Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels

Janiffer Mwende Nthiwa; Ali Salim Islam; Pius Nderitu Kihara

doi:doi:10.11648/j.ajtas.20200904.20

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Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels

Janiffer Mwende Nthiwa, Ali Salim Islam, Pius Nderitu Kihara

Published in American Journal of Theoretical and Applied Statistics (Volume 9, Issue 4)

Received: 16 July 2020 Accepted: 8 August 2020 Published: 19 August 2020

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Abstract

Nonparametric methods are rich classes of statistical tools that have gained acceptance in most areas of statistics. They have been used in the past by researchers to fit missing values in the presence of auxiliary variables in a sampling survey. Nonparametric methods have been preferred to parametric methods because they make it possible to analyze data, estimate trends and conduct inference without having to fully specify a parametric model for the data. This study, therefore, presents some new attempts in the complex survey through the nonparametric imputation of missing values by the use of both penalized splines and neural networks. More precisely, the study adopted a neural network and penalized splines to estimate the functional relationship between the survey variable and the auxiliary variables. This complex survey data was sampled through a cluster - strata design where a population is divided into clusters which are in turn subdivided into strata. Once missing values have been imputed, this study performs a model calibration with auxiliary information assumed completely available at the cluster level. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, then it is expected that the fitted values of the variable of interest should satisfy such constraints too. The population total estimators are derived by treating the calibration problems at cluster level as optimization problems and solving it by the method of penalty functions. A Monte Carlo simulation was run to assess the finite sample performance of the estimators under complex survey data. The efficiency of the estimator’s performance was then checked by MSE criterion. A comparison of the penalized spline model calibration and neural network model calibration estimators was done with Horvitz Thompson estimator. From the results, the two nonparametric estimator’s performances seem closer to that of Horvitz Thompson estimator and are both unbiased and consistent.

Published in	American Journal of Theoretical and Applied Statistics (Volume 9, Issue 4)
DOI	10.11648/j.ajtas.20200904.20
Page(s)	162-172
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), 2020. Published by Science Publishing Group

Keywords

Nonparametric Model, Auxiliary Information, Neural Network, Penalized Splines, 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]	Cochran, W. G.. (1977). Sampling techniques (3^rd ed.)., New york: John Wiley & sons.
[4]	Deville J. C. and Sarndal C. E. (1992). Calibration Estimators in Survey Sampling. Journal of the American Statistical Association, 87: 376-382.
[5]	Kihara, P. N. (2012). Estimation of Finite Population Total in the Face of Missing Values Using Model Calibration and Model Assistance on Semiparametric and Nonparametric Models. PhD thesis, JKUAT.
[6]	Montanari, G. E. and Ranalli, S. (2003). Nonparametric Model Calibration Estimation in Survey Sampling. Journal of Official Statistics, 2: 1-40.
[7]	Nordbotten, S. (1996). Neural Network imputation applied to the Norwegian 1990 population census data. Journal of Official Statistics, 12: 385-401.
[8]	Otieno et al., (2007). Nonparametric Model Assisted Model Calibrated Estimation in Two Stage Survey Sampling. The East African Journal of Statistics, 3: 261-281.
[9]	Rao, S. S. (1984). Optimization Theory and Applications. Wiley Eastern Limited Sahar, Z. Z. (2012). Model-based methods for robust finite population inference in the presence of external information. The University of Michigan.
[10]	Sahar, Z. Z. (2012). Model-based methods for robust finite population inference in the presence
[11]	of external information. The University of Michigan.
[12]	Sarndal, C. E., Swensson B. and Wretman J. (1992). Model Assisted Survey Sampling. Springer-Verlag, New York.
[13]	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.

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APA Style

Janiffer Mwende Nthiwa, Ali Salim Islam, Pius Nderitu Kihara. (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, 9(4), 162-172. https://doi.org/10.11648/j.ajtas.20200904.20

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ACS Style

Janiffer Mwende Nthiwa; Ali Salim Islam; Pius Nderitu Kihara. Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels. Am. J. Theor. Appl. Stat. 2020, 9(4), 162-172. doi: 10.11648/j.ajtas.20200904.20

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AMA Style

Janiffer Mwende Nthiwa, Ali Salim Islam, Pius Nderitu Kihara. Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels. Am J Theor Appl Stat. 2020;9(4):162-172. doi: 10.11648/j.ajtas.20200904.20

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@article{10.11648/j.ajtas.20200904.20,
  author = {Janiffer Mwende Nthiwa and Ali Salim Islam and Pius Nderitu Kihara},
  title = {Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {4},
  pages = {162-172},
  doi = {10.11648/j.ajtas.20200904.20},
  url = {https://doi.org/10.11648/j.ajtas.20200904.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200904.20},
  abstract = {Nonparametric methods are rich classes of statistical tools that have gained acceptance in most areas of statistics. They have been used in the past by researchers to fit missing values in the presence of auxiliary variables in a sampling survey. Nonparametric methods have been preferred to parametric methods because they make it possible to analyze data, estimate trends and conduct inference without having to fully specify a parametric model for the data. This study, therefore, presents some new attempts in the complex survey through the nonparametric imputation of missing values by the use of both penalized splines and neural networks. More precisely, the study adopted a neural network and penalized splines to estimate the functional relationship between the survey variable and the auxiliary variables. This complex survey data was sampled through a cluster - strata design where a population is divided into clusters which are in turn subdivided into strata. Once missing values have been imputed, this study performs a model calibration with auxiliary information assumed completely available at the cluster level. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, then it is expected that the fitted values of the variable of interest should satisfy such constraints too. The population total estimators are derived by treating the calibration problems at cluster level as optimization problems and solving it by the method of penalty functions. A Monte Carlo simulation was run to assess the finite sample performance of the estimators under complex survey data. The efficiency of the estimator’s performance was then checked by MSE criterion. A comparison of the penalized spline model calibration and neural network model calibration estimators was done with Horvitz Thompson estimator. From the results, the two nonparametric estimator’s performances seem closer to that of Horvitz Thompson estimator and are both unbiased and consistent.},
 year = {2020}
}

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TY  - JOUR
T1  - Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels
AU  - Janiffer Mwende Nthiwa
AU  - Ali Salim Islam
AU  - Pius Nderitu Kihara
Y1  - 2020/08/19
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200904.20
DO  - 10.11648/j.ajtas.20200904.20
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  - 162
EP  - 172
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200904.20
AB  - Nonparametric methods are rich classes of statistical tools that have gained acceptance in most areas of statistics. They have been used in the past by researchers to fit missing values in the presence of auxiliary variables in a sampling survey. Nonparametric methods have been preferred to parametric methods because they make it possible to analyze data, estimate trends and conduct inference without having to fully specify a parametric model for the data. This study, therefore, presents some new attempts in the complex survey through the nonparametric imputation of missing values by the use of both penalized splines and neural networks. More precisely, the study adopted a neural network and penalized splines to estimate the functional relationship between the survey variable and the auxiliary variables. This complex survey data was sampled through a cluster - strata design where a population is divided into clusters which are in turn subdivided into strata. Once missing values have been imputed, this study performs a model calibration with auxiliary information assumed completely available at the cluster level. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, then it is expected that the fitted values of the variable of interest should satisfy such constraints too. The population total estimators are derived by treating the calibration problems at cluster level as optimization problems and solving it by the method of penalty functions. A Monte Carlo simulation was run to assess the finite sample performance of the estimators under complex survey data. The efficiency of the estimator’s performance was then checked by MSE criterion. A comparison of the penalized spline model calibration and neural network model calibration estimators was done with Horvitz Thompson estimator. From the results, the two nonparametric estimator’s performances seem closer to that of Horvitz Thompson estimator and are both unbiased and consistent.
VL  - 9
IS  - 4
ER  -

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Author Information

Janiffer Mwende Nthiwa

Department of Mathematics, Egerton University, Nakuru, Kenya
Ali Salim Islam

Department of Mathematics, Egerton University, Nakuru, Kenya
Pius Nderitu Kihara

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

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APA Style

Janiffer Mwende Nthiwa, Ali Salim Islam, Pius Nderitu Kihara. (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, 9(4), 162-172. https://doi.org/10.11648/j.ajtas.20200904.20

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ACS Style

Janiffer Mwende Nthiwa; Ali Salim Islam; Pius Nderitu Kihara. Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels. Am. J. Theor. Appl. Stat. 2020, 9(4), 162-172. doi: 10.11648/j.ajtas.20200904.20

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AMA Style

Janiffer Mwende Nthiwa, Ali Salim Islam, Pius Nderitu Kihara. Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels. Am J Theor Appl Stat. 2020;9(4):162-172. doi: 10.11648/j.ajtas.20200904.20

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@article{10.11648/j.ajtas.20200904.20,
  author = {Janiffer Mwende Nthiwa and Ali Salim Islam and Pius Nderitu Kihara},
  title = {Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {4},
  pages = {162-172},
  doi = {10.11648/j.ajtas.20200904.20},
  url = {https://doi.org/10.11648/j.ajtas.20200904.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200904.20},
  abstract = {Nonparametric methods are rich classes of statistical tools that have gained acceptance in most areas of statistics. They have been used in the past by researchers to fit missing values in the presence of auxiliary variables in a sampling survey. Nonparametric methods have been preferred to parametric methods because they make it possible to analyze data, estimate trends and conduct inference without having to fully specify a parametric model for the data. This study, therefore, presents some new attempts in the complex survey through the nonparametric imputation of missing values by the use of both penalized splines and neural networks. More precisely, the study adopted a neural network and penalized splines to estimate the functional relationship between the survey variable and the auxiliary variables. This complex survey data was sampled through a cluster - strata design where a population is divided into clusters which are in turn subdivided into strata. Once missing values have been imputed, this study performs a model calibration with auxiliary information assumed completely available at the cluster level. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, then it is expected that the fitted values of the variable of interest should satisfy such constraints too. The population total estimators are derived by treating the calibration problems at cluster level as optimization problems and solving it by the method of penalty functions. A Monte Carlo simulation was run to assess the finite sample performance of the estimators under complex survey data. The efficiency of the estimator’s performance was then checked by MSE criterion. A comparison of the penalized spline model calibration and neural network model calibration estimators was done with Horvitz Thompson estimator. From the results, the two nonparametric estimator’s performances seem closer to that of Horvitz Thompson estimator and are both unbiased and consistent.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Population Total Estimation in a Complex Survey by Nonparametric Model Calibration Using Penalty Function Method with Auxiliary Information Known at Cluster Levels
AU  - Janiffer Mwende Nthiwa
AU  - Ali Salim Islam
AU  - Pius Nderitu Kihara
Y1  - 2020/08/19
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200904.20
DO  - 10.11648/j.ajtas.20200904.20
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  - 162
EP  - 172
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200904.20
AB  - Nonparametric methods are rich classes of statistical tools that have gained acceptance in most areas of statistics. They have been used in the past by researchers to fit missing values in the presence of auxiliary variables in a sampling survey. Nonparametric methods have been preferred to parametric methods because they make it possible to analyze data, estimate trends and conduct inference without having to fully specify a parametric model for the data. This study, therefore, presents some new attempts in the complex survey through the nonparametric imputation of missing values by the use of both penalized splines and neural networks. More precisely, the study adopted a neural network and penalized splines to estimate the functional relationship between the survey variable and the auxiliary variables. This complex survey data was sampled through a cluster - strata design where a population is divided into clusters which are in turn subdivided into strata. Once missing values have been imputed, this study performs a model calibration with auxiliary information assumed completely available at the cluster level. The reasoning behind model calibration is that if the calibration constraints are satisfied by the auxiliary variable, then it is expected that the fitted values of the variable of interest should satisfy such constraints too. The population total estimators are derived by treating the calibration problems at cluster level as optimization problems and solving it by the method of penalty functions. A Monte Carlo simulation was run to assess the finite sample performance of the estimators under complex survey data. The efficiency of the estimator’s performance was then checked by MSE criterion. A comparison of the penalized spline model calibration and neural network model calibration estimators was done with Horvitz Thompson estimator. From the results, the two nonparametric estimator’s performances seem closer to that of Horvitz Thompson estimator and are both unbiased and consistent.
VL  - 9
IS  - 4
ER  -

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