Efficient Class of Variance Estimators for Population using Supplementary Information in Stratified Random Sampling





This paper addresses an efficient class of variance estimators for population using stratified random sampling. The suggested class of estimators using supplementary information has been studied in different circumstances. The expressions of bias and mean square error (MSE) of the proposed estimators are derived up to the first degree of approximation. The theoretical comparison of the proposed and considered estimators is also discussed, which shows that the proposed estimators are more efficient than the existing estimators. Theoretical findings are validated by three different types of real data sets and simulation studies. The numerical results of the proposed and existing estimators are compared in terms of mean square error, percentage relative efficiency and diagrams. It is observed that all the proposed estimators outperform the existing estimators. For instance, the traditional unbiased estimator Ozel et.al [6] and other existing estimators. Lastly, appropriate recommendations have been provided for researchers to use these suggested estimators to solve real-world issues.


Ahmad, S. et al. (2022) ‘Improved estimation of finite population variance using dual supplementary information under stratified random sampling’, *Mathematical Problems in Engineering*, 2022.

Alomair, M. A. and Gardazi, S. A. H. S. (2024) ‘Hybrid class of robust type estimators for variance estimation using mean and variance of auxiliary variable’, *Heliyon*.

Bahl, S. and Tuteja, R. (1991) ‘Ratio and product type exponential estimators’, *Journal of Information and Optimization Sciences*, 12(1), pp. 159–164.

Cekim, H. O. and Kadilar, C. (2020a) ‘ln-type estimators for the population variance in stratified random sampling’, *Communications in Statistics—Simulation and Computation*, 49(7), pp. 1665–1677.

Cekim, H. O. and Kadilar, C. (2020b) ‘ln-type variance estimators in simple random sampling’, *Pakistan Journal of Statistics and Operation Research*, 16(4), pp. 689–696.

Cochran, W. G. (2007) *Sampling techniques*. 3rd edn. New York: John Wiley & Sons.

Etebong, P. C. (2018) ‘Improved family of ratio estimators of finite population variance in stratified random sampling’, *Biostatistics and Biometrics Open Access Journal*, 5(2), p. 556.

Hussain, M. et al. (2024) ‘Improved exponential type variance estimators for population utilizing supplementary information’, *Heliyon*.

Kadilar, C. and Cingi, H. (2006) ‘Ratio estimators for the population variance in simple and stratified random sampling’, *Applied Mathematics and Computation*, 173(2), pp. 1047–1059.

Kadilar, C. H. (2003) ‘A study on the chain ratio-type estimator’, *Hacettepe Journal of Mathematics and Statistics*, 32, pp. 105–108.

Pandey, M. K. et al. (2024) ‘Improved estimation of population variance in stratified successive sampling using calibrated weights under non-response’, *Heliyon*, 10(6).

Pandey, M. K. et al. (2024) ‘A general class of improved population variance estimators under non-sampling errors using calibrated weights in stratified sampling’, *Scientific Reports*, 14(1), p. 2948.

Powers, T. (2016) ‘Efficient estimator for population variance using auxiliary variable’, *American Journal of Operational Research*, 6(1), pp. 9–15.

Qian, J. (2020) ‘Variance estimation with complex data and finite population correction—a paradigm for comparing jackknife and formula-based methods for variance estimation’, *ETS Research Report Series*, 2020(1), pp. 1–16.

Shahzad, U. et al. (2021) ‘A novel family of variance estimators based on l-moments and calibration approach under stratified random sampling’, *Communications in Statistics—Simulation and Computation*, pp. 1–14.

Shahzad, U. et al. (2023) ‘Estimation of coefficient of variation using calibrated estimators in double stratified random sampling’, *Mathematics*, 11(1), p. 252.

Sidelel, E. B., Orwa, G. O. and Otieno, R. O. (2014) ‘Variance estimation in stratified random sampling in the presence of two auxiliary random variables’, *International Journal of Science and Research*, 3(9), pp. 2453–2459.

Singh, G. N., Bhattacharyya, D. and Bandyopadhyay, A. (2021) ‘Calibration estimation of population variance under stratified successive sampling in presence of random non response’, *Communications in Statistics—Theory and Methods*, 50(19), pp. 4487–4509.

Singh, N., Vishwakarma, G. K. and Gangele, R. K. (2021) ‘Variance estimation in the presence of measurement errors under stratified random sampling’, *REVSTAT*, 19(2), pp. 275–290.

Singh, R. and Mangat, N. S. (1996) *Elements of Survey Sampling*. London: Kluwer Academic Publishers.

Yasmeen, U., Noor-Ul Amin, M. and Hanif, M. (2019) ‘Exponential estimators of finite population variance using transformed auxiliary variables’, *Proceedings of the National Academy of Sciences, India Section A: Physical Sciences*, 89(1), pp. 185–191.

Zakari, Y. and Muhammad, I. (2023) ‘Modified estimator of finite population variance under stratified random sampling’, *Engineering Proceedings*, 56(1), p. 177.




How to Cite

Hussain, M., Khan, L., Zaman, Q., & Abdurrahman Sabir. (2024). Efficient Class of Variance Estimators for Population using Supplementary Information in Stratified Random Sampling. VFAST Transactions on Mathematics, 12(1), 264–279. https://doi.org/10.21015/vtm.v12i1.1794