Published

2014-01-01

Generalized Exponential Type Estimator for Population Variance in Survey Sampling

Estimadores tipo exponencial generalizado para la varianza poblacional en muestreo de encuestas

DOI:

https://doi.org/10.15446/rce.v37n1.44368

Keywords:

Auxiliary variable, Single-phase sampling, Mean square error, Bias (en)
Información auxiliar, muestras en dos fases, error cuadrático medio, sesgo (es)

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Authors

  • Amber Asghar Virtual University of Pakistan
  • Aamir Sanaullah NCBA & E
  • Muhammad Hanif NCBA & E

In this paper, generalized exponential-type estimator has been proposed for estimating the population variance using mean auxiliary variable in singlephase sampling. Some special cases of the proposed generalized estimator have also been discussed. The expressions for the mean square error and bias of the proposed generalized estimator have been derived. The proposed generalized estimator has been compared theoretically with the usual unbiased estimator, usual ratio and product, exponential-type ratio and product, and generalized exponential-type ratio estimators and the conditions under which the proposed estimators are better than some existing estimators have also been given. An empirical study has also been carried out to demonstrate the efficiencies of the proposed estimators. 

En este artículo, de tipo exponencial generalizado ha sido propuesto con el fin de estimar la varianza poblacional a través de una variables auxiliar en muestreo en dos fases. Algunos casos especiales del estimador medio y el sesgo del estimador generalizado propuesto son derivados. El estimador es comprado teóricamente con otros disponibles en la literatura y las condiciones bajos los cuales éste es mejor. Un estudio empírico es llevado a cabo para comprar la eficiencia de los estimadores propuestos.

https://doi.org/10.15446/rce.v37n1.44368

Generalized Exponential Type Estimator for Population Variance in Survey Sampling

Estimadores tipo exponencial generalizado para la varianza poblacional en muestreo de encuestas

AMBER ASGHAR1, AAMIR SANAULLAH2, MUHAMMAD HANIF3

1Virtual University of Pakistan, Department of Mathematics & Statistics, Lahore, Pakistan. Lecturer. Email: zukhruf10@gmail.com
2NCBA & E, Department of Statistics, Lahore, Pakistan. Lecturer. Email: chaamirsanaullah@yahoo.com
3NCBA & E, Department of Statistics, Lahore, Pakistan. Associate professor. Email: drmianhanif@gmail

Abstract

In this paper, generalized exponential-type estimator has been proposed for estimating the population variance using mean auxiliary variable in single-phase sampling. Some special cases of the proposed generalized estimator have also been discussed. The expressions for the mean square error and bias of the proposed generalized estimator have been derived. The proposed generalized estimator has been compared theoretically with the usual unbiased estimator, usual ratio and product, exponential-type ratio and product, and generalized exponential-type ratio estimators and the conditions under which the proposed estimators are better than some existing estimators have also been given. An empirical study has also been carried out to demonstrate the efficiencies of the proposed estimators.

Key words: Auxiliary variable, Single-phase sampling, Mean square error, Bias.

Resumen

En este artículo, de tipo exponencial generalizado ha sido propuesto con el fin de estimar la varianza poblacional a través de una variables auxiliar en muestreo en dos fases. Algunos casos especiales del estimador medio y el sesgo del estimador generalizado propuesto son derivados. El estimador es comprado teóricamente con otros disponibles en la literatura y las condiciones bajos los cuales éste es mejor. Un estudio empírico es llevado a cabo para comprar la eficiencia de los estimadores propuestos.

Palabras clave: Información auxiliar, muestras en dos fases, error cuadrático medio, sesgo.

Texto completo disponible en PDF

References

1. A. Sanaullah,, H. Khan,, A.H. Ali, & R. Singh, (2012), 'Improved ratio-type estimators in survey sampling', Journal of Reliability and Statistical Studies 5(2), 119-132.

2. B. K. Singh, & S. Choudhary, (2012), 'Exponential chain ratio and product type estimators for finite population mean under double sampling scheme', Journal of Science Frontier Research in Mathematics and Design Sciences 12(6), 0975-5896.

3. C. Isaki, (1983), 'Variance estimation using auxiliary information', Journal of the American Statistical Association 78, 117-123.

4. D.N. Gujarati, (2004), Basic Econometrics, 4 edn, The McGraw-Hill Companies.

5. H. P. Singh, & G. Vishwakarma, (2008), 'Some families of estimators of variance of stratified random sample mean using auxiliary information', Journal of Statistical Theory and Practice 2(1), 21-43.

6. H. P. Singh, & K. Vishwakarma, (2007), 'Modified exponential ratio and product estimators for finite population mean in double sampling', Australian Journal of Statistics 36, 217-225.

7. H. P. Singh, & N. Karpe, (2010), 'Estimation of mean, ratio and product using auxiliary information in the presence of measurement errors in sample surveys', Journal of Statistical Theory and Practice 4(1), 111-136.

8. H. P. Singh, & R. S. Solanki, (2009), 'Estimation of finite population variance using auxiliary information in presence of random non-response', Gujarat Statistical Review 1, 37-637.

9. H. P. Singh, & R. S. Solanki, (2010), 'Estimation of finite population variance using auxiliary information in presence of random non-response', Gujarat Statistical Review 2, 46-58.

10. H. P. Singh, & R. S. Solanki, (2013), 'A new procedure for variance estimation in simple random sampling using auxiliary information', Statistical Papers 54(2), 479-497.

11. J. Subramani, & G. Kumarapandiyan, (2012), 'Variance estimation using quartiles and their functions of an auxiliary variable', International Journal of Statistics and Applications 2(5), 67-72.

12. M. Murthy, (1967), Sampling Theory and Methods, Calcutta Statistical Publishing Society, Kolkatta, India.

13. M. Noor-ul-Amin, & M. Hanif, (2012), 'Some exponential estimators in survey sampling', Pakistan Journal of Statistics 28(3), 367-374.

14. M. S. Ahmed,, M. S. Raman, & M. I. Hossain, (2000), 'Some competitive estimators of finite population variance Multivariate Auxiliary Information', Information and Management Sciences 11(1), 49-54.

15. P. R. Dash, & G. Mishra, (2011), 'An improved class of estimators in two-phase sampling using two auxiliary variables', Communications in Statistics-Theory and Methods 40, 4347-4352.

16. P. S. Laplace, (1820), A Philosophical Essay on Probabilities, English Translation, Dover.

17. P. Sharma,, H .K. Verma,, A. Sanaullah, & R. Singh, (2013), 'Some Exponential Ratio- Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation', International Journal of Statistics and Economics 12(3), 58-66.

18. R. S. Singh,, P. Chauhan,, N. Sawan, & F. Smarandache, (2011), 'Improved exponential estimator for population variance using two auxiliary variables', Italian Journal of Pure and Applied Mathematics 28, 101-108.

19. R. S. Solanki, & H. P. Singh, (2013a), 'An improved class of estimators for the population variance', Model Assisted Statistics and Applications 8(3), 229-238.

20. R. S. Solanki, & H. P. Singh, (2013b), 'Improved estimation of population mean using population proportion of an auxiliary character', Chilean Journal of Statistics 4(1), 3-17.

21. S. Bahl, & R. K. Tuteja, (1991), 'Ratio and product type exponential estimator', Information and Optimization Sciences 12, 159-163.

22. S. Gupta, & J. Shabbir, (2008), 'Variance estimation in simple random sampling using auxiliary information', Hacettepe Journal of Mathematics and Statistics 37, 57-67.

23. S. K. Yadav, & C. Kadilar, (2013), 'Improved exponential type ratio estimator of population variance', Revista Colombiana de Estadística 36(1), 145-152.

24. W. G. Cochran, (1940), 'The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce', The Journal of Agricultural Science 30, 262-275.

[Recibido en noviembre de 2013. Aceptado en abril de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv37n1a14,
    AUTHOR  = {Asghar, Amber and Sanaullah, Aamir and Hanif, Muhammad},
    TITLE   = {{Generalized Exponential Type Estimator for Population Variance in Survey Sampling}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2014},
    volume  = {37},
    number  = {1},
    pages   = {213-224}
}

References

Ahmed, M. S., Raman, M. S. & Hossain, M. I. (2000), ‘Some competitive estimators of finite population variance multivariate auxiliary information’, Information and Management Sciences 11(1), 49–54.

Bahl, S. & Tuteja, R. K. (1991), ‘Ratio and product type exponential estimator’, Information and Optimization Sciences 12, 159–163.

Cochran, W. G. (1940), ‘The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce’, The Journal of Agricultural Science 30, 262–275.

Dash, P. R. & Mishra, G. (2011), ‘An improved class of estimators in two-phase sampling using two auxiliary variables’, Communications in Statistics-Theory and Methods 40, 4347–4352.

Gujarati, D. (2004), Basic Econometrics, 4 edn, The McGraw-Hill Companies.

Gupta, S. & Shabbir, J. (2008), ‘Variance estimation in simple random sampling using auxiliary information’, Hacettepe Journal of Mathematics and Statistics 37, 57–67.

Isaki, C. (1983), ‘Variance estimation using auxiliary information’, Journal of the American Statistical Association 78, 117–123.

Laplace, P. S. (1820), A Philosophical Essay on Probabilities, English Translation, Dover.

Murthy, M. (1967), Sampling Theory and Methods, Calcutta Statistical Publishing Society, Kolkatta, India.

Noor-ul Amin, M. & Hanif, M. (2012), ‘Some exponential estimators in survey sampling’, Pakistan Journal of Statistics 28(3), 367–374.

Sanaullah, A., Khan, H., Ali, A. & Singh, R. (2012), ‘Improved ratio-type estimators in survey sampling’, Journal of Reliability and Statistical Studies 5(2), 119–132.

Sharma, P., Verma, H. K., Sanaullah, A. & Singh, R. (2013), ‘Some exponential ratio- product type estimators using information on auxiliary attributes under second order approximation’, International Journal of Statistics and Economics 12(3), 58–66.

Singh, B. K. & Choudhary, S. (2012), ‘Exponential chain ratio and product type estimators for finite population mean under double sampling scheme’, Journal of Science Frontier Research in Mathematics and Design Sciences 12(6), 0975–5896.

Singh, H. P. & Karpe, N. (2010), ‘Estimation of mean, ratio and product using auxiliary information in the presence of measurement errors in simple surveys’, Journal of Statistical Theory and Practice 4(1), 111–136.

Singh, H. P. & Solanki, R. S. (2009), ‘Estimation of finite population variance using auxiliary information in presence of random non-response’, Gujarat Statistical Review 1, 37–637.

Singh, H. P. & Solanki, R. S. (2010), ‘Estimation of finite population variance using auxiliary information in presence of random non-response’, Gujarat Statistical Review 2, 46–58.

Singh, H. P. & Solanki, R. S. (2013), ‘A new procedure for variance estimation in simple random sampling using auxiliary information’, Statistical Papers 54(2), 479–497.

Singh, H. P. & Vishwakarma, G. (2008), ‘Some families of estimators of variance of stratified random sample mean using auxiliary information’, Journal of Statistical Theory and Practice 2(1), 21–43.

Singh, H. P. & Vishwakarma, K. (2007), ‘Modified exponential ratio and product estimators for finite population mean in double sampling’, Australian Journal of Statistics 36, 217–225.

Singh, R. S., Chauhan, P., Sawan, N. & Smarandache, F. (2011), ‘Improved exponential estimator for population variance using two auxiliary variables’, Italian Journal of Pure and Applied Mathematics 28, 101–108.

Solanki, R. S. & Singh, H. P. (2013a), ‘An improved class of estimators for the population variance’, Model Assisted Statistics and Applications 8(3), 229–238.

Solanki, R. S. & Singh, H. P. (2013b), ‘Improved estimation of population mean using population proportion of an auxiliary character’, Chilean Journal of Statistics 4(1), 3–17.

Subramani, J. & Kumarapandiyan, G. (2012), ‘Variance estimation using quartiles and their functions of an auxiliary variable’, International Journal of Statistics and Applications 2(5), 67–72.

Yadav, S. K. & Kadilar, C. (2013), ‘Improved exponential type ratio estimator of population variance’, Revista Colombiana de Estadística 36(1), 145–152.

How to Cite

APA

Asghar, A., Sanaullah, A. and Hanif, M. (2014). Generalized Exponential Type Estimator for Population Variance in Survey Sampling. Revista Colombiana de Estadística, 37(1), 213–224. https://doi.org/10.15446/rce.v37n1.44368

ACM

[1]
Asghar, A., Sanaullah, A. and Hanif, M. 2014. Generalized Exponential Type Estimator for Population Variance in Survey Sampling. Revista Colombiana de Estadística. 37, 1 (Jan. 2014), 213–224. DOI:https://doi.org/10.15446/rce.v37n1.44368.

ACS

(1)
Asghar, A.; Sanaullah, A.; Hanif, M. Generalized Exponential Type Estimator for Population Variance in Survey Sampling. Rev. colomb. estad. 2014, 37, 213-224.

ABNT

ASGHAR, A.; SANAULLAH, A.; HANIF, M. Generalized Exponential Type Estimator for Population Variance in Survey Sampling. Revista Colombiana de Estadística, [S. l.], v. 37, n. 1, p. 213–224, 2014. DOI: 10.15446/rce.v37n1.44368. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44368. Acesso em: 28 mar. 2025.

Chicago

Asghar, Amber, Aamir Sanaullah, and Muhammad Hanif. 2014. “Generalized Exponential Type Estimator for Population Variance in Survey Sampling”. Revista Colombiana De Estadística 37 (1):213-24. https://doi.org/10.15446/rce.v37n1.44368.

Harvard

Asghar, A., Sanaullah, A. and Hanif, M. (2014) “Generalized Exponential Type Estimator for Population Variance in Survey Sampling”, Revista Colombiana de Estadística, 37(1), pp. 213–224. doi: 10.15446/rce.v37n1.44368.

IEEE

[1]
A. Asghar, A. Sanaullah, and M. Hanif, “Generalized Exponential Type Estimator for Population Variance in Survey Sampling”, Rev. colomb. estad., vol. 37, no. 1, pp. 213–224, Jan. 2014.

MLA

Asghar, A., A. Sanaullah, and M. Hanif. “Generalized Exponential Type Estimator for Population Variance in Survey Sampling”. Revista Colombiana de Estadística, vol. 37, no. 1, Jan. 2014, pp. 213-24, doi:10.15446/rce.v37n1.44368.

Turabian

Asghar, Amber, Aamir Sanaullah, and Muhammad Hanif. “Generalized Exponential Type Estimator for Population Variance in Survey Sampling”. Revista Colombiana de Estadística 37, no. 1 (January 1, 2014): 213–224. Accessed March 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44368.

Vancouver

1.
Asghar A, Sanaullah A, Hanif M. Generalized Exponential Type Estimator for Population Variance in Survey Sampling. Rev. colomb. estad. [Internet]. 2014 Jan. 1 [cited 2025 Mar. 28];37(1):213-24. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44368

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CrossRef citations20

1. Nitesh K. Adichwal, Prayas Sharma, Rajesh Singh. (2017). Generalized Class of Estimators for Population Variance Using Information on Two Auxiliary Variables. International Journal of Applied and Computational Mathematics, 3(2), p.651. https://doi.org/10.1007/s40819-015-0119-6.

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6. Hatice Oncel Cekim, Cem Kadilar. (2020). In-type estimators for the population variance in stratified random sampling. Communications in Statistics - Simulation and Computation, 49(7), p.1665. https://doi.org/10.1080/03610918.2019.1577973.

7. Iqra Niaz, Aamir Sanaullah, Iram Saleem, Javid Shabbir. (2022). An improved efficient class of estimators for the population variance. Concurrency and Computation: Practice and Experience, 34(4) https://doi.org/10.1002/cpe.6620.

8. Saman Hanif Shahbaz, Aisha Fayomi, Muhammad Qaiser Shahbaz. (2023). Estimation of the general population parameter in single- and two-phase sampling. AIMS Mathematics, 8(7), p.14951. https://doi.org/10.3934/math.2023763.

9. Shashi Bhushan, Anoop Kumar, Abdelaziz Alsubie, Showkat Ahmad Lone. (2023). Variance estimation under an efficient class of estimators in simple random sampling. Ain Shams Engineering Journal, 14(8), p.102012. https://doi.org/10.1016/j.asej.2022.102012.

10. Tanveer Akhlaq, Muhammad Ismail, Muhammad Qaiser Shahbaz. (2019). On Efficient Estimation of Process Variability. Symmetry, 11(4), p.554. https://doi.org/10.3390/sym11040554.

11. Amber Asghar, Aamir Sanaullah, Azhar Mehmood Abbasi, Muhammad Hanif. (2023). Advancing Sampling Techniques: Multivariate Ratio Estimation for Variance Vector in Two-Phase Sampling. Bulletin of Business and Economics (BBE), 12(3), p.473. https://doi.org/10.61506/01.00056.

12. Javid Shabbir, Zabihullah Movaheedi, Rab Nawaz. (2024). Use of generalized randomized response model for enhancement of finite population variance: A simulation approach. PLOS ONE, 19(12), p.e0315658. https://doi.org/10.1371/journal.pone.0315658.

13. Shahid Hussain, Sun Mei, Muhammad Riaz, Saddam Akber Abbasi. (2020). On Phase-I Monitoring of Process Location Parameter with Auxiliary Information-Based Median Control Charts. Mathematics, 8(5), p.706. https://doi.org/10.3390/math8050706.

14. Javid Shabbir, Sat Gupta. (2015). A Note on Generalized Exponential Type Estimator for Population Variance in Survey Sampling. Revista Colombiana de Estadística, 38(2), p.385. https://doi.org/10.15446/rce.v38n2.51667.

15. Housila P. Singh, Diksha Arya, Subhash Kumar Yadav. (2023). Optimal Strategy for Improved Estimation of Population Variance Using Known Auxiliary Parameters. International Journal of Applied and Computational Mathematics, 9(5) https://doi.org/10.1007/s40819-023-01559-7.

16. Nitesh K. Adichwal, Jitendra Kumar, Rajesh Singh, Guohua Zou. (2018). An improved generalized class of estimators for population variance using auxiliary variables. Cogent Mathematics & Statistics, 5(1), p.1454579. https://doi.org/10.1080/25742558.2018.1454579.

17. Aamir Sanaullah, Iqra Niaz, Javid Shabbir, Iqra Ehsan. (2022). A class of hybrid type estimators for variance of a finite population in simple random sampling. Communications in Statistics - Simulation and Computation, 51(10), p.5609. https://doi.org/10.1080/03610918.2020.1776873.

18. Amber Asghar, Aamir Sanaullah, Hina Khan, Muhammad Hanif. (2023). Comparative Analysis of Variance Estimation Methods in Two-Phase Sampling: A Focus on Regression-cum-Exponential Estimators with Multiple Auxiliaries. Bulletin of Business and Economics (BBE), 12(3), p.573. https://doi.org/10.61506/01.00071.

19. Hatice Oncel Cekim, Cem Kadilar. (2020). ln-Type Variance Estimators in Simple Random Sampling. Pakistan Journal of Statistics and Operation Research, , p.689. https://doi.org/10.18187/pjsor.v16i4.3072.

20. Hameed Ali, Zafar Mahmood, T.H. AlAbdulaal. (2025). On the enhancement of estimator efficiency of population variance through stratification, transformation, and formulation with application to COVID-19 data. Alexandria Engineering Journal, 113, p.480. https://doi.org/10.1016/j.aej.2024.11.044.

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