Exponential Ratio Type Estimators of Population Mean under Non-Response
ABSTRACT
This paper proposes some exponential ratio type estimators of population mean under the situations when certain observations for some sampling units are missing. These missing observations may be for either auxiliary variable or study variable. The biases and mean square errors of the proposed estimators have been derived, up to the first order of approximation. The proposed estimators are compared theoretically with that of the existing ratio type estimators defined by [1]. It has been found that the proposed exponential ratio type estimators perform better than the mean per unit estimator even for the low positive correlation between study variable and auxiliary variable. Moreover, we obtained the conditions for which our proposed estimators are better than the corresponding ratio type estimators of [1]. To verify the theoretical results obtained, a simulation study is carried out finally.
Cite this paper
L. Grover and P. Kaur, "Exponential Ratio Type Estimators of Population Mean under Non-Response," Open Journal of Statistics, Vol. 4 No. 1, 2014, pp. 97-100. doi: 10.4236/ojs.2014.41010.
L. Grover and P. Kaur, "Exponential Ratio Type Estimators of Population Mean under Non-Response," Open Journal of Statistics, Vol. 4 No. 1, 2014, pp. 97-100. doi: 10.4236/ojs.2014.41010.
References
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http://dx.doi.org/10.1007/PL00003973 [2] W. G. Cochran, “The Estimation of Yields of Cereal Experiments by Sampling for the Ratio Gain to Total Produce,” Journal of Agriculture Society, Vol. 30, No. 2, 1940, pp. 262-275.
http://dx.doi.org/10.1017/S0021859600048012 [3] D. S. Tracy and S. S. Osahan, “Random Non-Response on Study Variable versus on Study as Well as Auxiliary Variables,” Statistica, Vol. 54, No. 2, 1994, pp. 163-168. [4] B. B. Khare and S. Srivastava, “Transformed Ratio Type Estimators for the Population Mean in the Presence of Non Response,” Communications in Statistics—Theory and Methods, Vol. 26, No. 7, 1997, pp. 1779-1791.
http://dx.doi.org/10.1080/03610929708832012 [5] H. Toutenberg and V. K. Srivastava, “Efficient Estimation of Population Mean Using Incomplete Survey Data on Study and Auxiliary Characteristics,” Statistica, Vol. 63, No. 2, 2003, pp. 223-236. [6] H. J. Chang and K. C. Huang, “Ratio Estimation in Survey Sampling When Some Observations Are Missing,” Information and Management Sciences, Vol. 12, No. 2, 2001, pp. 1-9.
http://dx.doi.org/10.1016/S0378-7206(01)00075-1 [7] C. N. Bouza, “Estimation of the Population Mean with Missing Observations Using Product Type Estimators,” Revista Investigación Operacional, Vol. 29, No. 3, 2008, pp. 207-223. [8] M. K. Chaudhary, R. Singh, R. K. Shukla, M. Kumar and F. Smarandache, “A Family of Estimators for Estimating Population Mean in Stratified Sampling under Non-Response,” Pakistan Journal of Statistics and Operational Research, Vol. 5, No. 1, 2009, pp. 47-54. [9] M. Ismail, M. Q. Shahbaz and M. Hanif, “A General Class of Estimators of Population Mean in Presence of Non-Response,” Pakistan Journal of Statistics, Vol. 27, No. 4, 2011, pp. 467-476. [10] C. Kadilar and H. Cingi, “Estimators for the Population Mean in the Case of Missing Data,” Communications in Statistics— Theory and Methods, Vol. 37, No. 14, 2008, pp. 2226-2236.
http://dx.doi.org/10.1080/03610920701855020 [11] S. Kumar and H. P. Singh, “Estimation of Mean Using Multi Auxiliary Information in Presence of Non-Response,” Communications of the Korean Statistical Society, Vol. 17, No. 3, 2010, pp. 391-411.
http://dx.doi.org/10.5351/CKSS.2010.17.3.391 [12] N. Nangsue, “Adjusted Ratio and Regression Type Estimators for Estimation of Population Mean When Some Observations Are Missing,” World Academy of Science, Engineering and Technology, Vol. 53, 2009, pp. 787-790. [13] M. M. Rueda, S. Gonza’lez and A. Arcos, “Indirect Methods of Imputation of Missing Data Based on Available Units,” Applied Mathematics and Computation, Vol. 164, No. 1, 2005, pp. 249-261. http://dx.doi.org/10.1016/j.amc.2004.04.102 [14] M. M. Rueda, S. Gonza’lez and A. Arcos, “Estimating the Difference between Two Means with Missing Data in Sample Surveys,” Model Assisted Statistics and Application, Vol. 1, No. 1, 2005, pp. 51-56. [15] M. M. Rueda, S. Gonza’lez and A. Arcos, “A General Class of Estimators with Auxiliary Information Based on Available Units,” Applied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 131-148.
http://dx.doi.org/10.1016/j.amc.2005.07.018 [16] M. Rueda, S. Martínez, H. Martínez and A. Arcos, “Mean Estimation with Calibration Techniques in Presence of Missing Data,” Computational Statistics and Data Analysis, Vol. 50, No. 11, 2006, pp. 3263-3277.
http://dx.doi.org/10.1016/j.csda.2005.06.003 [17] D. Shukla, N. S. Thakur and D. S. Thakur, “Utilization of Non-Response Auxiliary Population Mean in Imputation for Missing Observations,” Journal of Reliability and Statistical Studies, Vol. 2, No. 1, 2009, pp. 28-40. [18] H. P. Singh and S. Kumar, “A General Family of Estimators of Finite Population Ratio, Product and Mean Using Two Phase Sampling Scheme in the Presence of Non-Response,” Journal of Statistical Theory and Practice, Vol. 2, No. 4, 2008, pp. 677692.
http://dx.doi.org/10.1080/15598608.2008.10411902 [19] H. P. Singh and S. Kumar, “A General Procedure of Estimating the Population Mean in the Presence of Non-Response under Double Sampling Using Auxiliary Information,” Statistics and Operations Research Transactions, Vol. 33, No. 1, 2009, pp. 71-84. [20] H. P. Singh and S. Kumar, “Improved Estimation of Population Mean under Two-Phase Sampling with Sub Sampling the NonRespondents,” Journal of Statistical Planning and Inference, Vol. 140, No. 9, 2010, pp. 2536-2550.
http://dx.doi.org/10.1016/j.jspi.2010.03.023 [21] H. P. Singh and S. Kumar, “Combination of Regression and Ratio Estimate in Presence of Nonresponse,” Brazilian Journal of Probability and Statistics, Vol. 25, No. 2, 2011, pp. 205-217. http://dx.doi.org/10.1214/10-BJPS117 [22] S. Singh, H. P. Singh, R. Tailor, J. Allen and M. Kozak, “Estimation of Ratio of Two Finite-Population Means in the Presence of Non-Response,” Communications in Statistics—Theory and Methods, Vol. 38, No. 19, 2009, pp. 3608-3621.
http://dx.doi.org/10.1080/03610920802610100 [23] H. P. Singh and R. S. Solanki, “Estimation of Finite Population Means Using Random Non-Response in Survey Sampling,” Pakistan Journal of Statistics and Operation Research, Vol. 7, No. 1, 2011, pp. 21-41. [24] H. P. Singh and R. S. Solanki, “A General Procedure for Estimating the Population Parameter in the Presence of Random NonResponse,” Pakistan Journal of Statistics, Vol. 27, No. 4, 2011, pp. 427-465. [25] S. Bahl and R. K. Tuteja, “Ratio and Product Type Exponential Estimators,” Journal of Information and Optimization Sciences, Vol. 12, No. 1, 1991, pp. 159-163.
http://dx.doi.org/10.1080/02522667.1991.10699058 [26] V. N. Reddy, “A Study on the Use of Prior Knowledge on Certain Population Parameters in Estimation,” Sankhya, Sr C, Vol. 40, 1978, pp. 29-37. [27] D. Singh and F. S. Chaudhary, “Theory and Analysis of Sample Survey Designs,” New Age International (P) Limited Publishers, New Delhi, 2002.
[1] H. Toutenberg and V. K. Srivastava, “Estimation of Ratio of Population Means in Survey Sampling When Some Observations Are Missing,” Metrika, Vol. 48, No. 3, 1998, pp. 177-187.
http://dx.doi.org/10.1007/PL00003973 [2] W. G. Cochran, “The Estimation of Yields of Cereal Experiments by Sampling for the Ratio Gain to Total Produce,” Journal of Agriculture Society, Vol. 30, No. 2, 1940, pp. 262-275.
http://dx.doi.org/10.1017/S0021859600048012 [3] D. S. Tracy and S. S. Osahan, “Random Non-Response on Study Variable versus on Study as Well as Auxiliary Variables,” Statistica, Vol. 54, No. 2, 1994, pp. 163-168. [4] B. B. Khare and S. Srivastava, “Transformed Ratio Type Estimators for the Population Mean in the Presence of Non Response,” Communications in Statistics—Theory and Methods, Vol. 26, No. 7, 1997, pp. 1779-1791.
http://dx.doi.org/10.1080/03610929708832012 [5] H. Toutenberg and V. K. Srivastava, “Efficient Estimation of Population Mean Using Incomplete Survey Data on Study and Auxiliary Characteristics,” Statistica, Vol. 63, No. 2, 2003, pp. 223-236. [6] H. J. Chang and K. C. Huang, “Ratio Estimation in Survey Sampling When Some Observations Are Missing,” Information and Management Sciences, Vol. 12, No. 2, 2001, pp. 1-9.
http://dx.doi.org/10.1016/S0378-7206(01)00075-1 [7] C. N. Bouza, “Estimation of the Population Mean with Missing Observations Using Product Type Estimators,” Revista Investigación Operacional, Vol. 29, No. 3, 2008, pp. 207-223. [8] M. K. Chaudhary, R. Singh, R. K. Shukla, M. Kumar and F. Smarandache, “A Family of Estimators for Estimating Population Mean in Stratified Sampling under Non-Response,” Pakistan Journal of Statistics and Operational Research, Vol. 5, No. 1, 2009, pp. 47-54. [9] M. Ismail, M. Q. Shahbaz and M. Hanif, “A General Class of Estimators of Population Mean in Presence of Non-Response,” Pakistan Journal of Statistics, Vol. 27, No. 4, 2011, pp. 467-476. [10] C. Kadilar and H. Cingi, “Estimators for the Population Mean in the Case of Missing Data,” Communications in Statistics— Theory and Methods, Vol. 37, No. 14, 2008, pp. 2226-2236.
http://dx.doi.org/10.1080/03610920701855020 [11] S. Kumar and H. P. Singh, “Estimation of Mean Using Multi Auxiliary Information in Presence of Non-Response,” Communications of the Korean Statistical Society, Vol. 17, No. 3, 2010, pp. 391-411.
http://dx.doi.org/10.5351/CKSS.2010.17.3.391 [12] N. Nangsue, “Adjusted Ratio and Regression Type Estimators for Estimation of Population Mean When Some Observations Are Missing,” World Academy of Science, Engineering and Technology, Vol. 53, 2009, pp. 787-790. [13] M. M. Rueda, S. Gonza’lez and A. Arcos, “Indirect Methods of Imputation of Missing Data Based on Available Units,” Applied Mathematics and Computation, Vol. 164, No. 1, 2005, pp. 249-261. http://dx.doi.org/10.1016/j.amc.2004.04.102 [14] M. M. Rueda, S. Gonza’lez and A. Arcos, “Estimating the Difference between Two Means with Missing Data in Sample Surveys,” Model Assisted Statistics and Application, Vol. 1, No. 1, 2005, pp. 51-56. [15] M. M. Rueda, S. Gonza’lez and A. Arcos, “A General Class of Estimators with Auxiliary Information Based on Available Units,” Applied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 131-148.
http://dx.doi.org/10.1016/j.amc.2005.07.018 [16] M. Rueda, S. Martínez, H. Martínez and A. Arcos, “Mean Estimation with Calibration Techniques in Presence of Missing Data,” Computational Statistics and Data Analysis, Vol. 50, No. 11, 2006, pp. 3263-3277.
http://dx.doi.org/10.1016/j.csda.2005.06.003 [17] D. Shukla, N. S. Thakur and D. S. Thakur, “Utilization of Non-Response Auxiliary Population Mean in Imputation for Missing Observations,” Journal of Reliability and Statistical Studies, Vol. 2, No. 1, 2009, pp. 28-40. [18] H. P. Singh and S. Kumar, “A General Family of Estimators of Finite Population Ratio, Product and Mean Using Two Phase Sampling Scheme in the Presence of Non-Response,” Journal of Statistical Theory and Practice, Vol. 2, No. 4, 2008, pp. 677692.
http://dx.doi.org/10.1080/15598608.2008.10411902 [19] H. P. Singh and S. Kumar, “A General Procedure of Estimating the Population Mean in the Presence of Non-Response under Double Sampling Using Auxiliary Information,” Statistics and Operations Research Transactions, Vol. 33, No. 1, 2009, pp. 71-84. [20] H. P. Singh and S. Kumar, “Improved Estimation of Population Mean under Two-Phase Sampling with Sub Sampling the NonRespondents,” Journal of Statistical Planning and Inference, Vol. 140, No. 9, 2010, pp. 2536-2550.
http://dx.doi.org/10.1016/j.jspi.2010.03.023 [21] H. P. Singh and S. Kumar, “Combination of Regression and Ratio Estimate in Presence of Nonresponse,” Brazilian Journal of Probability and Statistics, Vol. 25, No. 2, 2011, pp. 205-217. http://dx.doi.org/10.1214/10-BJPS117 [22] S. Singh, H. P. Singh, R. Tailor, J. Allen and M. Kozak, “Estimation of Ratio of Two Finite-Population Means in the Presence of Non-Response,” Communications in Statistics—Theory and Methods, Vol. 38, No. 19, 2009, pp. 3608-3621.
http://dx.doi.org/10.1080/03610920802610100 [23] H. P. Singh and R. S. Solanki, “Estimation of Finite Population Means Using Random Non-Response in Survey Sampling,” Pakistan Journal of Statistics and Operation Research, Vol. 7, No. 1, 2011, pp. 21-41. [24] H. P. Singh and R. S. Solanki, “A General Procedure for Estimating the Population Parameter in the Presence of Random NonResponse,” Pakistan Journal of Statistics, Vol. 27, No. 4, 2011, pp. 427-465. [25] S. Bahl and R. K. Tuteja, “Ratio and Product Type Exponential Estimators,” Journal of Information and Optimization Sciences, Vol. 12, No. 1, 1991, pp. 159-163.
http://dx.doi.org/10.1080/02522667.1991.10699058 [26] V. N. Reddy, “A Study on the Use of Prior Knowledge on Certain Population Parameters in Estimation,” Sankhya, Sr C, Vol. 40, 1978, pp. 29-37. [27] D. Singh and F. S. Chaudhary, “Theory and Analysis of Sample Survey Designs,” New Age International (P) Limited Publishers, New Delhi, 2002.