An Efficient Class of Estimators of Population Mean in Two-Phase Sampling Using Two Auxiliary Variables

This study is devoted to obtaining an efficient estimator for population mean in two-phase sampling using two auxiliary variables following Vishwakarma and Kumar (2014). Expression for bias and mean squared error   MSE are obtained to the first order of approximation. The new proposed estimator is compared with some competitor estimators both theoretically and numerically using eight different data sets. It has been shown that the new proposed estimator gives more efficient results as compared to its competitor estimators.


Introduction
In sample surveys, it is common practice to use the auxiliary information either at selection stage or on estimation stage, or at both stages, to improve precision of the estimators of the population means.Several methods of using the auxiliary variable at the estimation stage are described, which include linear regression estimator, ratio estimator and product estimator.According to Cochran (1940), ratio method of estimation is most preferred when the study variable is positively correlated with the auxiliary variable.Robson (1957) and Murthy (1964) proposed product method of estimation in case when there is negative correlation between the study variable and the auxiliary variable.
The use of ratio and product strategies in sample survey solely depends upon the knowledge of population mean of the auxiliary variable x .In case when the population mean of the auxiliary variable x is not known in advance, then in such situation a first phase sample of size n is selected from the population of size N and only the auxiliary variable   x is measured to get an estimate of population mean   .
X Then in second phase a sample of a sample of size n is drawn from the first phase sample of size n on which both the study variable y and auxiliary variable x are measured.This technique of selecting the samples from given population is known as two-phase or double sampling.Sukhatme (1962), Hidiroglou and Sarndal (1998), Singh and Vishwakarma (2007) and Sahoo et al. (2010) have suggested some improved ratio, product and regression type estimators in two-phase sampling.
The chain regression type estimator was first introduced by Swain (1970).Chand (1975), Sukhatme and Chand (1977) and Kiregyera (1980) suggested some chain ratio and regression type estimators based on two auxiliary variables in two-phase sampling.Prasad et al. (1996), Singh and Espejo (2007), Singh and Choudhury (2012), Vishwakarma and Gangele (2014) and several authors have proposed some improved chain ratio, product and regression type estimators in two-phase sampling based on two auxiliary variables.
Following Vishwakarma and Kumar (2014), we suggest a class of estimators for the population mean in two-phase sampling using two auxiliary variables.It is shown that the proposed class of estimators outperforms as compared to the Vishwakarma and Kumar (2014) and several other competitors.Also, some special cases of the proposed class are considered in Table 2 (see Appendix).

Notations and some existing estimates
To obtain the bias and MSE of the estimators, we define: The variance of the usual sample mean y is given by The usual chain type ratio estimator of Y under two-phase sampling scheme using two auxiliary variables x and z , is given by where x and z are the sample means based on the first-phase sample of size n .Also y and x are the sample means based on the second phase sample of size n .
The bias and MSE of d R y , up to the first degree of approximation, are given by and The chain type product estimator of Y under two-phase sampling scheme using two auxiliary variables x and z , is given by The bias and MSE of d P y , up to the first degree of approximation, are given by and Singh and Choudhury (2012) suggested exponential chain type ratio estimator of Y under two-phase sampling scheme using two auxiliary variables x and z , is given by Re exp .
The bias and MSE of Re d y , up to the first degree of approximation, are given by and Singh and Choudhury (2012) suggested exponential chain type product estimator of Y under two-phase sampling scheme using two auxiliary variables x and z , is given by e exp .
The bias and MSE of e d P y , up to the first degree of approximation, are given by 11 , 82 and Singh and Espejo (2007) suggested the ratio-product type estimator of Y under two- phase sampling scheme using single auxiliary variable x as where  is a real constant.
Vishwakarma and Kumar (2014) suggested exponential chain ratio-product type estimator of Y under two-phase sampling scheme using two auxiliary variables x and z , is given by where  is a real constant. and

Proposed class of estimators
Following Vishwakarma and Kumar (2014), we propose a class of estimators of the population mean Y under two-phase sampling scheme using two auxiliary variables x and z , is given by where 1

 and 2
 are constants, whose values are to be determined.
The proposed estimator d M y can be written in terms of es  as Squaring both sides of Equation ( 21) and ignoring higher order terms of es  , we have Taking the expectation of both sides of the above equation, we obtain the MSE of d M y , as From Eq (23), the optimum values of and

Efficiency comparison
We have obtained the conditions under which the proposed class of estimators is more efficient than its competitor estimators.

Empirical study
We have taken eight natural populations for numerical study.
The results, based on populations 1 to 8 are given in Table 1.

Conclusion
In this paper, we have proposed a class of estimators in two-phase sampling using two auxiliary variables.The bias and mean squared error   MSE are obtained up to the first order of approximation.Theoretically, it has been shown that the proposed estimator is more efficient than the conventional estimator y , two-phase ratio estimator of both sides of the above equation, we get bias of d M y , is given by above conditions are obviously true.
chain type ratio estimator Re d y , exponential chain type product estimator d Pe y , Singh and Espejo (2007) ratio-product type estimator d RP y , Vishwakarma and Kumar (2014) exponential chain ratio-product type estimator d RPe y .The numerical results clearly show that the proposed estimator d M y performs much better than Vishwakarma and Kumar (2014) and all other competitor estimators.Also, d P y and Re d y show the poor performances in all data sets.


Which is always true.
The minimum bias and MSE of d RPe y at optimum value of  .. 1

res. Vol.XI No.3 2015 pp397-408 405
Number of paralytic polio cases in the not inoculated group.