, it is good practice to provide variance estimates for the estimator used in the study. The estimator of the variance can further be used to construct confidence interval, assuming that the sampling distribution of estimator is approximately normal. This study proposes estimation of standard error and confidence interval for a nonparametric regression estimator for a finite population using bootstrapping method. The idea behind bootstrapping is to carry out computations on the collected data. Computation activity assists in estimating the disparity of statistics that are themselves computed from the same data. The variance of the Nadaraya-Watson estimator is derived, based on bootstrap procedure. This operation has led to the derivation of confidence interval associated with Nadaraya-Watson estimator of the population total. A simulation study has been carried out. The overall conclusion is that the confidence interval associated with Nadaraya-Watson estimator is tighter than all the other estimators (Horvitz-Thompson estimator, Local linear estimator, and Ratio estimator).},
     year = {2017}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - On Bootstrap Confidence Intervals Associated with Nonparametric Regression Estimators for A Finite Population Total
    AU  - Nicholas Makumi
    AU  - Romanus Odhiambo
    AU  - George Otieno Orwa
    AU  - Stellamaris Adhiambo
    Y1  - 2017/03/21
    PY  - 2017
    N1  - https://doi.org/10.11648/j.ajtas.20170602.17
    DO  - 10.11648/j.ajtas.20170602.17
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
    SP  - 117
    EP  - 122
    PB  - Science Publishing Group
    SN  - 2326-9006
    UR  - https://doi.org/10.11648/j.ajtas.20170602.17
    AB  - The precision of an estimator is at times discussed regarding the variance. Usually, the exact value of the variance is unknown. The discussion relies on unknown populace quantities. When a researcher obtains the survey data, an estimate of the variance can, therefore, be calculated. When survey results are presented, it is good practice to provide variance estimates for the estimator used in the study. The estimator of the variance can further be used to construct confidence interval, assuming that the sampling distribution of estimator is approximately normal. This study proposes estimation of standard error and confidence interval for a nonparametric regression estimator for a finite population using bootstrapping method. The idea behind bootstrapping is to carry out computations on the collected data. Computation activity assists in estimating the disparity of statistics that are themselves computed from the same data. The variance of the Nadaraya-Watson estimator is derived, based on bootstrap procedure. This operation has led to the derivation of confidence interval associated with Nadaraya-Watson estimator of the population total. A simulation study has been carried out. The overall conclusion is that the confidence interval associated with Nadaraya-Watson estimator is tighter than all the other estimators (Horvitz-Thompson estimator, Local linear estimator, and Ratio estimator).
    VL  - 6
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • Department of Statistics and Actuarial Sciences, College of Pure and Applied Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, College of Pure and Applied Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, College of Pure and Applied Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, College of Pure and Applied Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Sections