New Efficient Estimators of Population Mean Using Non-Traditional Measures of Dispersion

One of the aims in survey sampling is to search for the estimators with highest efficiency. In the present paper, three improved estimators of population mean have been proposed using some non-traditional measures of dispersion of auxiliary variable such as Gini’s mean difference, Downton’s method and probability weighted moments early given by Abid [1] with a special population parameter of auxiliary variable. The large sample properties that are biased and mean squared errors of the proposed estimators have been derived up to the first order of approximation. A theoretical comparison of the proposed estimators has been made with the other existing estimators of population mean using auxiliary information. The conditions under which the proposed estimators perform better than the other existing estimators of population mean have been given. A numerical study is also carried out to see the performances of the proposed and existing estimators of population mean and verify the conditions under which proposed estimators are better than other estimators. It has been shown that the proposed estimators perform better than the existing estimators as they are having lesser mean squared error.


Introduction
Sampling is done when the population is very large and we have to get the result very soon.The population parameters are estimated by the corresponding statistics in a natural sense.As it has been mentioned that the most suitable estimator for the estimation of population parameter is the corresponding statistics so to estimate population mean the most suitable estimator is the sample mean.Although he sample mean is an unbiased estimator of population mean and it has reasonably large variance and our aim is to search for the estimator with minimum variance or may be biased but with minimum mean squared error.This purpose is solved through the use of auxiliary information.Auxiliary information is obtained from auxiliary variable which is highly positively or negatively correlated with main variable under study.When the auxiliary variable is positively correlated with the main variable under study, ratio type estimators are used for improved estimation of population parameters.When it is negatively with the main variable under consideration, product type estimators are used for improved estimation of population parameters.In the present manuscript, we have confined our study to positively correlated populations only and proposed three ratio type estimators for improved estimation of population mean with higher efficiencies.
Let the population under consideration consists of N distinct and identifiable units and let ( ) , , 1, 2, , i i x y i n =  be a two variable sample of size n taken from bivariate variables (X, Y) through simple random sampling without sampling scheme.Let X and Y be the population means of the auxiliary and the study variables respectively, and let x and y be the respective sample means and both are unbiased estimators of X and Y respectively.Let the correlation coefficient between the variables X and Y be denoted by ρ .

Existing Estimators under Review
As mentioned above most appropriate estimator of population mean is the sample mean y given by,

Proposed Estimators
Motivated by Abid et al. [1] and Subramani [19] and searching for the improved estimators, we have used a specific parameter as the ratio of correlation coefficient and coefficient of skewness of auxiliary variable along with some non-traditional parameters of auxiliary variable given by Abid et al. [1] as, ( ) where, To study the large sample approximations, we have used the following approximations as, ( ) ( ) and ( ) , and Using above approximation and up to the first order of approximations, the biases and the mean squared errors of proposed estimators are given by, ( ) where,

Efficiency Comparison
In this section, the proposed estimators have been compared theoretically with the other existing estimators of population mean in terms of theirs variances and mean squared errors under simple random sampling without replacement scheme.
From Equation (3) and the from the Equation ( 1), the proposed estimators performs better than the mean per unit estimator if, ( ) ( ) The proposed estimators ( ) 3) are better than the ratio estimator by Cochran [2] r t in Equation ( 2) under the condition if, ( ) From Equation (3) and the mean squared error of the estimators given by Kadilar and Cingi [14] in Table 1, the proposed estimators perform better than the Kadilar and Cingi [14] estimators under the condition if, ( ) From the mean squared errors of proposed estimators and Kadilar and Cingi [15] estimators respectively in Equation (3) and in Table 1, the proposed estimators are better than the Kadilar and Cingi [15] estimators if, ( ) From Equation (3) and the mean squared error of the estimators given by Yan and Tian [16] in Table 1, the proposed estimators are better than Yan and Tian [16] estimators if, ( ) From Equation (3) and the mean squared errors of the estimators given by Subramani and Kumarpandiyan [17] in Table 1, the proposed estimators perform better than Subramani and Kumarpandiyan [17] estimators if, ( ) , 1, 2,3 , 13,14,15,16 The proposed estimators are better than the estimators by Jeelani et al. [18] in From MSE of the proposed estimators in Equation ( 3) and the estimators given by Abid et al. [1], it is found the proposed estimators are better than Abid et al. [1] estimators if, ( )

Empirical Example
To judge the performances of the proposed and the existing estimators of population mean and to verify the conditions under which proposed estimators performs better than the existing estimators, we have considered the population given by Kadilar and Cingi [14].The numerical values of the constants, biases and the mean squared error of the proposed and the existing estimators have been calculated for this data.The population parameters for the above popula-

Results
Form Table 2, we see that the proposed estimators are having lesser biases and mean squared errors as compared to all existing estimators.So the proposed estimators are more efficient than the other estimators for estimating population mean.Our purpose to search for the estimator with higher efficiency is achieved.

Conclusion
This paper deals with the estimation of population mean of the study variable using auxiliary variable in the form of a special parameter along with some non-traditional measures of dispersion of auxiliary variable used by Abid et al. [1].The expressions for the biases and mean squared errors of these proposed estimators have been derived up to the first order of approximation.A theoretical comparison of the proposed estimators has been made with the existing estimators of population mean under simple random sampling scheme.An empirical study is also carried out to judge the performances of the proposed and existing estimators of population mean.Through this numerical study, it has been found that the proposed estimators are more efficient than the other existing estimators.As proposed estimators are more efficient estimators for population mean, so they should be used for the improved estimation of population mean of study variable using auxiliary variable under simple random sampling scheme.

Table 1 .
Various estimators of population mean, bias, mean squared error and constant.

Table 2
represents the numerical values of constants, biases and the mean squared errors of proposed and other existing estimators of population mean using auxiliary variable for the above data.

Table 2 .
Constants, Biases and MSE of Proposed and other estimators.Further it is to be mentioned that among the proposed estimators,