Document Type : Article

**Authors**

^{1}
School of Mathematical Sciences, Institute of Statistics, Zhejiang University, Hangzhou 310058, China

^{2}
- Department of Statistics, Faculty of Physical Sciences, Government College University Faisalabad, Allama Iqbal Road, Faisalabad 38000, Pakistan. - School of Mathematical Sciences, Shanghai Jiao Tong University, Minhang Campus, 800 Dongchuan Road, Shanghai 200240, China.

^{3}
Department of Statistics, Faculty of Physical Sciences, Government College University Faisalabad, Allama Iqbal Road, Faisalabad 38000, Pakistan

**Abstract**

Several auxiliary information-based estimators of the population variance are available in the existing literature of survey sampling. Mostly, these estimators are based on conventional dispersion measures of the auxiliary variable. In this study, a generalized class of ratio-product type exponential estimators of the population variance is proposed which integrates the auxiliary information on non-conventional dispersion measures under simple random sampling in the ratio-type exponential class of estimators. The performance of the proposed estimators is compared, theoretically and numerically, with the several existing estimators of the population variance. It is established that the proposed class of estimators outperforms the existing estimators in terms of the lower mean square and relative root mean square errors. Moreover, the percentage relative efficiency of the proposed estimators is much higher as compared to their counterparts.

**Keywords**

References:

1. Cochran, W.G. "The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce", The Journal of Agricultural Science, 30(2), pp. 262-275 (1940).

2. 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).

3. Ogliore, R.C., Huss, G.R., and Nagashima, K. "Ratio estimation in SIMS analysis", Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 269(17), pp. 1910-1918 (2011).

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

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

6. 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, 27(3), pp. 1593-1605 (2020).

7. Sanusi, R.A., Abbas, N., and Riaz, M. "On efficient CUSUM-type location control charts using auxiliary information", Quality Technology & Quantitative Management, 15(1), pp. 87-105 (2018).

8. Naz, F., Nawaz, T., Pang, T., et al. "Use of nonconventional dispersion measures to improve the efficiency of ratio-type estimators of variance in the presence of outliers", Symmetry, 12(1), 16 (2020).

9. Ozel, G., C ingi, H., and Oguz, M. "Separate ratio estimators for the population variance in stratified random sampling", Communications in Statistics-Theory and Methods, 43(22), pp. 4766-4779 (2014).

10. 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).

11. Bahl, S. and Tuteja, R.K. "Ratio and product type exponential estimators", Journal of Information and Optimization Sciences, 12(1), pp. 159-164 (1991).

12. Das, A.K. and Tripathi, T.P. "Use of auxiliary information in estimating the finite population variance", Sankhya C, 40, pp. 139-148 (1978).

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

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

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. 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), pp. 5-5 (2017).

17. Shabbir, J. and Gupta, S. "On improvement in variance estimation using auxiliary information", Communications in Statistics-Theory and Methods, 36(12), pp. 2177-2185 (2007).

18. Singh, H.P. and Solanki, R.S. "Improved estimation of finite population variance using auxiliary information", Communications in Statistics-Theory and Methods, 42(15), pp. 2718-2730 (2013).

19. Singh, H.P., Upadhyaya, L.N., and Namjoshi, U.D. "Estimation of finite population variance", Current Science, 57(24), pp. 1331-1334 (1988).

20. Singh, R. and Malik, S. "Improved estimation of population variance using information on auxiliary attribute in simple random sampling", Applied Mathematics and Computation, 235, pp. 43-49 (2014).

21. 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).

22. 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).

23. 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).

24. Swain, A.K.P.C. "Generalized estimator of finite population variance", Journal of Statistical Theory and Applications, 14(1), pp. 45-51 (2015).

25. 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).

26. Yadav, S.K., Kadilar, C., Shabbir, J., et al. "Improved family of estimators of population variance in simple random sampling", Journal of Statistical Theory and Practice, 9(2), pp. 219-226 (2015).

27. 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).

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

29. Abid, M., Nazir, H.Z., Riaz, M., et al. "Improved ratio estimators using some robust measures", Hacettepe Journal of Mathematics and Statistics, 47(5), pp. 1375-1393 (2018).

30. Singh, H.P. and Yadav, A. "A new exponential approach for reducing the mean squared errors of the estimators of population mean using conventional and non-conventional location parameters", Journal of Modern Applied Statistical Methods, 18(1), 26 (2020).

31. Subzar, M., Maqbool, S., Raja, T.A., et al. "Improved family of ratio type estimators for estimating population mean using conventional and non conventional location parameters", Revista Investigacion Operacional, 38(5), pp. 499-513 (2017).

32. Abid, M., Sherwani, R.A.K., Abbas, N., et al. "Some improved modified ratio estimators based on decile mean of an auxiliary variable", Pakistan Journal of Statistics and Operation Research, 12(4), pp. 787-797 (2016).

33. Abid, M., Ahmed, S., Tahir, M., et al. "Improved ratio estimators of variance based on robust measures", Scientia Iranica, 26(4), pp. 2484-2494 (2019).

34. Naz, F., Abid, M., Nawaz, T., et al. "Enhancing efficiency of ratio-type estimators of population variance by a combination of information on robust location measures", Scientia Iranica, 27(4), pp. 2040-2056 (2020).

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

36. 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).

37. Yadav, S.K. and Kadilar, C. "A two parameter variance estimator using auxiliary information", Applied Mathematics and Computation, 226, pp. 117-122 (2014).

38. Gini, C. "Variabilita e mutabilita", in Memorie di Metodologica Statistica Ed., Libreria Eredi Virgilio Veschi, Rome, (1912).

39. David, H.A. "Miscellanea: Gini's mean difference rediscovered", Biometrika, 55(3), pp. 573-575 (1968).

40. Downton, F. "Linear estimates with polynomial coeficients", Biometrika, 53(1-2), pp. 129-141 (1966).

41. Muhammad, F., Ahmad, S., and Abiodullah, M. "Use of probability weighted moments in the analysis of means", Biometrical Journal, 35(3), pp. 371-378 (1993).

42. Hampel, F.R. "The influence curve and its role in robust estimation", Journal of the American Statistical Association, 69(346), pp. 383-393 (1974).

43. Shamos, M.I. "Geometry and Statistics: Problems at the Interface", In New Directions and Recent Results in Algorithms and Complexity, Traub, J.F., Ed., Academic Press, New York, pp. 251-280 (1976).

44. Bickel, P.J. and Lehmann, E.L. "Descriptive statistics for nonparametric models IV. Spread", In Contributions to Statistics, Jureckova, J., Ed., Hajek Memorial Volume, Academia, Prague, pp. 33-40 (1979).

45. Rousseeuw, P.J. and Croux, C. "Alternatives to the median absolute deviation", Journal of the American Statistical Association, 88(424), pp. 1273-1283 (1993).

46. Croux, C. and Rousseeuw, P.J. "A class of highbreakdown scale estimators based on subranges", Communications in Statistics-Theory and Methods, 21(7), pp. 1935-1951 (1992).

47. Rousseeuw, P.J. and Croux, C. "Explicit scale estimators with high breakdown point", In L1-Statistical Analysis and Related Methods, Dodge, Y., Ed., North- Holland, Amsterdam, pp. 77-92 (1992).

48. Muneer, S., Khalil, A., Shabbir, J., et al. "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).

49. Singh, H.P., Singh, A.K., and Solanki, R.S. "Estimation of finite population variance using auxiliary information in sample surveys", Statistica, 74(1), pp. 99-116 (2014).

50. Murthy, M.N., Sampling Theory and Methods, Statistical Publishing Society, Calcutta (1967).

2. 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).

3. Ogliore, R.C., Huss, G.R., and Nagashima, K. "Ratio estimation in SIMS analysis", Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 269(17), pp. 1910-1918 (2011).

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

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

6. 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, 27(3), pp. 1593-1605 (2020).

7. Sanusi, R.A., Abbas, N., and Riaz, M. "On efficient CUSUM-type location control charts using auxiliary information", Quality Technology & Quantitative Management, 15(1), pp. 87-105 (2018).

8. Naz, F., Nawaz, T., Pang, T., et al. "Use of nonconventional dispersion measures to improve the efficiency of ratio-type estimators of variance in the presence of outliers", Symmetry, 12(1), 16 (2020).

9. Ozel, G., C ingi, H., and Oguz, M. "Separate ratio estimators for the population variance in stratified random sampling", Communications in Statistics-Theory and Methods, 43(22), pp. 4766-4779 (2014).

10. 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).

11. Bahl, S. and Tuteja, R.K. "Ratio and product type exponential estimators", Journal of Information and Optimization Sciences, 12(1), pp. 159-164 (1991).

12. Das, A.K. and Tripathi, T.P. "Use of auxiliary information in estimating the finite population variance", Sankhya C, 40, pp. 139-148 (1978).

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

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

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. 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), pp. 5-5 (2017).

17. Shabbir, J. and Gupta, S. "On improvement in variance estimation using auxiliary information", Communications in Statistics-Theory and Methods, 36(12), pp. 2177-2185 (2007).

18. Singh, H.P. and Solanki, R.S. "Improved estimation of finite population variance using auxiliary information", Communications in Statistics-Theory and Methods, 42(15), pp. 2718-2730 (2013).

19. Singh, H.P., Upadhyaya, L.N., and Namjoshi, U.D. "Estimation of finite population variance", Current Science, 57(24), pp. 1331-1334 (1988).

20. Singh, R. and Malik, S. "Improved estimation of population variance using information on auxiliary attribute in simple random sampling", Applied Mathematics and Computation, 235, pp. 43-49 (2014).

21. 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).

22. 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).

23. 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).

24. Swain, A.K.P.C. "Generalized estimator of finite population variance", Journal of Statistical Theory and Applications, 14(1), pp. 45-51 (2015).

25. 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).

26. Yadav, S.K., Kadilar, C., Shabbir, J., et al. "Improved family of estimators of population variance in simple random sampling", Journal of Statistical Theory and Practice, 9(2), pp. 219-226 (2015).

27. 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).

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

29. Abid, M., Nazir, H.Z., Riaz, M., et al. "Improved ratio estimators using some robust measures", Hacettepe Journal of Mathematics and Statistics, 47(5), pp. 1375-1393 (2018).

30. Singh, H.P. and Yadav, A. "A new exponential approach for reducing the mean squared errors of the estimators of population mean using conventional and non-conventional location parameters", Journal of Modern Applied Statistical Methods, 18(1), 26 (2020).

31. Subzar, M., Maqbool, S., Raja, T.A., et al. "Improved family of ratio type estimators for estimating population mean using conventional and non conventional location parameters", Revista Investigacion Operacional, 38(5), pp. 499-513 (2017).

32. Abid, M., Sherwani, R.A.K., Abbas, N., et al. "Some improved modified ratio estimators based on decile mean of an auxiliary variable", Pakistan Journal of Statistics and Operation Research, 12(4), pp. 787-797 (2016).

33. Abid, M., Ahmed, S., Tahir, M., et al. "Improved ratio estimators of variance based on robust measures", Scientia Iranica, 26(4), pp. 2484-2494 (2019).

34. Naz, F., Abid, M., Nawaz, T., et al. "Enhancing efficiency of ratio-type estimators of population variance by a combination of information on robust location measures", Scientia Iranica, 27(4), pp. 2040-2056 (2020).

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

36. 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).

37. Yadav, S.K. and Kadilar, C. "A two parameter variance estimator using auxiliary information", Applied Mathematics and Computation, 226, pp. 117-122 (2014).

38. Gini, C. "Variabilita e mutabilita", in Memorie di Metodologica Statistica Ed., Libreria Eredi Virgilio Veschi, Rome, (1912).

39. David, H.A. "Miscellanea: Gini's mean difference rediscovered", Biometrika, 55(3), pp. 573-575 (1968).

40. Downton, F. "Linear estimates with polynomial coeficients", Biometrika, 53(1-2), pp. 129-141 (1966).

41. Muhammad, F., Ahmad, S., and Abiodullah, M. "Use of probability weighted moments in the analysis of means", Biometrical Journal, 35(3), pp. 371-378 (1993).

42. Hampel, F.R. "The influence curve and its role in robust estimation", Journal of the American Statistical Association, 69(346), pp. 383-393 (1974).

43. Shamos, M.I. "Geometry and Statistics: Problems at the Interface", In New Directions and Recent Results in Algorithms and Complexity, Traub, J.F., Ed., Academic Press, New York, pp. 251-280 (1976).

44. Bickel, P.J. and Lehmann, E.L. "Descriptive statistics for nonparametric models IV. Spread", In Contributions to Statistics, Jureckova, J., Ed., Hajek Memorial Volume, Academia, Prague, pp. 33-40 (1979).

45. Rousseeuw, P.J. and Croux, C. "Alternatives to the median absolute deviation", Journal of the American Statistical Association, 88(424), pp. 1273-1283 (1993).

46. Croux, C. and Rousseeuw, P.J. "A class of highbreakdown scale estimators based on subranges", Communications in Statistics-Theory and Methods, 21(7), pp. 1935-1951 (1992).

47. Rousseeuw, P.J. and Croux, C. "Explicit scale estimators with high breakdown point", In L1-Statistical Analysis and Related Methods, Dodge, Y., Ed., North- Holland, Amsterdam, pp. 77-92 (1992).

48. Muneer, S., Khalil, A., Shabbir, J., et al. "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).

49. Singh, H.P., Singh, A.K., and Solanki, R.S. "Estimation of finite population variance using auxiliary information in sample surveys", Statistica, 74(1), pp. 99-116 (2014).

50. Murthy, M.N., Sampling Theory and Methods, Statistical Publishing Society, Calcutta (1967).

Transactions on Industrial Engineering (E)

July and August 2022Pages 2115-2133