An improved and robust class of variance estimator

Document Type : Article


1 Department of Statistics, Government College University, Faisalabad, 38000, Pakistan

2 College of Statistical and Actuarial Sciences, University of the Punjab Lahore, Pakistan

3 Department of Statistics, University of Sargodha, Sargodha, Pakistan

4 Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia


The ratio, product, and regression estimators are commonly constructed based on the conventional measures such as mean, median, quartiles, semi-interquartile range, semi-interquartile average, coefficient of skewness, and coefficient of kurtosis. In case of the presence of outliers, these conventional measures lose their efficiency/performance ability and hence are of less efficient as compared to those measures which performed efficiently in the presence of outliers. This study offers improved class of estimators for estimating the population variance using robust dispersion measures such as probability-weighted moments, Gini’s, Downton’s and Bickel and Lehmann measures of an auxiliary variable. Bias, Mean square error (MSE) and minimum MSE of the suggested class of estimators have been derived. Application with two natural data sets is also provided to explain the proposal for practical considerations. In addition, a robustness study is also carried out to evaluate the performance of the proposed estimators in the presence of outliers by using an environment protection data. The results reveal that the proposed estimators perform better than its competitors and are robust, not only in simple conditions but also in the presence of outliers.


1. Solanki, R.S., Singh, H.P., and Pal, S.K. Improved
ratio-type estimators of  nite population variance using
quartiles", Hacettepe Journal of Mathematics and
Statistics, 44(3), pp. 747{754 (2015).
2. Cochran, W.G., Sampling Techniques, 3rd Edn., John
Wiley and Sons, New York (1977).
3. Isaki, C.T. Variance estimation using auxiliary information",
Journal of the American Statistical Association,
78, pp. 117{123 (1983).
4. Upadhyaya, L.N. and Singh., H.P. An estimator for
population variance that utilizes the kurtosis of an
auxiliary variable in sample surveys", Vikram Mathematical
Journal, 19, pp. 14{17 (1999).
5. Kadilar, C. and Cingi, H. Ratio estimators for population
variance in simple and strati ed sampling", Applied
Mathematics and Computation, 173, pp. 1047{
1058 (2006).
3600 M. Abid et al./Scientia Iranica, Transactions E: Industrial Engineering 28 (2021) 3589{3601
6. Subramani, J. and Kumarapandiyan, G. Variance
estimation using median of the auxiliary variable",
International Journal of Probability and Statistics,
1(3), pp. 36{40 (2012).
7. Subramani, J. and Kumarapandiyan, G. Variance
estimation using quartiles and their functions of an
auxiliary variable", International Journal of Statistics
and Applications, 2(5), pp. 67{72 (2012).
8. Subramani, J. and Kumarapandiyan, G. Estimation
of variance using deciles of an auxiliary variable",
Proceedings of International Conference on Frontiers
of Statistics and Its Applications, 33, pp. 143{149
9. Subramani, J. and Kumarapandiyan, G. Estimation
of variance using known coecient of variation and
median of an auxiliary variable", Journal of Modern
Applied Statistical Methods, 12(1), pp. 58{64 (2013).
10. Khan, M. and Shabbir, J. A ratio type estimators for
the estimation of population variance using quartiles
of an auxiliary variable", Journal of Statistics Applications
and Probability, 2(3), pp. 319{325 (2013).
11. Hussain, Z. and Shabbir, J. Estimation of the mean
of a socially undesirable characteristics", Scientia Iranica,
20(3), pp. 839{845 (2013).
12. Zamanzade, E. and Vock, M. Variance estimation in
ranked set sampling using a concomitant variable",
Statistics and Probability Letters, 105, pp. 1{5 (2015).
13. Yaqub, M. and Shabbir, J. An improved class of
estimators for  nite population variance", Hacettepe
Journal of Mathematics and Statistics, 45(5), pp.
1641{1660 (2016).
14. Abid, M., Abbas, N., and Riaz, M. Improved modi ed
ratio estimators of population mean based on deciles",
Chiang Mai Journal of Science, 43(1), pp. 11311{1323
15. Maqbool, S. and Javaid, S. Variance estimation using
linear combination of tri-mean and quartile average",
American Journal of Biological and Environmental
Statistics, 3(1), p. 5 (2017).
16. Adichwal, N.K., Sharma, P., and Singh, R. Generalized
class of estimators for population variance using
information on two auxiliary variables", International
Journal of Applied and Computational Mathematics,
3(2), pp. 651{661 (2017).
17. Maji, R., Singh, G.N., and Bandyopadhyay, A. E ective
estimation strategy of  nite population variance
using multi-auxiliary variables in double sampling",
Journal of Modern Applied Statistical Methods, 16(1),
pp. 158{178 (2017).
18. Zamanzade, E. and Wang, X. Estimation of population
proportion for judgment post-strati cation",
Computational Statistics and Data Analysis, 112, pp.
257{269 (2017).
19. Zamanzade, E. and Mahdizadeh, M. A more ecient
proportion estimator in ranked set sampling", Statistics
and Probability Letters, 129, pp. 28{33 (2017).
20. Singh, H.P., Pal, S.K., and Yadav, A. A study on the
chain ratio-ratio-type exponential estimator for  nite
population variance", Communications in Statistics-
Theory and Methods, 47(6), pp. 1442{1458 (2018).
21. Zamanzade, E. and Wang, X. Proportion estimation
in ranked set sampling in the presence of tie information",
Computational Statistics, 33(3), pp. 1349{1366
22. Hussain, S., Song, L., Ahmad, S., and Riaz, M. On
auxiliary information based improved EWMA median
control charts", Scientia Iranica, 25(2), pp. 954{982
23. Muneer, S., Khalil, A., Shabbir, J., and Narjis, G.
A new improved ratio-product type exponential estimator
of  nite population variance using auxiliary
information", Journal of Statistical Computation and
Simulation, 88(16), pp. 3179{3192 (2018).
24. Mahdizadeh, M. and Zamanzade, E. Ecient body
fat estimation using multistage pair ranked set sampling",
Statistical Methods in Medical Research, 28(1),
pp. 223{234 (2019).
25. Abid, M., Ahmed, S., Tahir, M., Nazir, H.Z., and Riaz,
M. Improved ratio estimators of variance based on
robust measures", Scientia Iranica, 26(4), pp. 2484{
2494 (2019).
26. Abid, M., Abbas, N., Sherwani, R.A.K., and Nazir,
Z.A. Improved ratio estimators for the population
mean using non-conventional measures of dispersion",
Pakistan Journal of Statistics and Operation Research,
12(2), pp. 353{367 (2016).
27. Naz, F., Abid, M., Nawaz, T., and Pang, T. Enhancing
the eciency of the ratio-type estimators of
population variance with a blend of information on
robust location measures", Scientia Iranic, 27, pp.
2040{2056 (2020).
28. Muhammad, F. and Riaz, M. Probability weighted
moments approach to quality control charts", Economic
Quality Control, 21, pp. 251{260 (2006).
29. Gini, C. Variabilita e mutabilita, contributo allo
studio delle distribuzioni e delle relazioni statistiche",
Studi Economico-Giuridici della R. Universita di
Cagliari, 3, pp. 153{159 (1912).
30. Downton, F. Linear estimates with polynomial coef-
 cients", Biometrika, 53, pp. 129{141 (1966).
31. Bickel, P.J. and Lehmann, E.L. Descriptive statistics
for nonparametric models III: dispersion", In Contributions
to Statistics, Hajek Memorial Volume, J.
Jurekova, Ed., Academia, Prague, pp. 33{40 (1979).
32. David, H.A. Gini's mean di erence rediscovered",
Biometrika, 55, pp. 573{575 (1968).
33. Shamos, M.I. Geometry and statistics: problems at
the interface", In New York Directions and Recent
Results in Algorithms and Complexity, J.F. Traub, Ed.,
Academic Press, New York, pp. 251{280 (1976).
34. Murthy, M.N., Sampling Theory and Methods, Statistical
Publishing Society, Calcutta, India (1967).
35. ISPRA. (2004).