Modified Estimation Using Different Linear Combination

The present study was taken under consideration in order to propose new modified estimators using different linear combinations for population mean using the auxiliary information of MidRange with coefficient of correlation, coefficient of variation and coefficient of skewness in order to achieve more precision in estimates than the already existing estimators. The properties associated with the proposed estimators are assessed by mean square error and bias and compared with the existing estimators. In the support of the theoretical proposed work we have given numerical illustration.


INTRODUCTION
An important purpose of sampling theory is to make sampling more efficient. It attempts to develop methods of sample selection and of estimation that provides, at the lowest possible cost, estimates that are precise enough for our purpose. This principle of specified precision at minimum cost recurs repeatedly in the presentation of theory. Estimation theory is an important part of statistical studies, whereby, population parameters are obtained using sample statistics. In any survey work, the experimenter's interest is to make use of methods that will improve precisions of estimates of the population parameters both at the design stage and estimation stage. These parameters can be totals, means or proportions of some desired characters. In sample surveys, auxiliary information is used at selection as well as estimation stages to improve the design as well as obtaining more efficient estimators. Increased precision can be obtained when study variable Y is highly correlated with auxiliary variable X. Usually, in a class of efficient estimators, the estimator with minimum variance or mean square error is regarded as the most efficient estimator. Linear combination 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 linear combination estimators provide better efficiency in comparison to simple mean estimator if the study variable and auxiliary variable are positively correlated.
Further, had taken initiative by proposed modified linear combinations 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 propose modified estimators for estimating the population mean by using their linear combinations with the correlation coefficient and the coefficient of skewness of the auxiliary variable.

Notations Used
The following are the notations used in the paper: Then the classical ratio estimator is defined.
Where X , the population mean of the auxiliary variable x is known.
The mean square 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.
Where MR is the Mid-range of the auxiliary variable X.
To the first degree of approximation, we have obtained the expression of bias and mean squared error (MSE) of the proposed estimator as Kadilar and Cingi (2004) Kadilar and Cingi (2006) Kadilar and Cingi (2006) And The, MSE will be minimum when by substituting the minimum value of  in the proposed estimators, one can get the asymptotically optimum estimators (AOE) as . From the above conditions, it is noted that the proposed estimators are 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. We have used the Murthy (1967) where Y is output for 34 factories in a region and X is Data on number of workers. The descriptions of the population are given below. Here, we have computed mean squared error (MSE) and the Bias of the estimators.
The results are given in the following table.  Kadilar and Cingi (2006) 53.98248 13.76056 T4; Kadilar and Cingi (2006) 52.63652 13.58105 T5; Kadilar and Cingi (2006) 50.78761 13.33051 T6; Kadilar and Cingi (2006) 42.40512 12.12991 Proposed 1 0.855269351 0.332265112 Proposed 2 0.855269334 0.312863365 Proposed 3 0.855269328 0.257737132 CONCLUSION In this paper we have proposed modified estimators based on simple random sampling without replacement by using the auxiliary variable, under the situation when mid-range, coefficient of skewness and correlation coefficient is known. We found that the performances of our proposed estimators in terms of mean square error are more efficient than all other existing estimators in the literature. Hence we strongly recommend that our proposed estimators preferred over the existing estimators for use in practical application.