Optimized generalized estimator for valuation of population mean under probability proportional to size sampling using auxiliary variables

S. Alghamdi Abdulaziz; Ahmad Sohaib; S. Alghamdi Abdulaziz; Ahmad Sohaib

doi:10.3934/math.2026811

AIMS Mathematics

2026, Volume 11, Issue 7: 19983-20006. doi: 10.3934/math.2026811

Previous Article Next Article

Research article Special Issues

Optimized generalized estimator for valuation of population mean under probability proportional to size sampling using auxiliary variables

S. Alghamdi Abdulaziz ¹,
Ahmad Sohaib ^{2
,
,}

1.
Department of Mathematics, College of Science &Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia
2.
Department of Statistics, Abdul Wali Khan University, Mardan, Pakistan

Received: 07 May 2026 Revised: 11 June 2026 Accepted: 16 June 2026 Published: 08 July 2026
MSC : 62D05, 62L05, 62L10, 62L12, 62F12, 62F25, 62A86, 62G07

The key issue in survey sampling is the efficient estimation of population mean under probability proportional to size sampling (PPS), especially when population units vary considerably in size. In this article, an optimization of generalized estimator of population mean under PPS sampling was proposed by including two auxiliary variables. The proposed class of estimators was designed in such a way that it was flexible in using known population parameters of auxiliary variables, thereby enhancing the efficiency of the estimators. The bias and the mean squared error (MSE) were derived up to the first-order approximation for all the considered estimators. Theoretical comparisons showed that in realistic conditions, the proposed estimator performed better than existing estimators. To validate the theoretical results, an empirical study using real and simulated data was carried out, demonstrating significant improvements in efficiency. The numerical findings showed that the incorporation of auxiliary information in the PPS framework can greatly increase the precision of population mean estimation. The proposed methodology offers a useful contribution to the field of survey sampling and can be effectively implemented, when the population size varies from one individual to another.
- PPS sampling,
- auxiliary variables,
- population mean,
- bias,
- mean squared error,
- efficiency
Citation: S. Alghamdi Abdulaziz, Ahmad Sohaib. Optimized generalized estimator for valuation of population mean under probability proportional to size sampling using auxiliary variables[J]. AIMS Mathematics, 2026, 11(7): 19983-20006. doi: 10.3934/math.2026811

Related Papers:

Abstract

The key issue in survey sampling is the efficient estimation of population mean under probability proportional to size sampling (PPS), especially when population units vary considerably in size. In this article, an optimization of generalized estimator of population mean under PPS sampling was proposed by including two auxiliary variables. The proposed class of estimators was designed in such a way that it was flexible in using known population parameters of auxiliary variables, thereby enhancing the efficiency of the estimators. The bias and the mean squared error (MSE) were derived up to the first-order approximation for all the considered estimators. Theoretical comparisons showed that in realistic conditions, the proposed estimator performed better than existing estimators. To validate the theoretical results, an empirical study using real and simulated data was carried out, demonstrating significant improvements in efficiency. The numerical findings showed that the incorporation of auxiliary information in the PPS framework can greatly increase the precision of population mean estimation. The proposed methodology offers a useful contribution to the field of survey sampling and can be effectively implemented, when the population size varies from one individual to another.

References

[1]	A. R. El-Saeed, S. Ahmad, B. Aloraini, Robust estimator for estimation of population mean under PPS sampling: application to radiation data, J. Radiat. Res. Appl. Sci., 18 (2025), 101384. https://doi.org/10.1016/j.jrras.2025.101384 doi: 10.1016/j.jrras.2025.101384
[2]	S. M. Alghamdi, S. Ahmad, S. M. Almarzouki, B. Aloraini, M. M. Badr, M. A. Abdelkawy, Constructing a new estimator for estimating population mean utilizing auxiliary information in probability proportional to size sampling, Alex. Eng. J., 110 (2025), 506–511. https://doi.org/10.1016/j.aej.2024.10.029 doi: 10.1016/j.aej.2024.10.029
[3]	M. Azeem, S. Iftikhar, M. Ijaz, N. Salahuddin, M. Ilyas, An improved estimator of population mean under PPS sampling with application to radiation data sets, J. Radiat. Res. Appl. Sci., 18 (2025), 101543. https://doi.org/10.1016/j.jrras.2025.101543 doi: 10.1016/j.jrras.2025.101543
[4]	H. P. Singh, A. C. Mishra, S. K. Pal, Improved estimator of population total in PPS sampling, Commun. Stat. Theory Methods, 47 (2018), 912–934. https://doi.org/10.1080/03610926.2017.1313987 doi: 10.1080/03610926.2017.1313987
[5]	A. B. Ghorbal, Robust estimation of mean under probability proportional to size sampling: An application to radiation science and engineering data, J. Radiat. Res. Appl. Sci., 19 (2026), 102404. https://doi.org/10.1016/j.jrras.2026.102404 doi: 10.1016/j.jrras.2026.102404
[6]	M. Hussein Mohamud, F. A. Mohamud, Estimation of the mean using robust regression and probability proportional to size sampling, Stat. Theory Relat. Fields, 9 (2025), 213–222. https://doi.org/10.1080/24754269.2025.2516339 doi: 10.1080/24754269.2025.2516339
[7]	S. Shah, E. Mahmoudi, H. Iftikhar, P. C. Rodrigues, R. I. Gonzales Medina, J. L. López-Gonzales, A novel family of CDF estimators under PPS sampling: computational, theoretical, and applied perspectives, Axioms, 14 (2025), 796. https://doi.org/10.3390/axioms14110796 doi: 10.3390/axioms14110796
[8]	S. Ahmad, J. Shabbir, E. Zahid, M. Aamir, Improved family of estimators for the population mean using supplementary variables under PPS sampling, Sci. Prog., 106 (2023), 00368504231180085. https://doi.org/10.1177/00368504231180085 doi: 10.1177/00368504231180085
[9]	M. Azeem, An optimal quantitative randomized response technique under PPS sampling design, Commun. Stat. Theory Methods, 55 (2025), 3534–3546. https://doi.org/10.1080/03610926.2025.2580525 doi: 10.1080/03610926.2025.2580525
[10]	M. Mustafa, S. Ahmad, E. Zahid, J. Shabbir, S. Masood, Novel methods for estimation of population mean using auxiliary information under PPS sampling: application with real and simulated data sets, Kurd. Stud., 12 (2024), 1553–1562. https://doi.org/10.53555/ks.v12i5.3508 doi: 10.53555/ks.v12i5.3508
[11]	J. Wang, S. Ahmad, M. Arslan, S. A. Lone, A. H. Abd Ellah, M. A. Aldahlan, et al., Estimation of finite population mean using double sampling under probability proportional to size sampling in the presence of extreme values, Heliyon, 9 (2023), e21418. https://doi.org/10.1016/j.heliyon.2023.e21418 doi: 10.1016/j.heliyon.2023.e21418
[12]	S. Ahmad, E. Zahid, J. Shabbir, M. Aamir, R. Onyango, Enhanced estimation of the population mean using two auxiliary variables under probability proportional to size sampling, Math. Probl. Eng., 2023 (2023), 5564360. https://doi.org/10.1155/2023/5564360 doi: 10.1155/2023/5564360
[13]	S. Khan, M. Farooq, S. Ahmad, S. Khan, Improved estimator for the estimation of population mean using a predictive approach under PPS sampling, VFAST Trans. Math., 12 (2024), 1–16.
[14]	S. Yang, D. Meng, H. Yang, C. Luo, X. Su, Enhanced soft Monte Carlo simulation coupled with support vector regression for structural reliability analysis, Proc. Inst. Civ. Eng. Transp., 178 (2025), 459–474. https://doi.org/10.1680/jtran.24.00128 doi: 10.1680/jtran.24.00128
[15]	S. Yang, D. Meng, B. Keshtegar, A. M. P. De Jesus, M. Alfouneh, S. P. Zhu, ASVR-GPSO: A novel hybrid active support vector regression and global-best partial swarm optimization for structural reliability analysis, Struct. Multidiscip. Optim., 68 (2025), 196. https://doi.org/10.1007/s00158-025-04134-4 doi: 10.1007/s00158-025-04134-4
[16]	S. Yang, D. Meng, H. Yang, B. Keshtegar, A. M. P. De Jesus, S. Zhu, Adaptive Kriging‑assisted enhanced sparrow search with augmented‑Lagrangian first‑order reliability method for highly efficient structural reliability analysis, Reliab. Eng. Syst. Saf., 267 (2026), 111916. https://doi.org/10.1016/j.ress.2025.111916 doi: 10.1016/j.ress.2025.111916
[17]	B. Elkalzah, M. El-Morshedy, H. S. Shahen, M. Elgarhy, A new generalized class of estimators for estimation of population mean under PPS sampling: application with radiation science, J. Radiat. Res. Appl. Sci., 18 (2025), 101738. https://doi.org/10.1016/j.jrras.2025.101738 doi: 10.1016/j.jrras.2025.101738
[18]	G. Alomani, M. Kayid, S. Ahmad, Application to radiation data sets by suggesting an improved mean estimator under probability proportional to size sampling, J. Radiat. Res. Appl. Sci., 18 (2025), 101354. https://doi.org/10.1016/j.jrras.2025.101354 doi: 10.1016/j.jrras.2025.101354
[19]	C. Kadilar, H. Cingi, An improvement in estimating the population mean by using the correlation coefficient, Hacet. J. Math. Stat., 35 (2006), 103–109.
[20]	L. N. Upadhyaya, H. P. Singh, Use of transformed auxiliary variable in estimating the finite population mean, Biom. J., 41 (1999), 627–636. Available from: https://api.semanticscholar.org/CorpusID:120685207.
[21]	P. Mishra, A. Sharma, N. K. Adichwal, S. Rai, R. Singh, Log-product-type estimator for estimation of population variance using auxiliary information, Thail. Stat., 22 (2024), 610–617. Available from: https://ph02.tci-thaijo.org/index.php/thaistat/article/view/254771.
[22]	S. Singh, Advanced sampling theory with applications: how Michael "selected" Amy, Dordrecht: Springer Netherlands, 2003.
[23]	C. Kadilar, H. Cingi, Ratio estimators in stratified random sampling, Biom. J., 45 (2003), 218–25. https://doi.org/10.1002/bimj.200390007 doi: 10.1002/bimj.200390007
[24]	M. N. Murthy, Sampling theory and methods, Calcutta: Statistical Publishing Society, 1967.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)