An Alternative Estimator for Estimating the Finite Population Mean Using Auxiliary Information in Sample Surveys

This paper suggests a class of estimators for estimating the finite population mean Y of the study variable y using known population mean X of the auxiliary variable x. Asymptotic expressions of bias and variance of the suggested class of estimators have been obtained. Asymptotic optimum estimator AOE in the class is identified along with its variance formula. It has been shown that the proposed class of estimators is more efficient than usual unbiased, usual ratio, usual product, Bahl and Tuteja 1991 , and Kadilar and Cingi 2003 estimators under some realistic conditions. An empirical study is carried out to judge the merits of suggested estimator over other competitors practically.


Introduction
The literature on survey sampling describes a great variety of techniques for using auxiliary information to obtain more efficient estimators.Ratio, product, and regression methods of estimation are good examples in this context see Singh 1 .If the correlation between the study variable y and the auxiliary variable x is positive high , the ratio method of estimation is quite effective.On the other hand, if this correlation is negative high the product method of estimation envisaged by Robson 2 and rediscovered by Murthy 3 can be employed quite effectively.The classical ratio and product estimators are considered to be the most practicable in many situations, but they have the limitation of having at most the same efficiency as that of the linear regression estimator.In the situation where the relation between the study variable y and the auxiliary variable x is a straight line and passing through the origin, the ratio and product estimators have efficiencies equal to the usual regression estimator.But in many practical situations, the line does not pass through the origin, and in such circumstances the ratio and product estimators do not perform equally well to the regression estimator.Keeping this fact in view, a large number of authors have paid their attention toward the formulation of modified ratio and product estimators, for instance, see Singh 4,5  In this paper, we have envisaged a new class of estimators for population mean of study variable y using information on an auxiliary variable x which is highly correlated with the study variable and have shown that the suggested class of estimators is more efficient than some existing estimators.
Consider a finite population U u 1 , u 2 , . . ., u N of size N from which a sample of size n is drawn according to simple random sampling without replacement SRSWOR .Let y and x be the sample mean estimators of the population means Y and X, respectively, of the study variable y and the auxiliary variable x.Let C y S y /Y , C x S x /X , ρ S yx /S x S y correlation coefficient between the variables y and x and k ρC y /C x , where 1.1 The remaining part of the paper is organized as follows.Section 2 gives the brief review of some estimators for the population mean of study variable y with its properties.In Section 3, a new class of estimators for the population mean is described, and the expressions for the asymptotic bias and variance are obtained.Asymptotic optimum estimator AOE in the suggested class is obtained with its variance formula.Section 4 addresses the problem of efficiency comparisons, while in Section 5 an empirical study is carried out to evaluate the performance of different estimators.

Reviewing Estimators
It is very well known that the sample mean y is an unbiased estimator of population mean Y , and under SRSWOR , its variance is given by where f n/N .
The usual ratio and product estimators of population mean Y of the study variable y are, respectively, defined as y R y X x , y P y x X .

2.2
Bahl and Tuteja 25 suggested exponential ratio-type and product-type estimators for population mean Y , respectively, as

2.3
Kadilar and Cingi 31 suggested a chain ratio-type estimator for population mean Y as Following the procedure adopted by Kadilar and Cingi 31 , one may define a chain product-type estimator for population mean Y as

2.5
To the first degree of approximation, the biases and variances of estimators y R , y p , y Re , y Pe , y CR , and y CP are, respectively, given by Var From 2.1 and 2.12 -2.17 , we have made some efficiency comparisons between the estimators y, y R , y p , y Re , y Pe , y CR , and y CP , as shown in Table 1.

Suggested Class of Estimators
We define the following class of estimators for the population mean Y as where α, δ are suitable chosen scalars.It is to be mentioned that for δ 0, the class of estimators t α,δ reduces to the following class of estimators which is due to Sahai and Ray 13 .
While for α 0, it reduces to the new transformed class of estimators defined as

3.3
To obtain the bias and variance of suggested class of estimators t α,δ , we write 3.4 such that 3.5

ISRN Probability and Statistics
Expressing 3.1 in terms of e's, we have

3.7
Neglecting the terms of e's having greater than two in 3.7 , we have

3.9
Taking expectation of both sides of 3.9 , we get the bias of the class of estimators t α,δ to the first degree of approximation as

3.10
Squaring both sides of 3.9 and neglecting terms e's having power greater than two, we have

3.11
Taking expectation of both sides of 3.11 , we get the variance of t α,δ to the first degree of approximation as which is minimum when 2α δ 2k.

3.13
Thus, the resulting minimum variance of t α,δ is obtained as

3.14
The minimum variance of t α,δ equals to the approximate variance of the usual linear regression estimator defined as It is interesting to note that if we set δ 0 and α 0 in 3.12 , we get the variances of the classes of estimators t α,0 and t 0,δ , respectively, as

3.16
If we assume the value of α is specified by α o say , then the variances of t α,0 and t α,δ are, respectively, given by

3.17
If the value of δ is specified δ o , then the variances of the estimators t 0,δ and t α,δ are, respectively, given by

3.18
To illustrate our general results, we consider a particular case of the proposed class of estimators t α,δ with its properties.If we set α, δ 1, 1 in 3.1 , we get an estimator of population mean Y as Putting α, δ 1, 1 in 3.10 and 3.12 , we get the bias and variance of t 1,1 , to the first degree of approximation, respectively, as Expression 3.20 clearly indicates that the proposed estimator t 1,1 is better than conventional unbiased estimator y if k > 3/4 , a condition which is usually met in survey situations.

Efficiency Comparisons
From 3.17 , we have From 3.18 , we have From 2.1 , 2.12 -2.17 , and 3.12 , we have

Empirical Study
To judge the merits of the suggested estimator t 1,1 over usual unbiased estimator y, usual ratio estimator y R , Bahl and Tuteja 25 exponential ratio-type estimator y Re and Kadilar and Cingi 31 chain ratio-type estimator y CR we have considered three populations.Descriptions of the populations are given below.Population II (source: Gupta and Shabbir [32]) y : the level of apple production amount 1 unit 100 tones , x : the number of apple trees 1 unit 100 trees , Population III (Source: Kadilar and Cingi [33]) y : the level of apple production amount, x : the number of apple trees, and results are summarized in Table 3. Table 3 exhibits that the proposed estimator t 1,1 is more efficient than usual unbiased estimator y, usual ratio estimator y R , Bahl and Tuteja 25 estimator y Re , and Kadilar and Cingi 31 estimator y CR in the sense of having the largest PRE in all three populations.In populations II and III, the proposed estimator t 1,1 has the largest gain in efficiency over all the estimators, while in population I, there is marginal gain in efficiency as compared to y CR .It is further noted that the condition 5/4 < k < 7/4 has been satisfied in population I 1.25 < 1.9221 < 2.33 , II 1.25 < 1.6968 < 2.33 , and III 1.25 < 1.3955 < 2.33 .Thus, we recommend the proposed estimator t 1,1 for its use in practice wherever the condition 5/4 < k < 7/4 is satisfied.

Table 1 :
Efficiency comparisons between different estimators.
Re , and y CR , as shown in Table 2.It is clearly indicated from Table 2 that the estimator t 1,1 is more efficient than estimators y, y R , y Re , and y CR if

Table 2 :
Efficiency comparisons between different estimators.

Table 3 :
PREs of different estimators of population mean y.