Science Journal of Applied Mathematics and Statistics

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Nonparametric Penalized Spline Model Calibrated Estimator in Complex Survey with Known Auxiliary Information at Both Cluster and Element Levels

Received: 27 November 2020    Accepted: 16 February 2021    Published: 10 March 2021
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Abstract

The present study uses penalized splines (p- spline) to estimate the functional relationship between the survey variable and the auxiliary variable in a complex survey design; where a population divided into clusters is in turn subdivided into strata. This study has considered a case of auxiliary information present at two levels; at both cluster and element levels. The study further applied model calibration technique by penalty function to estimate the population total. The calibration problems at both levels have been treated as optimization problems and solved using penalty functions to derive the estimators for this study. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, the study expects that the variable of interest’s fitted values meets such constraints. This study runs a Monte Carlo simulation to assess the finite sample performance of the penalized spline model calibrated estimator under complex survey data. Simulation studies were conducted to compare the efficiency of p-spline model calibrated estimator with Horvitz Thompson estimator (HT) by mean squared error (MSE) criterion. This study shows that the p-spline model-based estimator is generally more efficient than the HT in terms of the mean squared error. The results have also shown that the estimator obtained is unbiased, consistent and very robust because it does not fail if the model is misspecified for the data.

DOI 10.11648/j.sjams.20210901.13
Published in Science Journal of Applied Mathematics and Statistics (Volume 9, Issue 1, February 2021)
Page(s) 20-32
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), 2024. Published by Science Publishing Group

Keywords

Penalized Spline, Nonparametric Model, Auxilliary Information and Optimization Problem

References
[1] Breidt, F. J. and Opsomer, J. D. (2000). Local Polynomial Regression Estimation in Survey Sampling. Annals of Statistics, 28: 1026-1053.
[2] Clair, l. (2016). Nonparametric kernel estimation methods using Complex survey data, PhD thesis, mcmaster university, Main St. West, Hamilton Ontario.
[3] De Boor, C. (2001). A Practical Guide To Splines (Revised Edition). Springer, New York.
[4] Deville, J. C. and Sarndal C. E. (1992). Calibration Estimators in Survey Sampling. Journal of the American Statistical Association, 87: 376-382.
[5] Eilers, P. H. C. and Marx, B. D. (1996). Flexible Smoothing with B-Splines and Penalties (with discussion). Statistical Science, 11: 89-121.
[6] Eubank, R. L. (1988). Spline smoothing and Nonparametric regression. New york and Basel: Marcel Dekker.
[7] Horvitz, D. G and Thompson, D. J. (1952). A Generalization of sampling without Replacement from Finite Universe. Journal of American Statistical Association, 47: 663-685.
[8] Nthiwa Janiffer Mwende et al. (2020). Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels. American Journal of Theoretical and Applied Statistics. 4: 162-172.
[9] Pinheiro, J. C. and Bates, D. M (2000). Mixed-effects models in S and S-PLUS, Springer: New York.
[10] Rao, S. S. (1984). Optimization Theory and Applications. Wiley Eastern Limited.
[11] Sahar, Z. Z. (2012). Model-based methods for robust finite population inference in the presence of external information. The University of Michigan.
[12] Sayed, A. M. (2010). Nonparametric kernel density estimation using auxiliary information from complex survey data. Masters thesis.
[13] Wahba, G. (1990). A comparison of GCV and GLM for Choosing the Smoothing parameters in the Generalized Spline smoothing problem. Annals of Statistics, 4: 1378-1407.
[14] Wu, C. and Sitter, R. R. (2001). A Model Calibration Approach to Using Complete Auxiliary Information from Survey Data. Journal of American Statistical Association, 96: 185-193.
[15] Zheng and Little (2003): Penalized Spline Model-Based Estimation of the Finite Populations Total Journal of Official Statistics, 19: 99-117.
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    Nthiwa Janiffer Mwende, Ali Salim Islam, Pius Nderitu Kihara. (2021). Nonparametric Penalized Spline Model Calibrated Estimator in Complex Survey with Known Auxiliary Information at Both Cluster and Element Levels. Science Journal of Applied Mathematics and Statistics, 9(1), 20-32. https://doi.org/10.11648/j.sjams.20210901.13

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    ACS Style

    Nthiwa Janiffer Mwende; Ali Salim Islam; Pius Nderitu Kihara. Nonparametric Penalized Spline Model Calibrated Estimator in Complex Survey with Known Auxiliary Information at Both Cluster and Element Levels. Sci. J. Appl. Math. Stat. 2021, 9(1), 20-32. doi: 10.11648/j.sjams.20210901.13

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    AMA Style

    Nthiwa Janiffer Mwende, Ali Salim Islam, Pius Nderitu Kihara. Nonparametric Penalized Spline Model Calibrated Estimator in Complex Survey with Known Auxiliary Information at Both Cluster and Element Levels. Sci J Appl Math Stat. 2021;9(1):20-32. doi: 10.11648/j.sjams.20210901.13

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  • @article{10.11648/j.sjams.20210901.13,
      author = {Nthiwa Janiffer Mwende and Ali Salim Islam and Pius Nderitu Kihara},
      title = {Nonparametric Penalized Spline Model Calibrated Estimator in Complex Survey with Known Auxiliary Information at Both Cluster and Element Levels},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {9},
      number = {1},
      pages = {20-32},
      doi = {10.11648/j.sjams.20210901.13},
      url = {https://doi.org/10.11648/j.sjams.20210901.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20210901.13},
      abstract = {The present study uses penalized splines (p- spline) to estimate the functional relationship between the survey variable and the auxiliary variable in a complex survey design; where a population divided into clusters is in turn subdivided into strata. This study has considered a case of auxiliary information present at two levels; at both cluster and element levels. The study further applied model calibration technique by penalty function to estimate the population total. The calibration problems at both levels have been treated as optimization problems and solved using penalty functions to derive the estimators for this study. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, the study expects that the variable of interest’s fitted values meets such constraints. This study runs a Monte Carlo simulation to assess the finite sample performance of the penalized spline model calibrated estimator under complex survey data. Simulation studies were conducted to compare the efficiency of p-spline model calibrated estimator with Horvitz Thompson estimator (HT) by mean squared error (MSE) criterion. This study shows that the p-spline model-based estimator is generally more efficient than the HT in terms of the mean squared error. The results have also shown that the estimator obtained is unbiased, consistent and very robust because it does not fail if the model is misspecified for the data.},
     year = {2021}
    }
    

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    AU  - Nthiwa Janiffer Mwende
    AU  - Ali Salim Islam
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    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
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    UR  - https://doi.org/10.11648/j.sjams.20210901.13
    AB  - The present study uses penalized splines (p- spline) to estimate the functional relationship between the survey variable and the auxiliary variable in a complex survey design; where a population divided into clusters is in turn subdivided into strata. This study has considered a case of auxiliary information present at two levels; at both cluster and element levels. The study further applied model calibration technique by penalty function to estimate the population total. The calibration problems at both levels have been treated as optimization problems and solved using penalty functions to derive the estimators for this study. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, the study expects that the variable of interest’s fitted values meets such constraints. This study runs a Monte Carlo simulation to assess the finite sample performance of the penalized spline model calibrated estimator under complex survey data. Simulation studies were conducted to compare the efficiency of p-spline model calibrated estimator with Horvitz Thompson estimator (HT) by mean squared error (MSE) criterion. This study shows that the p-spline model-based estimator is generally more efficient than the HT in terms of the mean squared error. The results have also shown that the estimator obtained is unbiased, consistent and very robust because it does not fail if the model is misspecified for the data.
    VL  - 9
    IS  - 1
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Author Information
  • Department of Mathematics, Egerton University, Nakuru, Kenya

  • Department of Mathematics, Egerton University, Nakuru, Kenya

  • Department of Financial and Actuarial Mathematics, Technical University of Kenya, Nairobi, Kenya

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