An Estimator of the Mean Estimation of Study Variable Using Median of Auxiliary Variable

In the present study, I propose a modified estimator for estimating the population mean of the study variable auxiliary information when the population mean and the population median of the auxiliary variable is known. The expression of bias and mean squared error (MSE) of the proposed estimator is derived. Some existing estimators are also discussed. Comparisons of the proposed estimator with the other estimators are carried out. The results obtained are illustrated numerically by using three natural populations considered in the literature.


Introduction to the Research Problem
Use of auxiliary information in the estimation of population parameters such as population mean, ratio of two population means, product of two population means, coefficient of variation etc. has been in practice. Ratio, product and regression type estimators are good examples in this context. Cochran (1940) initiated the use of auxiliary information at estimation stage and proposed ratio estimator for population mean. It is well established fact that ratio type estimators provide better efficiency in comparison to simple mean estimator if the study variable and auxiliary variable are positively correlated. If the correlation between the study variable and auxiliary variables negative, product estimator given by Robson (1957) is more efficient than simple mean estimator. Further improvements are also achieved on the classical ratio estimator by introducing a large number of modified ratio estimators with the use of known parameters like, coefficient of variation, coefficient of kurtosis, coefficient of skewness and population correlation coefficient. For more detailed discussion one may refer to Cochran (1977), Cingi (2004, 2006), Koyuncu and Kadilar (2009), Murthy (1967), Prasad (1989), Rao (1991), Singh (2003), Singh andTailor (2003, 2005), Singh et al (2004), Sisodia and Dwivedi (1981), Upadhyaya and Singh (1999) and Yan and Tian (2010). Further, Subramani and Kumarapandiyan (2013) had taken initiative by proposed modified ratio estimator for estimating the population mean of the study variable by using the population median of the auxiliary variable. The objective of the paper is to proposed modified estimator for estimating the population mean by using the population median of the auxiliary variable.

Notations Used
The following are the notations used in the paper: : is the coefficient of Skewness of the auxiliary variable, ) is the coefficient of Kurtosis of the auxiliary variable, and is the population median of the auxiliary variable.

Procedure and Definitions
Consider a finite population ( ) of size . Let and denote the study variable and the auxiliary variable taking values and respectively on the i th unit (i = 1, 2,…,N). For estimating the population mean ̅ of a simple random IASSL ISSN 2424-6271 109 sample of size is draw without replacement from the population . Then the classical ratio estimator is defined as and the product estimator is given by ̅ ̅ ̅ ; where ̅ , the population mean of the auxiliary variable x is known. The mean squared error expressions of the ratio and product estimators are Further, a list of modified ratio estimators is given in table 1 is used for assessing the performance of the proposed estimator along with their bias and mean squared error expressions.

Proposed Estimator
Following Subramani and Kumarapandiyan (2013), I have proposed an estimator for estimating the population mean when the population mean and population median of the auxiliary variable is known where is the population median of the auxiliary variable X. To the first degree of approximation, I have obtained the expression of bias and mean squared error (MSE) of the proposed estimator as The ( ) will be minimum when

Efficiency Comparison
For comparison of proposed estimator with the existing estimators, I have derived the conditions for which the proposed estimator is more efficient than the existing modified ratio estimators as From the above conditions, it is noted that the proposed estimator is more efficient among other discussed estimators if the above conditions holds true.

Empirical Study
To demonstrate the performance of the suggested estimator empirically in comparison to other estimators. I have used three natural population data sets.   From the above table, it is envisaged that the proposed optimum estimator is more efficient than other existing estimator mentioned in Table 1 in terms of less mean squared error (MSE).

Conclusion
In this paper, I have proposed a modified estimator based on simple random sampling without replacement by using auxiliary variable, under the situation when population mean and median of the auxiliary variable is known. It is found that the performance of the proposed estimator in terms of bias and mean squared error is more efficient than all other existing estimator for certain known population parameters of auxiliary variable. The above results are supported theoretically and empirically by three natural populations.