Document Type : Article

**Authors**

^{1}
Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 310027, People’s Republic of China

^{2}
Faculty of Physical Sciences, Department of Statistics, Government College University Faisalabad, Pakistan

^{3}
Faculty of Physical Sciences, Department of Statistics, Government College University Faisalabad, Pakistan. ;School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China.

^{4}
Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 310027, People’s Republic of China.

**Abstract**

Numerous ratio-type estimators of the population variance are proposed in the existing literature based on different characteristics of the study as well as the auxiliary variable. However, mostly the existing estimators are based on the conventional measures of the population characteristics and their efficiency is dubious in the presence of outliers in the data. This study presents improved families of variance estimators under simple random sampling without replacement assuming that the information on some robust non-conventional location parameters of the auxiliary variable is known besides the usual conventional parameters. The bias and mean square error of the proposed families of estimators are obtained and the efficiency conditions are derived mathematically. The theoretical results are supplemented with the numerical illustrations by using real datasets which indicates the supremacy of the suggested families of estimators.

**Keywords**

**Main Subjects**

References:

1. Solanki, R.S., Singh, H.P., and Pal, S.K. "Improved ratio-type estimators of finite population variance using quartiles", Hacettepe Journal of Mathematics and Statistics, 44(3), pp. 747-754 (2015).

2. Cochran, W.G., Sampling Techniques, 3rd Ed., John Wiley and Sons, New York (1977).

3. Sanusi, R.A., Abujiya, M.R., Riaz, M., and Abbas, N. "Combined Shewhart CUSUM charts using auxiliary variable", Computers and Industrial Engineering, 105, pp. 329-337 (2017).

4. Sanusi, R.A., Abbas, N., and Riaz, M. "On efficient CUSUM-type location control charts using auxiliary information", Quality Technology and Quantitative Management, 15(1), pp. 80-105 (2018).

5. Temesgen, H., Monleon, V., Weiskittel, A., and Wilson, D. "Sampling strategies for efficient estimation of tree foliage biomass", Forest Science, 57(2), pp. 153- 163 (2011).

6. Stehman, S.V. "Use of auxiliary data to improve the precision of estimators of thematic map accuracy", Remote Sensing of Environment, 58(2), pp. 169-176 (1996).

7. 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 (2018).

8. Raza, M.A., Nawaz, T., and Aslam, M. "On designing CUSUM charts using ratio-type estimators for monitoring the location of normal processes", Scientia Iranica, Trans. E, 27(3), pp. 1593-1605 (2020).

9. Isaki, C.T. "Variance estimation using auxiliary information", Journal of the American Statistical Association, 78, pp. 117-123 (1983).

10. 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).

11. Kadilar, C. and Cingi, H. "Ratio estimators for population variance in simple and stratified sampling", Applied Mathematics and Computation, 173, pp. 1047- 1058 (2006).

12. Subramani, J. and Kumarapandiyan, G. "Estimation of variance using deciles of an auxiliary variable", International Conference on Frontiers of Statistics and Its Applications, Bonfring Publisher, pp. 143-149 (2012).

13. 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).

14. 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).

15. Khan, M. and Shabbir, J. "A ratio type estimator for the estimation of population variance using quartiles of an auxiliary variable", Journal of Statistics Application and Probability, 2(3), pp. 319-325 (2013).

16. Yaqub, M. and Shabbir, J. "An improved class of estimators for finite population variance", Hacettepe Journal of Mathematics and Statistics, 45(5), pp. 1641-1660 (2016).

17. 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).

18. 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).

19. Maji, R., Singh, G.N., and Bandyopadhyay, A. "Effective estimation strategy of finite population variance using multi-auxiliary variables in double sampling", Journal of Modern Applied Statistical Methods, 16(1), pp. 158-178 (2017).

20. Abid, M., Ahmed, S., Tahir, M., Nazir, H.Z., and Riaz, M. "Improved ratio estimators of variance based on robust measures", Scientia Iranica, Trans. E, 26(4), pp. 2484-2494 (2019).

21. Singh, H.P., Pal, S.K., and Yadav, A. "A study on the chain ratio-ratio-type exponential estimator for finite population variance", Communications in Statistics - Theory and Methods, 47(6), pp. 1442-1458 (2018).

22. Muneer, S., Khalil, A., Shabbir, J., and Narjis, G. "A new improved ratio-product type exponential estimator of finite population variance using auxiliary information", Journal of Statistical Computation and Simulation, 88(16), pp. 3179-3192 (2018).

23. Vinson, W.D., An Investigation of Measures of Central Tendency Used in Quality Control, The University of North Carolina, Chapel Hill (1951).

24. Johnstone, C.D., Statistical Methods in Quality Control, Prentice-Hall(1957).

25. Ferrell, E. "Control charts using midranges and medians", Industrial Quality Control, 9(5), pp. 30-34 (1953).

26. Wang, T., Li, Y., and Cui, H. "On weighted randomly trimmed means", Journal of Systems Science and Complexity, 20(1), pp. 47-65 (2007).

27. Hettmansperger, T. and McKean, J.W., Robust Nonparametric Statistical Methods, 2nd Ed., Chapman & Hall /CRC Press, Boca Raton (2011).

28. Rana, S., Siraj-Ud-Doulah, M., Midi, H., and Imon, A.H.M.R. "Decile mean: A new robust measure of central tendency", Chiang Mai Journal of Science, 39(3), pp. 478-485 (2012).

29. Subramani, J. and Kumarapandiyan, G. "Estimation of variance using known coefficient of variation and median of an auxiliary variable", Journal of Modern Applied Statistical Methods, 12(1), pp. 58-64 (2013).

30. Upadhyaya, L.N. and Singh, H.P. "Estimation of population standard deviation using auxiliary information", American Journal of Mathematics and Management Sciences, 21(3-4), pp. 345-358 (2001).

31. Subramani, J. and Kumarapandiyan, G. "A class of modified ratio estimators for estimation of population variance", Journal of Applied Mathematics, Statistics and Informatics, 11(1), pp. 91-114 (2015).

32. Abid, M., Abbas, N., Nazir, H.Z., and Lin, Z. "Enhancing the mean ratio estimators for estimating population mean using non-conventional location parameters", Revista Colombiana de Estadstica, 39(1), pp. 63-79 (2016).

33. Searls, D.T. "Utilization of known coefficient of kurtosis in the estimation procedure of variance", Journal of American Statistical Association, 59, pp. 1225-1226 (1964).

34. Singh, D. and Chaudhary, F.S., Theory and Analysis of Sample Survey Designs, New Age International Publisher, New Dehli (1986).

35. Shabbir, J. and Gupta, S. "Estimation of the finite population mean in two phase sampling when auxiliary variables are attributes", Hacettepe Journal of Mathematics and Statistics, 39(1), pp. 121-129 (2010).

36. Yadav, S.K., Mishra, S.S., and Shukla, K.A. "Improved ratio estimators for population mean based on median using linear combination of population mean and median of an auxiliary variable", American Journal of Operational Research, 4(2), pp. 21-27 (2014).

37. Abid, M., Abbas, N., and Riaz, M. "Improved modified ratio estimators of population mean based on deciles", Chiang Mai Journal of Science, 43(1), pp. 11311-1323 (2016).

38. Shabbir, J. and Gupta, S. "Estimation of finite population mean in simple and stratified random sampling using two auxiliary variables", Communications in Statistics - Theory and Methods, 46(20), pp. 10135- 10148 (2017).

2. Cochran, W.G., Sampling Techniques, 3rd Ed., John Wiley and Sons, New York (1977).

3. Sanusi, R.A., Abujiya, M.R., Riaz, M., and Abbas, N. "Combined Shewhart CUSUM charts using auxiliary variable", Computers and Industrial Engineering, 105, pp. 329-337 (2017).

4. Sanusi, R.A., Abbas, N., and Riaz, M. "On efficient CUSUM-type location control charts using auxiliary information", Quality Technology and Quantitative Management, 15(1), pp. 80-105 (2018).

5. Temesgen, H., Monleon, V., Weiskittel, A., and Wilson, D. "Sampling strategies for efficient estimation of tree foliage biomass", Forest Science, 57(2), pp. 153- 163 (2011).

6. Stehman, S.V. "Use of auxiliary data to improve the precision of estimators of thematic map accuracy", Remote Sensing of Environment, 58(2), pp. 169-176 (1996).

7. 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 (2018).

8. Raza, M.A., Nawaz, T., and Aslam, M. "On designing CUSUM charts using ratio-type estimators for monitoring the location of normal processes", Scientia Iranica, Trans. E, 27(3), pp. 1593-1605 (2020).

9. Isaki, C.T. "Variance estimation using auxiliary information", Journal of the American Statistical Association, 78, pp. 117-123 (1983).

10. 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).

11. Kadilar, C. and Cingi, H. "Ratio estimators for population variance in simple and stratified sampling", Applied Mathematics and Computation, 173, pp. 1047- 1058 (2006).

12. Subramani, J. and Kumarapandiyan, G. "Estimation of variance using deciles of an auxiliary variable", International Conference on Frontiers of Statistics and Its Applications, Bonfring Publisher, pp. 143-149 (2012).

13. 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).

14. 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).

15. Khan, M. and Shabbir, J. "A ratio type estimator for the estimation of population variance using quartiles of an auxiliary variable", Journal of Statistics Application and Probability, 2(3), pp. 319-325 (2013).

16. Yaqub, M. and Shabbir, J. "An improved class of estimators for finite population variance", Hacettepe Journal of Mathematics and Statistics, 45(5), pp. 1641-1660 (2016).

17. 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).

18. 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).

19. Maji, R., Singh, G.N., and Bandyopadhyay, A. "Effective estimation strategy of finite population variance using multi-auxiliary variables in double sampling", Journal of Modern Applied Statistical Methods, 16(1), pp. 158-178 (2017).

20. Abid, M., Ahmed, S., Tahir, M., Nazir, H.Z., and Riaz, M. "Improved ratio estimators of variance based on robust measures", Scientia Iranica, Trans. E, 26(4), pp. 2484-2494 (2019).

21. Singh, H.P., Pal, S.K., and Yadav, A. "A study on the chain ratio-ratio-type exponential estimator for finite population variance", Communications in Statistics - Theory and Methods, 47(6), pp. 1442-1458 (2018).

22. Muneer, S., Khalil, A., Shabbir, J., and Narjis, G. "A new improved ratio-product type exponential estimator of finite population variance using auxiliary information", Journal of Statistical Computation and Simulation, 88(16), pp. 3179-3192 (2018).

23. Vinson, W.D., An Investigation of Measures of Central Tendency Used in Quality Control, The University of North Carolina, Chapel Hill (1951).

24. Johnstone, C.D., Statistical Methods in Quality Control, Prentice-Hall(1957).

25. Ferrell, E. "Control charts using midranges and medians", Industrial Quality Control, 9(5), pp. 30-34 (1953).

26. Wang, T., Li, Y., and Cui, H. "On weighted randomly trimmed means", Journal of Systems Science and Complexity, 20(1), pp. 47-65 (2007).

27. Hettmansperger, T. and McKean, J.W., Robust Nonparametric Statistical Methods, 2nd Ed., Chapman & Hall /CRC Press, Boca Raton (2011).

28. Rana, S., Siraj-Ud-Doulah, M., Midi, H., and Imon, A.H.M.R. "Decile mean: A new robust measure of central tendency", Chiang Mai Journal of Science, 39(3), pp. 478-485 (2012).

29. Subramani, J. and Kumarapandiyan, G. "Estimation of variance using known coefficient of variation and median of an auxiliary variable", Journal of Modern Applied Statistical Methods, 12(1), pp. 58-64 (2013).

30. Upadhyaya, L.N. and Singh, H.P. "Estimation of population standard deviation using auxiliary information", American Journal of Mathematics and Management Sciences, 21(3-4), pp. 345-358 (2001).

31. Subramani, J. and Kumarapandiyan, G. "A class of modified ratio estimators for estimation of population variance", Journal of Applied Mathematics, Statistics and Informatics, 11(1), pp. 91-114 (2015).

32. Abid, M., Abbas, N., Nazir, H.Z., and Lin, Z. "Enhancing the mean ratio estimators for estimating population mean using non-conventional location parameters", Revista Colombiana de Estadstica, 39(1), pp. 63-79 (2016).

33. Searls, D.T. "Utilization of known coefficient of kurtosis in the estimation procedure of variance", Journal of American Statistical Association, 59, pp. 1225-1226 (1964).

34. Singh, D. and Chaudhary, F.S., Theory and Analysis of Sample Survey Designs, New Age International Publisher, New Dehli (1986).

35. Shabbir, J. and Gupta, S. "Estimation of the finite population mean in two phase sampling when auxiliary variables are attributes", Hacettepe Journal of Mathematics and Statistics, 39(1), pp. 121-129 (2010).

36. Yadav, S.K., Mishra, S.S., and Shukla, K.A. "Improved ratio estimators for population mean based on median using linear combination of population mean and median of an auxiliary variable", American Journal of Operational Research, 4(2), pp. 21-27 (2014).

37. Abid, M., Abbas, N., and Riaz, M. "Improved modified ratio estimators of population mean based on deciles", Chiang Mai Journal of Science, 43(1), pp. 11311-1323 (2016).

38. Shabbir, J. and Gupta, S. "Estimation of finite population mean in simple and stratified random sampling using two auxiliary variables", Communications in Statistics - Theory and Methods, 46(20), pp. 10135- 10148 (2017).

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July and August 2020Pages 2040-2056