CSIRO PUBLISHING The South Pacific Journal of Natural and Applied Sciences, 31, 39-44, 2013 www.publish.csiro.au/journals/spjnas 10.1071/SP13003 Modified ratio estimators of population mean using linear combination of co-efficient of skewness and quartile deviation M. Iqbal Jeelani and S. Maqbool Division of Agricultural Statistics, Faculty of Agriculture, Sher-e-Kashmir University of Agricultural Sciences and Technology of Kashmir, Shalimar-Srinagar, Jammu and Kashmir, India. Abstract The present paper deals with the estimation of population mean of the study variable using the linear combination of known population values of coefficient of skewness and quartile deviation of auxiliary variable. Two modified ratio estimators for estimation of population mean of the study variable involving the above linear combinations are being used. Mean squared errors and biases up to the first degree of approximation are derived and compared with the proposed modified ratio estimators. The proposed modified ratio estimators perform better than the existing ratio estimators. The empirical study has been carried out in support of the results. Keywords: Modified ratio estimators, simple random sampling, co-efficient of skewness, quartile deviation 1. Introduction Sampling is not mere substitution of a partial coverage for a total coverage. Sampling is the science and art of controlling and measuring the reliability of useful statistical information through the theory of probability. The simplest and the most common method of sampling is simple random sampling in which a sample is drawn unit by unit, with equal probability of selection for each unit at each draw, there is no additional information available. Most of the times in sample surveys, along with the variable of interest, information on auxiliary variable, which is highly correlated with is also collected. This information on auxiliary variable may well be utilized to obtain a more efficient estimator of population mean. Ratio method of estimation is one such example which utilizes the information on auxiliary variable, which is positively correlated with the variable of interest, in order to improve the precision of the estimate of population mean. Early historical developments of the ratio method of estimation are being well presented by Sen (1993). Some of the modified Ratio estimators given by Cochran (1977), Kadilar and Cingi (2004, 2006), Koyuncu and Kadilar (2009), Murthy (1967), Prasad (1989), Singh and Tailor (2003, 2005), Singh et.al (2004), Upadhyaya and Singh (1999), Yan and Tian (2010) are available in literature. The lists of these modified Ratio estimators along with their mean squared errors, biases and constants available in literature are classified into two classes class A and class B, which are given in Tables 1 and 2. These modified ratio estimators are although biased, but have minimum mean squared errors as compared to the classical ratio estimator. But till date no attempt has been made which utilizes the linear combination of known values of Co-Efficient of Skewness and Quartile Deviation of the auxiliary variable, hence an attempt has been made to propose some modified based on the above linear combination of the auxiliary variable which performs better than the already existing modified Ratio estimators, and are given in the Tables 1 and 2. The notations to be used in the paper are described Population size; Sample size; Sampling fraction; Study variable; Auxiliary variable; Population means; Sample means; Population standard deviations; Coefficient of variations; Co-efficient of correlation; auxiliary variable;, Co-efficient of Skewness of, Co-efficient of kurtosis; Median of auxiliary variable; Quartile Deviation (, is 3 rd quartile and is I st quartile of auxiliary variable; Bias of the estimator; Mean squared error of estimator Existing (proposed) modified estimator of. The Ratio estimator for estimating the population mean of the study variable is defined as, (1.1) is the estimate of 2. Suggested Modified Ratio Estimators Two modified ratio estimators using the linear combination of Co-Efficient of Skewness and Quartile Deviation of the auxiliary variable are being proposed. University of the South Pacific (2013)
40 M. I. Jeelani and S. Maqbool: Modified ratio estimators of population Table 1. Existing modified ratio estimators (Class A) with their biases, mean squared errors and their constants. Estimator Bias - (.) Mean squared error MSE(.) Constant Upadhyaya and Singh ( 1999) Upadhyaya and Singh (1999) Yan and Tian (2010) Yan and Tian (2010) Table 2. Existing modified ratio estimators (Class B) with their biases, mean squared errors and their constants. Estimator Bias - (.) Mean squared error MSE (.) Constant Kadilar and Cingi (2004) ) Kadilar and Cingi (2004) Yan and Tian (2010) Kadilar and Cingi (2005) Kadilar and Cingi (2005) The proposed estimators for estimating the population mean together with the first degree of approximation, biases, mean squared errors and constants are given (2.1) (2.4) (2.5) (2.2) (2.6) (2.3)
41 M. I. Jeelani and S. Maqbool: Modified ratio estimators of population 3. Comparison of Efficiency Existing modified Ratio estimators are given in Table 1 and Table 2 grouped in two classes and are given below. The proposed estimator is compared with that of existing modified Ratio estimators which is grouped in class A, while as proposed estimator is compared with that of existing modified Ratio estimators grouped in class B. 3.1.4 (Proposed Estimator 2) (3.9) (3.10) 3.1.1 (Class A) Biases, mean squared errors and constant of existing estimators from to listed in Table 1 are given Conditions for which the above proposed estimator is more efficient than existing modified ratio estimators given in class 1, if is given (3.1) (3.2) Conditions for which the above proposed estimator is more efficient than existing modified ratio estimators given in class 2, if is given,,, (3.3) 3.1.2 (Class B) Biases, mean squared errors and constant of existing estimators from to listed in Table 2 are given,,, (3.4) (3.5), (3.6) 3.1.3 (Proposed Estimator 1) The biases mean squared errors and constant of the suggested modified ratio estimators are given as under: Where (3.7) (3.8) 4. Numerical Illustration To examine the merits of the proposed estimators over its competitors numerically, we have considered two natural populations. Population 1 is taken from Singh and Chaudhary (1986) and population 2 is taken from Cochran(1977). The population parameters and the constants computed from the above populations are given Table 3. The constants of the existing and proposed modified ratio estimators for the above populations are given in the Table 4 and Table 5. The biases of the existing and proposed modified ratio estimators for the above populations are given in the Table 6 and Table 7. The mean squared errors of the existing and proposed modified ratio estimators for the above populations are given in the Table 8 and Table 9. From the values of Table 6 and Table 7, it is observed that the bias of the proposed modified ratio estimator is less than the biases of the existing modified ratio estimators given in Class A and the bias of the proposed modified ratio estimator is less than the biases of the existing modified ratio estimators given in Class B. Similarly from the values of Table 8 and Table 9, it is observed that the mean squared error of the proposed modified ratio estimator is less than the mean squared errors of the existing modified ratio estimators given in Class A and the mean squared error of the proposed modified ratio estimator is less than the mean squared errors of the existing modified ratio estimators given in Class B.
42 M. I. Jeelani and S. Maqbool: Modified ratio estimators of population Table 3. Parameters and constants of populations. 22 49 5 20 22.6209 116.1633 1467.5455 98.6735 0.9022 0.6904 33.0469 98.8286 1.4609 0.8508 2562.1449 102.9709 1.7459 1.0436 13.3694 5.9878 3.3914 2.4224 534.5 64 1035 78.50 Table 4. Constants of class A existing and proposed estimator ( ). Upadhyaya and Singh ( 1999) 0.9948 0.945 Upadhyaya and Singh (1999) 0.9999 0.9982 Yan and Tian (2010) 0.9998 0.9959 Yan and Tian (2010) 0.9973 0.9756 Proposed Estimator 0.8278 0.7527 Table 5. Constants of class B existing and proposed estimator. Kadilar and Cingi (2004) 0.0154 1.1752 Kadilar and Cingi (2004) 0.0153 1.1126 Yan and Tian (2010) 0.0154 1.1485 Kadilar and Cingi (2006) 0.0154 1.1759 Kadilar and Cingi (2006) 0.0153 1.0821 Proposed Estimator 0.0127 0.8862
43 M. I. Jeelani and S. Maqbool: Modified ratio estimators of population Table 6. Biases of class A existing and proposed estimator Upadhyaya and Singh ( 1999) 2.5432 1.3519 Upadhyaya and Singh (1999) 2.6106 1.6268 Yan and Tian (2010) 2.6095 1.6144 Yan and Tian (2010) 2.5763 1.507 Proposed Estimator 0.6195 0.5328 Table 7. Biases of class B existing and proposed estimator Kadilar and Cingi (2004) 10.654 3.7302 Kadilar and Cingi (2004) 10.5456 3.3433 Yan and Tian (2010) 10.5989 3.5627 Kadilar and Cingi (2006) 10.6549 3.7347 Kadilar and Cingi (2006) 10.4439 3.163 Proposed Estimator 7.3049 2.1217 Table 8. Mean squared errors of class A existing and proposed estimator. Upadhyaya and Singh ( 1999) 45.2894 214.7486 Upadhyaya and Singh (1999) 45.8857 233.6573 Yan and Tian (2010) 45.8758 232.7813 Yan and Tian (2010) 45.5814 225.2956 Proposed Estimator 32.604 166.708 Table 9. Mean squared errors of class B existing and proposed estimator Kadilar and Cingi (2004) 272.4185 584.5606 Kadilar and Cingi (2004) 269.9654 539.612 Yan and Tian (2010) 271.1716 565.0981 Kadilar and Cingi (2006) 272.4393 585.0781 Kadilar and Cingi (2006) 267.666 518.6688 Proposed Estimator 196.79 397.755
44 M. I. Jeelani and S. Maqbool: Modified ratio estimators of population 5. Conclusions From theoretical discussion and numerical example, we infer that the proposed estimators are more efficient than the existing estimators, as the constants, biases and mean squared errors of the proposed estimators are less than the constants, biases and mean squared errors of the existing modified ratio estimators for certain known populations. Hence we strongly recommend that the proposed modified estimators may be preferred over the existing modified ratio estimators for the use of practical applications. Acknowledgement The authors are grateful to the Editor and the referee for their valuable comments, suggestions and corrections that helped to improve the manuscript to its present form. References Cochran, W. G.1977. Sampling Techniques. Third Edition, Wiley Eastern Limited, India. Kadilar, C. and Cingi, H. 2004. Ratio estimators in simple random sampling. Applied Mathematics and Computation 151, 893-902. Kadilar, C. and Cingi, H. 2006. An improvement in estimating the population mean by using the correlation co-efficient. Hacettepe. Journal of Mathematics and Statistics 35, 103-109. Koyuncu, N. and Kadilar, C. 2009. Efficient estimators for the population mean. Hacettepe Journal of Mathematics and Statistics 38, 217-225. Murthy, M.N. 1967. Sampling Theory and Methods. Statistical Publishing Society, Calcutta, India. Prasad, B. 1989. Some improved ratio type estimators of population mean and ratio in finite population sample surveys. Communications in Statistics: Theory and Methods 18, 379-392. Sen, A.R. 1993. Some early developments in ratio estimation. Biometric Journal 35, 3-13. Singh, D. and Chaudhary, F.S. 1986. Theory and Analysis of Sample Survey Designs. New Age International Publisher, India. Singh, H.P. and Tailor, R. 2003. Use of known correlation co-efficient in estimating the finite population means. Statistics in Transition 6, 555-560. Singh, H.P. and Tailor, R. 2005. Estimation of finite population mean with known co-efficient of variation of an auxiliary variable. Statistica LXV, 301-313. Singh, H.P., Tailor, R., Tailor, R. and Kakran, M.S. 2004. An improved estimator of population mean using power transformation. Journal of the Indian Society of Agricultural Statistics 58, 223-230. Upadhyaya, L.N. and Singh, H.P. 1999. Use of transformed auxiliary variable in estimating the finite population mean. Biometrical Journal 41, 627-636. Yan, Z. and Tian, B. 2010. Ratio method to the mean estimation using co-efficient of skewness of auxiliary variable. Information Computing and Applications: Communications in Computer and Information Science 106, 103-110. Correspondence to: M. Iqbal Jeelani Email: jeelani.miqbal@gmail.com and S. Maqbool Email: showkatmaq@gmail.com http://www.publish.csiro.au/journals/spjnas