An Efficient Class of Exponential Chain Ratio Type Estimator for Finite Population Mean in Double Sampling

This paper presents a class of exponential chain ratio type estimator in double sampling for estimating finite population mean of the study variable, when the information on another additional auxiliary variable is known along with the main auxiliary variable. The property of proposed class of estimator has been studied. Comparison has been made with other competitive estimators. The proposed estimator is found to be more efficient both theoretically and empirically.


Introduction
It is well established fact that the uses of auxiliary information in sample survey increases the precision of the estimate of population mean of study variable. W. G. Cochran [5] used auxiliary information and proposed the classical ratio estimator of population mean or total of the study variable y . Ratio method of estimation is quite effective when there is a high positive correlation between study and auxiliary variables. In the ratio method of estimation, it is assumed that the auxiliary information is known in advance. However, there are some situations of practical importance where the population mean of auxiliary information X is not known before start of the surveys. In such a situation, a sample of size 1 n is selected initially by using a suitable sampling design and population mean X is estimated, then a sample of size n ( ) 1 n n < is selected to estimate the population mean of the study and auxiliary variables. The second phase sample can either be a subsample of the first phase sample or it can be directly drawn from the given population. This technique is known as double or two phase sampling and was first introduced by Neyman [7].
However, if the population mean Z of another auxiliary variable z , closely related to x but compared to x remotely related to y is known (i.e. yx yz ρ ρ > ), it is preferable to estimate X by 1 1 X x Z z = , which would provide better estimate of X than 1 x to the terms of order ( ) yx yz ρ ρ , xz ρ are coefficient of variation of x , z and correlation coefficient between y and x ; y and z ; x and z respectively.
This technique is known as chaining. The chain regression estimator was first introduced by Swain [18].Chand [4], Sukhatme and Chand [16], Kiregyera [9] proposed some chain ratio type estimators based on two auxiliary variates. Srivastava and Jhajj [14], Al-Jararha and Ahmed [1], Pradhan [11], Amin and Hanif [2], Singh and Choudhury [12], Kalita et. al [8] and many authors have suggested some improved chain ratio type estimators using single (or more) auxiliary variable(s) in order to provide better efficiency. Let us consider a finite population ( ) be the population means of the study variable y and the auxiliary variable x respectively. For estimating the population mean Y of y , a simple random sample of size n is drawn without replacement from the population U .
Then the classical ratio estimator is defined by where y and x are the sample means of y and x respectively based on a sample of size n out of the population of size N units and X is the known as population mean of x . With known population mean X , Bahl & Tuteja [3] suggested the exponential ratio type estimator as ˆe xp for estimating the population mean Y .
If the population mean X of the auxiliary variable x is not known before start of the survey, a first-phase sample of size 1 n is drawn from the population, on which only the auxiliary variable x is observed. Then a second phase sample of size n is drawn, on which both study variable y and auxiliary variable x are observed. Let sampling Sukhatme [17] for estimating population mean Y is given by Singh &Vishwakarma [13] suggested the exponential ratio type estimators for Y in double sampling as Chand [4] developed a chain ratio estimator for the population mean in double sampling as, Singh & Choudhury [12] suggested the exponential chain ratio type estimators for Y in double sampling as In this paper, under SRSWOR scheme, a class of exponential chain ratio type estimators has been proposed in double sampling. The bias and the mean square error (MSE) of proposed class of estimators have been obtained to the first order approximation. Numerical illustrations are given to show the performance of the proposed estimator over other estimators.

The Proposed Class of Estimators
The exponential chain ratio type estimators in double sampling as follows where, ( ) 0 a ≠ and b are scalar constants.

Remark:
, the estimator 1 t of equation (1) reduces to the 'exponential chain ratio-type estimator' Singh & Choudhury [12] in double sampling as The bias and MSE of the proposed class of estimators are obtained for the following two cases.
Case I: When the second phase sample is a subsample of the first phase sample.
Case II: When the second phase sample is drawn independently of the first phase sample.

Bias and MSE of Case I
In case I, let us write where 1 1 , Expanding the right hand side ofequation (1) in terms of e's, multiplying out and neglecting the terms of e's of power greater than two, ( ) e e e e e e e e e e e λ λ λ e e e e e e λ λ λ Taking expectations in equation (3) and using the results of the equation (2), the Bias of the estimator 1 t to the first order approximation as From equation (3), Squaring both the sides of equation (4), taking expectations of its terms and using the results of equation (2) Optimization of equation (5) with respect to λ yields its optimum value as where, Substituting the value of λ in equation (5), the optimum MSE of 1 t as ( ) Efficiency Comparisons of 1 t (i) with sample mean per unit estimator y The MSE of sample mean per unit estimator is ( ) From equations (7) and (8), it is found that the proposed estimator is more efficient than y , since (ii) with chain ratio estimator in double sampling The MSE of chain ratio estimator in double sampling is From equations (9) and (7), Therefore, the proposed estimator is better than chain ratio type estimator if 1 2 0 (iii) with exponential chain ratio estimator in double sampling The MSE of exponential chain ratio estimator in double sampling is ( ) From equations (10) and (7), Therefore, the proposed estimator is more efficient than chain ratio estimator in double sampling.

Bias and MSE of Case II
Taking expectations in equation (3) and using the results of the equation (11), the bias of the estimators 1 t to the first order approximation as Squaring both the sides of equation (4), taking expectations of its terms and using the results of equation (11), the MSE of 1 t as ( ) where, ** 1 Optimization of equation (12) with respect to λ yields its optimum value as Finite Population Mean in Double Sampling 1 .
Efficiency Comparisons of 1 t (i) With sample mean per unit estimator y From equations (8) and (14), Therefore, the proposed estimator is better than sample mean per unit estimator if1 2 0 (ii) With chain ratio estimator in double sampling The MSE of chain ratio estimator in double sampling is From equations (15) and (14), we have Therefore, the proposed estimator is better than the chain ratio estimator if 1 2 0 and 1 2 0 (iii) With exponential chain ratio estimator in double sampling The MSE of exponential chain ratio estimator in double sampling is 2 2 ** 2 2 2 2 1 1 Re 1 1 From equations (14) and (16) Therefore, the proposed estimator is more efficient than exponential chain ratio estimator in double sampling if 0 and 0 yx xz

Empirical Study
To examine the merits of the proposed estimator, we have considered five natural population data sets. The sources of populations, nature of the variates y , x and z ;and the values of the various parameters are given as follows.

Conclusions
From Table 1 and Table 2, it is seen that the percentage relative efficiencies of proposed class of estimator is higher than the other estimators in all the population data sets considered in this paper. So it is evident that the proposed class of exponential chain ratio type estimators 1 t is more efficient than the sample mean per unit estimator y , chain ratio type estimator ( ) dc R Y proposed by Chand [4] and exponential chain ratio type estimator ( ) dc Re Y proposed by Singh & Choudhury [12] for both the cases under double sampling. Thus, the uses of the proposed estimators are preferable over other competitive estimators.