A NEW CHAIN RATIO ESTIMATOR USING INFORMATION ON AUXILIARY ATTRIBUTE

Abstract: In this paper, we develop to ratio estimator suggested by Naik-Gupta [J. Indian Soc. Agric. Stat., 48 (2), 151-158] [1] and obtain its MSE equation. We prove that the proposed chain ratio estimator is more efficient than the Naik-Gupta estimator under certain conditions. In addition, this theoretical result is supported by an application with original data sets.


Introduction
The Naik and Gupta estimator for the population mean of the variate of study, which make use of information regarding the population proportion possessing certain attribute, is defined by where it is assumed that the population proportion of the form of attribute is known.
Let be th characteristic of the population and is the case of possessing certain attributes.
If th unit has the desired characteristic, it takes the value 1, if not then the value 0. That is; Let and be the the total count of the units that possess certain attribute in population and sample, respectively.And and shows the ratio of these units, respectively.
The MSE of the Naik and Gupta estimator is where, ; N is the number of units in the population; is the population ratio; is the population variance of the form of attribute and is the population variance of the study variable [1].

The Proposed Chain Estimator
Following Kadılar and Cingi (2003) [2], We propose a chain estimator using information about population proportion possessing certain attributes.When in (1.1) is replaced with , the proposed chain estimator is obtained as We can re-write (2.1) using (1.1) as, where is real numbers.MSE of this estimator can be found using Taylor series method defined as; where, and [3].Where, .and denote the population of variances of the study variable and unit ratios possessing certain attributes, respectively.denotes the population covariance between units ratio possessing certain attributes and study variable.
According to this definition, we obtain d for this estimator as follows; We obtain the MSE equation of this estimator using (2.3) as follows; where, , , , .
We can have the optimal values of (2.4) by following equations: where .
We can obtain minimum MSE of the proposed chain estimator using the optimal equations of in (2.5).

Efficiency Comparisons
In this section, we compare the MSE of the proposed chain estimator, given in (2.2), with the MSE of the Naik-Gupta estimators, given in (1.1).We have the condition;  For Populations 1 and 2, We take the sample sizes as and using simple random sampling [6] .The MSE of the Naik-Gupta and proposed chain estimators are computed as given in (1.2) and (2.6), respectively, and these estimators are compared to each other with respect to their MSE values.
In tables 1 and 2, There are the statistics about the population for data 1, data 2 sets.Note that the correlations between the variate are 0.766 and 0.878, respectively.; and Thus, the condition mentioned in section 3 is satisfied for Population 1 and 2 data sets.

Conclusion
We have analyzed the proposed chain estimator and obtained its MSE equation.According to the theoretical discussion in Section 3 and the results of the numerical examples, we infer that the proposed chain estimator are more efficient than the Naik-Gupta ratio estimator.In forthcoming studies, we hope to adapt the proposed chain estimators in stratified random sampling.

For
condition (3.1) or (3.2) is satisfied, the proposed chain estimator given in (2.2), are more efficient than the Naik-Gupta estimator, given in (1.1).

Table 2 :
Population 2 Data Statistics