Document Type : Article

**Authors**

^{1}
a. Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 310027, China. b. Department of Statistics, Government College University, Faisalabad, Pakistan

^{2}
Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou 310027, China.

**Abstract**

In survey sampling, most of the research work based on the fact that utilizing the information of auxiliary variable(s) boosts the efficiency of estimators. Keeping this fact in mind we used the information of two auxiliary variables to propose a family of Hartley-Ross type unbiased estimators for estimating population mean under simple random sampling without replacement. Minimum variance of the new family is derived up to first order of approximation. Three real data sets are used to verify that the new family acts efficiently than the usual unbiased, Hartley and Ross (1954), Grover and Kaur (2014), Singh et al. (2014), Cekim and Kadilar (2016), Muneer et al. (2017) and Shabbir and Gupta (2017) estimators.

**Keywords**

**Main Subjects**

References:

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2. Kadilar, C. and Cingi, H. "A new estimator using two auxiliary variables", Applied Mathematics and Computation, 162, pp. 901-908 (2005).

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5. Lu, J., Yan, Z., and Peng, X. "A new exponential ratio type estimator with linear combination of two auxiliary variables", PLOS ONE, 9(12), pp. 1-10 (2014).

6. Vishwakarma., G.K. and Kumar, M. "A general family of dual to ratio-cum-product estimators of population mean in simple random sampling", Chilean Journal of Statistics, 6(2), pp. 69-79 (2015).

7. Sharma, P. and Singh, R. "A class of exponential ratio estimators of finite population mean using two auxiliary variables", Pakistan Journal of Statistics and Operational Research, 11(2), pp. 221-229 (2015).

8. Yasmeen, U., Amin, N.M., and Hanif, M. "Exponential ratio and product type estimators of finite population mean", Journal of Statistics and Management Systems, 19(1), pp. 55-71 (2016).

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10. Muneer, S., Shabbir, J., and Khalil, A. "Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables", Communications in Statistics- Theory and Methods, 46(5), pp. 2181-2192 (2017).

11. 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).

12. Hartley, H.O. and Ross, A. "Unbiased ratio estimators", Nature, 174, pp. 270-272 (1954).

13. Robson, D.S. "Application of multivariate polykays to the theory of unbiased ratio type estimation", Journal of American Statistical Association, 50, pp. 1225-1226 (1957).

14. Murthy, M.N. and Nanjamma, N.S. "Almost unbiased estimator based on interpenetrating sub-sample estimates", Sankhya, 21, pp. 381-392 (1959).

15. Biradar, R.S. and Singh, H.P. "A note on almost unbiased ratio-cum-product estimator", Metron, 40(1-2), pp. 249-255 (1992).

16. Biradar, R.S. and Singh, H.P. "On a class of almost unbiased ratio estimators", Biomedical Journal, 34(8), pp. 937-944 (1992).

17. Biradar, R.S. and Singh, H.P. "A class of unbiased ratio estimators", Journal of Indian Society Agricultural Statistics, 47(3), pp. 230-239 (1995).

18. Sahoo, J., Sahoo, L.N., and Mohanty, S. "An alternative approach to estimation in two phase sampling using two auxiliary variables", Biometrical Journal, 36, pp. 293-298 (1994).

19. Singh, H.P., Sharma, B., and Tailor, R. "Hartley-Ross type estimators for population mean using known parameters of auxiliary variate", Communications in Statistics- Theory and Methods, 43, pp. 547-565(2014).

20. Cekim, H.O. and Kadilar, C. "New unbiased estimators with the help of Hartley-Ross type estimators", Pakistan Journal of Statistics, 32(4), pp. 247-260 (2016).

21. Khan, L., Shabbir, J., and Gupta, S. "Unbiased ratio estimators of the mean in stratified ranked set sampling", Hacettepe Journal of Mathematics and Statistics, 46(6), pp. 1151-1158 (2017).

22. Singh, R. and Mangat, N.S., Elements of Survey Sampling, Norwell, MA: Kluwer Academic Publishers (1996).

23. Grover, L.K. and Kaur, P. "A generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable", Communications in Statistics- Simulation and Computation, 43, pp. 1552-1574 (2014).

24. Gupta, S. and Shabbir, J. "On improvement in estimating the population mean in simple random sampling", Journal of Applied Statistics, 35(5), pp. 559- 566 (2008).

25. Shabbir, J. and Gupta, S. "On estimating finite population mean in simple and stratified random sampling", Communications in Statistics- Theory and Methods, 40(2), pp. 199-212 (2011).

26. Kadilar, C. and Cingi, H., A New Ratio Estimator Using Correlation Coefficient, Inter-Stat, pp. 1-11 (2006).

27. Upadhyaya, L.N. and Singh, H.P. "Use of transformed auxiliary variable in estimating the finite population mean", Biometrical Journal, 41(5), pp. 627-636 (1999).

28. Khoshnevisan, M., Singh, R., Chauhan, P., Sawan, N., and Smarandache, F. "A general family of estimators for estimating population mean using known value of some population parameter(s)", Far East Journal of Theoretical Statistics, 22, pp. 181-191 (2007).

29. Koyuncu, N. and Kadilar, C. "Efficient estimators for the population mean", Hacettepe Journal of Mathematics and Statistics, 38(2), pp. 217-225 (2009).

30. Singh, V.K. and Singh, R. "Performance of an estimator for estimating population mean using simple and stratified random sampling", SOP Transactions on Statistics and Analysis, 1(1), pp. 1-8 (2014).

31. Grover, L.K. and Kaur, P. "An improved estimator of the finite population mean in simple random sampling", Model Assisted Statistics and Application, 6(1), pp. 47-55 (2011).

32. Koyuncu, N. and Kadilar, C. "Family of estimators of population mean using two auxiliary variables in stratified sampling", Communications in Statistics- Theory and Methods, 38, pp. 2398-2417 (2009).

2. Kadilar, C. and Cingi, H. "A new estimator using two auxiliary variables", Applied Mathematics and Computation, 162, pp. 901-908 (2005).

3. Singh, H.P. and Tailor, R. "Estimation of finite population mean using known correlation coefficient between auxiliary characters", Statistica, 65, pp. 407- 418 (2005).

4. Lu, J. and Yan, Z. "A class of ratio estimators of a finite population mean using two auxiliary variables", PLOS ONE, 9(2), pp. 1-6 (2014).

5. Lu, J., Yan, Z., and Peng, X. "A new exponential ratio type estimator with linear combination of two auxiliary variables", PLOS ONE, 9(12), pp. 1-10 (2014).

6. Vishwakarma., G.K. and Kumar, M. "A general family of dual to ratio-cum-product estimators of population mean in simple random sampling", Chilean Journal of Statistics, 6(2), pp. 69-79 (2015).

7. Sharma, P. and Singh, R. "A class of exponential ratio estimators of finite population mean using two auxiliary variables", Pakistan Journal of Statistics and Operational Research, 11(2), pp. 221-229 (2015).

8. Yasmeen, U., Amin, N.M., and Hanif, M. "Exponential ratio and product type estimators of finite population mean", Journal of Statistics and Management Systems, 19(1), pp. 55-71 (2016).

9. Lu, J. "Efficient estimator of a finite population mean using two auxiliary variables and numerical application in agriculture, biomedical and power engineering", Mathematical Problems in Engineering, Article ID 8704734 (2017). https://doi.org/10.1155/2017/8704734.

10. Muneer, S., Shabbir, J., and Khalil, A. "Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables", Communications in Statistics- Theory and Methods, 46(5), pp. 2181-2192 (2017).

11. 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).

12. Hartley, H.O. and Ross, A. "Unbiased ratio estimators", Nature, 174, pp. 270-272 (1954).

13. Robson, D.S. "Application of multivariate polykays to the theory of unbiased ratio type estimation", Journal of American Statistical Association, 50, pp. 1225-1226 (1957).

14. Murthy, M.N. and Nanjamma, N.S. "Almost unbiased estimator based on interpenetrating sub-sample estimates", Sankhya, 21, pp. 381-392 (1959).

15. Biradar, R.S. and Singh, H.P. "A note on almost unbiased ratio-cum-product estimator", Metron, 40(1-2), pp. 249-255 (1992).

16. Biradar, R.S. and Singh, H.P. "On a class of almost unbiased ratio estimators", Biomedical Journal, 34(8), pp. 937-944 (1992).

17. Biradar, R.S. and Singh, H.P. "A class of unbiased ratio estimators", Journal of Indian Society Agricultural Statistics, 47(3), pp. 230-239 (1995).

18. Sahoo, J., Sahoo, L.N., and Mohanty, S. "An alternative approach to estimation in two phase sampling using two auxiliary variables", Biometrical Journal, 36, pp. 293-298 (1994).

19. Singh, H.P., Sharma, B., and Tailor, R. "Hartley-Ross type estimators for population mean using known parameters of auxiliary variate", Communications in Statistics- Theory and Methods, 43, pp. 547-565(2014).

20. Cekim, H.O. and Kadilar, C. "New unbiased estimators with the help of Hartley-Ross type estimators", Pakistan Journal of Statistics, 32(4), pp. 247-260 (2016).

21. Khan, L., Shabbir, J., and Gupta, S. "Unbiased ratio estimators of the mean in stratified ranked set sampling", Hacettepe Journal of Mathematics and Statistics, 46(6), pp. 1151-1158 (2017).

22. Singh, R. and Mangat, N.S., Elements of Survey Sampling, Norwell, MA: Kluwer Academic Publishers (1996).

23. Grover, L.K. and Kaur, P. "A generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable", Communications in Statistics- Simulation and Computation, 43, pp. 1552-1574 (2014).

24. Gupta, S. and Shabbir, J. "On improvement in estimating the population mean in simple random sampling", Journal of Applied Statistics, 35(5), pp. 559- 566 (2008).

25. Shabbir, J. and Gupta, S. "On estimating finite population mean in simple and stratified random sampling", Communications in Statistics- Theory and Methods, 40(2), pp. 199-212 (2011).

26. Kadilar, C. and Cingi, H., A New Ratio Estimator Using Correlation Coefficient, Inter-Stat, pp. 1-11 (2006).

27. Upadhyaya, L.N. and Singh, H.P. "Use of transformed auxiliary variable in estimating the finite population mean", Biometrical Journal, 41(5), pp. 627-636 (1999).

28. Khoshnevisan, M., Singh, R., Chauhan, P., Sawan, N., and Smarandache, F. "A general family of estimators for estimating population mean using known value of some population parameter(s)", Far East Journal of Theoretical Statistics, 22, pp. 181-191 (2007).

29. Koyuncu, N. and Kadilar, C. "Efficient estimators for the population mean", Hacettepe Journal of Mathematics and Statistics, 38(2), pp. 217-225 (2009).

30. Singh, V.K. and Singh, R. "Performance of an estimator for estimating population mean using simple and stratified random sampling", SOP Transactions on Statistics and Analysis, 1(1), pp. 1-8 (2014).

31. Grover, L.K. and Kaur, P. "An improved estimator of the finite population mean in simple random sampling", Model Assisted Statistics and Application, 6(1), pp. 47-55 (2011).

32. Koyuncu, N. and Kadilar, C. "Family of estimators of population mean using two auxiliary variables in stratified sampling", Communications in Statistics- Theory and Methods, 38, pp. 2398-2417 (2009).

Transactions on Industrial Engineering (E)

November and December 2019Pages 3835-3845