An Improved Class of Chain Ratio-Product Type Estimators in Two-Phase Sampling Using Two Auxiliary Variables

This paper presents a technique for estimating finite populationmean of the study variable in the presence of two auxiliary variables using two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin. The properties of the proposed class of estimators are studied under large sample approximation. In addition, bias and efficiency comparisons are carried out to study the performances of the proposed class of estimators over the existing estimators. It has also been shown that the proposed technique has greater applicability in survey research. An empirical study is carried out to demonstrate the performance of the proposed estimators.


Introduction
The use of auxiliary information for estimating population mean of the study variable has greater applicability in survey research.It is utilized at the estimation stage and design stage to obtain an improved estimator compared to those not utilizing auxiliary information.The use of ratio and product strategies in survey sampling solely depends upon the knowledge of population mean  of the auxiliary variable .
The ratio estimator was developed by Cochran [1] to estimate the population mean  of the study variable  by using information on auxiliary variable , positively correlated with .The ratio estimator is most effective when the relationship between  and  is linear through the origin and the variance of  is proportional to .Robson [2] defined a product estimator that was revisited by Murthy [3].The product estimator is used when the auxiliary variable  is negatively correlated with the study variable .
When the population mean  of the auxiliary variable  is not known before the start of a survey, then a firstphase sample of size   is selected from the population of size  on which only the auxiliary variable  is measured in order to furnish a good estimate of .And then a second-phase sample of size  is selected from the first-phase sample of size   on which both the study variable  and the auxiliary variable  are measured.This procedure of selecting the samples from the given population is known as two-phase sampling (or double sampling).The concept of double sampling was first introduced by Neyman [4].Some contribution to two-phase sampling has been made by Sukhatme [5], Hidiroglou and Sarndal [6], Fuller [7], Hidiroglou [8], Singh and Vishwakarma [9], and Sahoo et al. [10].
We can use either one or two (or more than two) auxiliary variables while estimating population mean of the study variable; keeping this fact, Chand [11] introduced chain ratio estimators.This led various authors including Kiregyera [12], Singh and Upadhyaya [13], Prasad et al. [14], Singh et al. [15], Singh and Choudhury [16], and Vishwakarma and Gangele [17] to modify the chain type estimators and discuss their properties.
When the population mean  of another auxiliary variable  which has a positive correlation with X (i.e.,   > 0) is known and if   >   > 0, then it is advisable to estimate  by  =   (/  ), which would provide a better estimate of  as compared to   .
The usual chain type ratio and product estimators of  under double sampling scheme using two auxiliary variables  and  are given, respectively, by Singh and Choudhury [16] suggested the following exponential chain type ratio and product estimators of  under double sampling scheme using two auxiliary variables  and : where   and   are the sample means of  and , respectively, based on the first-phase sample of size   drawn from the population of size  with the help of Simple Random Sampling Without Replacement (SRSWOR) scheme.Also,  and  are the sample means of  and , respectively, based on the second-phase sample of size  drawn from the first-phase sample of size   with the help of SRSWOR scheme.

Proposed Estimator
It has been theoretically established that, in general, the linear regression estimator is more efficient than the ratio (product) estimator except when the regression line of  on  passes through the neighborhood of the origin, in which the efficiencies of these estimators are almost equal.However, owing to stronger intuitive appeal, survey statisticians favour the use of ratio and product estimators.Further, we note that, in many practical situations, the regression line does not pass through the neighborhood of the origin.In these situations, the ratio estimator does not perform well as the linear regression estimator.Considering this fact, Singh and Ruiz Espejo [18] made an attempt to improve the performance of these estimators and suggested the following ratio-product type estimator for population mean  under double sampling scheme using single auxiliary variable : where  is a real constant.
We propose the following exponential chain ratioproduct type estimator for population mean  under double sampling scheme using two auxiliary variables  and : where  is a real constant to be determined such that the Mean Square Error (MSE) of the proposed estimator  dc RPe is minimum.For  = 1,  dc RPe →  dc Re , whereas, for  = 0,  dc RPe →  dc Pe .
Remark.It is noted that the proposed estimator in ( 4) is a special case of the class of estimators  class = (,   ) proposed by Srivastava [19], where (⋅) is a parametric function such that (  |  1 , ) = 1 and satisfies certain regularity conditions defined in Srivastava [19].

Bias and MSE of the Proposed Estimator
To obtain the Bias and Mean Square Error (MSE) of the proposed estimator  dc RPe , we consider such that where Let   ,   , and   be the coefficients of variation of , , and , respectively.Also, let   ,   , and   be the correlation coefficients between  and ,  and , and  and , respectively.Then, we have where Now, expressing the estimator  dc RPe in terms of  0 ,  1 ,   1 , and   2 and neglecting the terms of  0 ,  1 ,   1 , and   2 involving degree greater than two, we get To the first degree of approximation, the Bias and Mean Square Error (MSE) of the proposed estimator  dc RPe are given by To the first degree of approximation, the expressions for Bias and Mean Square Error (MSE) of the estimators  dc  ,  dc  ,  dc Re ,  dc Pe , and   RP are, respectively, given by The optimum value of , which minimizes the Mean Square Error (MSE) of the estimator   RP , is given by The optimum value of , which minimizes the Mean Square Error (MSE) of the estimator  dc RPe , is given by Substituting the value of  from ( 15) in ( 13), we get the minimum MSE of   RP as Substituting the value of  from ( 16) in (11), we get the minimum MSE of  dc RPe as

Efficiency Comparisons
It is well known that the Bias and variance of the usual unbiased estimator  for population mean in SRSWOR are From ( 11), (13), and (20), we have The range of  provides enough scope for choosing many estimators that are more efficient than the above considered estimators.

Empirical Study
To examine the merits of the proposed estimator of , we have considered the following natural population datasets.

Conclusion
It is observed from From Table 2, we see that the Percentage Relative Efficiency (PRE) of the proposed estimator  dc RPe , for populations I and II, is more as compared to all other existing estimators, that is, usual unbiased estimator , chain type ratio estimator  dc  , chain type product estimator  dc  , exponential chain type ratio estimator  dc Re , exponential chain type product estimator  dc Pe , and ratio-product type estimator   RP .Finally, from Tables 1 and 2, we conclude that the proposed estimator  dc RPe (based on two auxiliary variables  and ) is a more appropriate estimator in comparison to other existing estimators as it has appreciable efficiency as well as lower relative bias.

Table 1 :
Absolute Relative Bias (ARB) of different estimators of .

Table 2 :
Percentage Relative Efficiencies (PREs) of different estimators of  with respect to .
* Data is not applicable.