Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE

Samuel Joel Kamun; Richard Simwa; Stanley Sewe

doi:doi:10.11648/j.ajtas.20211004.14

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Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE

Samuel Joel Kamun, Richard Simwa, Stanley Sewe

Published in American Journal of Theoretical and Applied Statistics (Volume 10, Issue 4)

Received: 6 August 2021 Accepted: 21 August 2021 Published: 31 August 2021

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Abstract

The analysis of sample-based studies involving sampling designs for small sample size, is challenging because the sample selection probabilities (as well as the sample weights) is dependent on the response variable and covariates. The study has focused on using systems of weighted regression estimating equations, using different modified weights, to estimate the coefficients of Weighted Likelihood Estimators. Usually, the design-consistent (Weighted) estimators are obtained by solving (sample) weighted estimating equations. They are then used to construct estimates which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A. The purpose of our study is to compare derived Estimators of the weighted regression estimating equations for estimating the coefficients of Weighted Likelihood Estimators, the Semi-Parametric Weighted Likelihood Estimator, SPW and the Weighted Conditional Pseudo Likelihood Estimator, WCPE with the conventional Horvitz-Thompson Weighted Likelihood Estimator, using relative efficiency, sample bias and Standard Error for small sample size. The constructed estimates from the system of weighted regression estimating equations, using different modified weights, are actually the Weighted Likelihood Estimators. The study compared the two new estimators, the Semi-parametric weighted estimator, SPW and the Weighted Conditional Pseudo Likelihood estimator, WCPE, for both the unmodified and modified Weights, which were found to have better relative efficiency and smaller finite small sample bias than the estimates from conventional Horvitz-Thompson Weighted Estimator, for both generated and for real data. The outcome of the tests show strong similarity in performance to those obtained using the simulated data. Estimates were constructed which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A.

Published in	American Journal of Theoretical and Applied Statistics (Volume 10, Issue 4)
DOI	10.11648/j.ajtas.20211004.14
Page(s)	202-207
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

Semi Parametric, Imputation, Estimating Error, Small Samples, Estimators, Relative Efficiency, Sample Bias

References

[1]	Scott A. J, Wild CJ. Fitting logistic models under case-control or choice based sampling. JRSS, B 1986; 48: 170–182.
[2]	Graubard B. I, Fears TR, Gail MH. Effects of cluster sampling on epidemiologic analysis in population-based case-control studies. Biometrics 1989; 45: 1053–1071.
[3]	Korn EL, Graubard BI. Analysis of Health Surveys. Wiley: New York, NY, 1999.
[4]	Scott AJ, Wild CJ. Population-based case-control studies. In Handbook of Statistics: Sample Surveys: Inference and Analysis, Vol. 29B, Pfefferman D, Rao CR (eds). Elsevier: Amsterdam, 2009; 431–453.
[5]	Li Y, Graubard BI, DiGaetano R. Weighting methods for population-based case-control studies with complex sampling. JRSS (C) 2011; 60: 165–185.
[6]	Rust KF, Rao JNK. Variance estimation for complex surveys using replication techniques. Statistical Methods in Medical Research 1996; 5: 283–310.
[7]	Pfeffermann D, Sverchkov M. Inference under informative sampling. In Handbook of Statistics: Sample Surveys: Inference and Analysis, Vol. 29B, Pfefferman D, Rao CR (eds). Elsevier: Amsterdam, 2009; 455–487.
[8]	Robins J. M, Rotnitzky A, Zhao L-P. Estimation of regression coefficients when some regressors are not always observed. Journal of American Statistical Association 1994; 89: 846–866.
[9]	Lumley T, Shaw PA, Dai JY. Connections between survey calibration estimators and semiparametric models for incomplete data. International Statistical Review 2011; 97 (2): 200–220.
[10]	Pfeffermann D, Sverchkov M. Parametric and semi-parametric estimation of regression models fitted to survey data. Sankhya, Series B 1999; 61: 166–186.
[11]	Pfeffermann D, Krieger A, Rinott Y. Parametric distributions of complex survey data under informative probability sampling}. Statistica Sinica 1981; 8: 1087–1114.
[12]	Gong G, Samaniego F. Pseudo maximum likelihood estimation: theory and applications. The Annals of Statistics 1981; 9: 861–869.
[13]	Pfeffermann D, Sverchkov M. Fitting generalized linear models under informative sampling. In Analysis of Survey Data, Skinner C, Chambers R (eds). Wiley: New York, 2003; 175–195.
[14]	Yuan K-H, Jennrich R. Estimating equations with nuisance parameters: theory and applications. Annals of the Institute of Statistical Mathematics 2000; 52: 343–350.
[15]	Krewski D., Rao J. N. K. Inference from stratified samples: properties of linearization, jackknife and balanced repeated replication methods. The Annals of Statistics 1981; 9: 1010–1019.
[16]	Lumley T. Complex Surveys: A Guide to Analysis Using R. Wiley: Hoboken NJ, 2010.
[17]	Landsman, V. and Graudbard, B. I. Efficient analysis of case-control studies. Wiley on line library.com, DOI: 10.1002/SIM 5530, 2012.
[18]	J. F. Lawless, J. D. Kalbfleisch and C. J. Wild. Semiparametric Methods for Response-Selective and Missing Data Problems in Regression, Wiley, Journal of the Royal Statistical Society. Series B, Vol. 61, No. 2 (1999), pp. 413-438 (26 pages).
[19]	Edwin L. Bradley. The Equivalence of Maximum Likelihood and Weighted Least Squares Estimates in the Exponential class of Distribution, Wiley, Journal of the American Statistical Association; 68: 341, 199-200, DOI (1973).
[20]	Robert J. Gray. Weighted Estimating Equations for Linear Regression Analysis of Clustered Failure Time Data, Lifetime Data 9 (2): 123-138, (2003), DOI: 10.1023/A:1022932117951.
[21]	Adhidev Biswas, Suman Majumder, Pratim Guha Niyogi and Aye nendranath Basu. A Weighted Likelihood Approach to Problems in Survival Data, sankhaya (2019). http://doi.org/10.1007/513571-019-00214-W
[22]	Samuel J. Kamun, Richard O. Simwa and Stanley Sewe. On Derivation of the Semi-Parametric Weighted Likelihood Estimator, SPW, and the Weighted Conditional Pseudo Likelihood Estimator, WPCE, Far East Journal of Theoretical Statistics © 2021 Pushpa Publishing House, Prayagraj, India, http://www.pphmj.com, http://dx.doi.org/10.17654/

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Samuel Joel Kamun, Richard Simwa, Stanley Sewe. (2021). Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE. American Journal of Theoretical and Applied Statistics, 10(4), 202-207. https://doi.org/10.11648/j.ajtas.20211004.14

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ACS Style

Samuel Joel Kamun; Richard Simwa; Stanley Sewe. Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE. Am. J. Theor. Appl. Stat. 2021, 10(4), 202-207. doi: 10.11648/j.ajtas.20211004.14

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AMA Style

Samuel Joel Kamun, Richard Simwa, Stanley Sewe. Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE. Am J Theor Appl Stat. 2021;10(4):202-207. doi: 10.11648/j.ajtas.20211004.14

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@article{10.11648/j.ajtas.20211004.14,
  author = {Samuel Joel Kamun and Richard Simwa and Stanley Sewe},
  title = {Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {4},
  pages = {202-207},
  doi = {10.11648/j.ajtas.20211004.14},
  url = {https://doi.org/10.11648/j.ajtas.20211004.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211004.14},
  abstract = {The analysis of sample-based studies involving sampling designs for small sample size, is challenging because the sample selection probabilities (as well as the sample weights) is dependent on the response variable and covariates. The study has focused on using systems of weighted regression estimating equations, using different modified weights, to estimate the coefficients of Weighted Likelihood Estimators. Usually, the design-consistent (Weighted) estimators are obtained by solving (sample) weighted estimating equations. They are then used to construct estimates which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A. The purpose of our study is to compare derived Estimators of the weighted regression estimating equations for estimating the coefficients of Weighted Likelihood Estimators, the Semi-Parametric Weighted Likelihood Estimator, SPW and the Weighted Conditional Pseudo Likelihood Estimator, WCPE with the conventional Horvitz-Thompson Weighted Likelihood Estimator, using relative efficiency, sample bias and Standard Error for small sample size. The constructed estimates from the system of weighted regression estimating equations, using different modified weights, are actually the Weighted Likelihood Estimators. The study compared the two new estimators, the Semi-parametric weighted estimator, SPW and the Weighted Conditional Pseudo Likelihood estimator, WCPE, for both the unmodified and modified Weights, which were found to have better relative efficiency and smaller finite small sample bias than the estimates from conventional Horvitz-Thompson Weighted Estimator, for both generated and for real data. The outcome of the tests show strong similarity in performance to those obtained using the simulated data. Estimates were constructed which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A.},
 year = {2021}
}

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TY  - JOUR
T1  - Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE
AU  - Samuel Joel Kamun
AU  - Richard Simwa
AU  - Stanley Sewe
Y1  - 2021/08/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211004.14
DO  - 10.11648/j.ajtas.20211004.14
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  - 202
EP  - 207
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211004.14
AB  - The analysis of sample-based studies involving sampling designs for small sample size, is challenging because the sample selection probabilities (as well as the sample weights) is dependent on the response variable and covariates. The study has focused on using systems of weighted regression estimating equations, using different modified weights, to estimate the coefficients of Weighted Likelihood Estimators. Usually, the design-consistent (Weighted) estimators are obtained by solving (sample) weighted estimating equations. They are then used to construct estimates which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A. The purpose of our study is to compare derived Estimators of the weighted regression estimating equations for estimating the coefficients of Weighted Likelihood Estimators, the Semi-Parametric Weighted Likelihood Estimator, SPW and the Weighted Conditional Pseudo Likelihood Estimator, WCPE with the conventional Horvitz-Thompson Weighted Likelihood Estimator, using relative efficiency, sample bias and Standard Error for small sample size. The constructed estimates from the system of weighted regression estimating equations, using different modified weights, are actually the Weighted Likelihood Estimators. The study compared the two new estimators, the Semi-parametric weighted estimator, SPW and the Weighted Conditional Pseudo Likelihood estimator, WCPE, for both the unmodified and modified Weights, which were found to have better relative efficiency and smaller finite small sample bias than the estimates from conventional Horvitz-Thompson Weighted Estimator, for both generated and for real data. The outcome of the tests show strong similarity in performance to those obtained using the simulated data. Estimates were constructed which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A.
VL  - 10
IS  - 4
ER  -

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Author Information

Samuel Joel Kamun

Department of Mathematics and Computing Studies, Faculty of Science, Catholic University of Eastern Africa, Nairobi, Kenya
Richard Simwa

Department of Mathematics, Faculty of Science, University of Nairobi, Nairobi, Kenya
Stanley Sewe

Department of Mathematics and Computing Studies, Faculty of Science, Catholic University of Eastern Africa, Nairobi, Kenya

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APA Style

Samuel Joel Kamun, Richard Simwa, Stanley Sewe. (2021). Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE. American Journal of Theoretical and Applied Statistics, 10(4), 202-207. https://doi.org/10.11648/j.ajtas.20211004.14

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ACS Style

Samuel Joel Kamun; Richard Simwa; Stanley Sewe. Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE. Am. J. Theor. Appl. Stat. 2021, 10(4), 202-207. doi: 10.11648/j.ajtas.20211004.14

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AMA Style

Samuel Joel Kamun, Richard Simwa, Stanley Sewe. Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE. Am J Theor Appl Stat. 2021;10(4):202-207. doi: 10.11648/j.ajtas.20211004.14

Copy | Download

@article{10.11648/j.ajtas.20211004.14,
  author = {Samuel Joel Kamun and Richard Simwa and Stanley Sewe},
  title = {Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {4},
  pages = {202-207},
  doi = {10.11648/j.ajtas.20211004.14},
  url = {https://doi.org/10.11648/j.ajtas.20211004.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211004.14},
  abstract = {The analysis of sample-based studies involving sampling designs for small sample size, is challenging because the sample selection probabilities (as well as the sample weights) is dependent on the response variable and covariates. The study has focused on using systems of weighted regression estimating equations, using different modified weights, to estimate the coefficients of Weighted Likelihood Estimators. Usually, the design-consistent (Weighted) estimators are obtained by solving (sample) weighted estimating equations. They are then used to construct estimates which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A. The purpose of our study is to compare derived Estimators of the weighted regression estimating equations for estimating the coefficients of Weighted Likelihood Estimators, the Semi-Parametric Weighted Likelihood Estimator, SPW and the Weighted Conditional Pseudo Likelihood Estimator, WCPE with the conventional Horvitz-Thompson Weighted Likelihood Estimator, using relative efficiency, sample bias and Standard Error for small sample size. The constructed estimates from the system of weighted regression estimating equations, using different modified weights, are actually the Weighted Likelihood Estimators. The study compared the two new estimators, the Semi-parametric weighted estimator, SPW and the Weighted Conditional Pseudo Likelihood estimator, WCPE, for both the unmodified and modified Weights, which were found to have better relative efficiency and smaller finite small sample bias than the estimates from conventional Horvitz-Thompson Weighted Estimator, for both generated and for real data. The outcome of the tests show strong similarity in performance to those obtained using the simulated data. Estimates were constructed which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE
AU  - Samuel Joel Kamun
AU  - Richard Simwa
AU  - Stanley Sewe
Y1  - 2021/08/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211004.14
DO  - 10.11648/j.ajtas.20211004.14
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  - 202
EP  - 207
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211004.14
AB  - The analysis of sample-based studies involving sampling designs for small sample size, is challenging because the sample selection probabilities (as well as the sample weights) is dependent on the response variable and covariates. The study has focused on using systems of weighted regression estimating equations, using different modified weights, to estimate the coefficients of Weighted Likelihood Estimators. Usually, the design-consistent (Weighted) estimators are obtained by solving (sample) weighted estimating equations. They are then used to construct estimates which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A. The purpose of our study is to compare derived Estimators of the weighted regression estimating equations for estimating the coefficients of Weighted Likelihood Estimators, the Semi-Parametric Weighted Likelihood Estimator, SPW and the Weighted Conditional Pseudo Likelihood Estimator, WCPE with the conventional Horvitz-Thompson Weighted Likelihood Estimator, using relative efficiency, sample bias and Standard Error for small sample size. The constructed estimates from the system of weighted regression estimating equations, using different modified weights, are actually the Weighted Likelihood Estimators. The study compared the two new estimators, the Semi-parametric weighted estimator, SPW and the Weighted Conditional Pseudo Likelihood estimator, WCPE, for both the unmodified and modified Weights, which were found to have better relative efficiency and smaller finite small sample bias than the estimates from conventional Horvitz-Thompson Weighted Estimator, for both generated and for real data. The outcome of the tests show strong similarity in performance to those obtained using the simulated data. Estimates were constructed which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A.
VL  - 10
IS  - 4
ER  -

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