A New Difference-Cum-Exponential Type Estimator of Finite Population Mean in Simple Random Sampling

Auxiliary information is frequently used to improve the accuracy of the estimators when estimating the unknown population parameters. In this paper, we propose a new difference-cum-exponential type estimator for the finite population mean using auxiliary information in simple random sampling. The expressions for the bias and mean squared error of the proposed estimator are obtained under first order of approximation. It is shown theoretically, that the proposed estimator is always more efficient than the sample mean, ratio, product, regression and several other existing estimators considered here. An empirical study using 10 data sets is also conducted to validate the theoretical findings.


Introduction
In sample surveys, auxiliary information can be used either at the design stage or at the estimation stage or at both stages to increase precision of the estimators of population parameters.The ratio, product and regression methods of estimation are commonly used in this context.Recently many research articles have appeared where authors have tried to modify existing estimators or construct new hybrid type estimators.Some contribution in this area are due to Bahl & Tuteja (1991), Singh, Chauhan & Sawan (2008), Singh, Chauhan, Sawan & Smarandache (2009), Yadav & Kadilar (2013), Haq & Shabbir (2013), Singh, Sharma & Tailor (2014) and Grover & Kaur, (2011, 2014).
Consider a finite population U = {U 1 , U 2 , . . ., U N }.We draw a sample of size n from this population using simple random sampling without replacement scheme.Let y and x respectively be the study and the auxiliary variables and y i and x i , respectively be the observations on the ith unit.Let ȳ = 1 X respectively be the coefficients of variation for y and x.
In order to obtain the bias and mean squared error (MSE) for the proposed estimator and existing estimators considered here, we define the following relative error terms: Let x and E(δ 0 δ 1 ) = λρ yx C y C x , where λ = 1 n − 1 N .In this paper, our objective is to propose an improved estimator of the finite population mean using information on a single auxiliary variable in simple random sampling.Expressions for the bias and mean squared error (MSE) of the proposed estimator are derived under first order of approximation.Based on both theoretical and numerical comparisons, we show that the proposed estimator outperforms several existing estimators.The outline of the paper is as follows: in Section 2, we consider several estimators of the finite population mean that are available in literature.The proposed estimators are given in Section 3 along with the corresponding bias and MSE expressions.In Section 4, we provide theoretical comparisons to evaluate the performances of the proposed and existing estimators.An empirical study is conducted in Section 5, and some concluding remarks are given in Section 6.

Some Existing Estimators
In this section, we consider several estimators of finite population mean.

Sample Mean Estimator
The variance of the sample mean ȳ, the usual unbiased estimator, is given by

Traditional Ratio and Product Estimators
Using information on the auxiliary variable, Cochran (1940) suggested a ratio estimator ŶR for estimating Ȳ .It is given by The MSE of ŶR , to first order of approximation, is given by On similar lines, Murthy (1964) suggested a product estimator ( ŶP ), given by The MSE of ŶP , to first order of approximation, is given by The ratio and product estimators are widely used when the correlation coefficient between the study and the auxiliary variable is positive and negative, respectively.Both of the estimators, ŶR and ŶP , show better performances in comparison with ȳ when ρ yx > Cx 2Cy and ρ yx < − Cx 2Cy , respectively.

Regression Estimator
The usual regression estimator ŶReg of Ȳ , is given by where b is the usual slope estimator of the population regression coefficient β (Cochran 1977).The estimator ŶReg is biased, but the bias approaches zero as the sample size n increases.
Asymptotic variance of ŶReg , is given by The regression estimator ŶReg performs better than the usual mean estimator ȳ, ratio estimator ŶR and product estimator 2.4.Bahl & Tuteja (1991) Estimators Bahl & Tuteja (1991) suggested ratio-and product type estimators of Ȳ , given respectively by and The MSEs of ŶBT,R and ŶBT,P , to first order of approximation, are given by and
The minimum MSE of ȲS,RP , up to first order of approximation, at optimum value of α, i.e., α (opt) = 1 2 + ρyxCy Cx , is given by The minimum MSE of ŶS,RP is exactly equal to variance of the linear regression estimator ( ŶReg ).
The minimum MSE of ŶGK , up to first order of approximation, at optimum values of d 1 and Grover & Kaur (2011) derived the result Equation ( 18) shows that ŶGK is more efficient than the linear regression estimator ŶReg .
Since regression estimator ŶReg is always better than ȳ, ŶR , ŶP , ŶBT,R , ŶBT,P , it can be argued that ŶGK is also always better than these estimators.

Proposed Estimator
In this section, an improved difference-cum-exponential type estimator of the finite population mean Ȳ using a single auxiliary variable is proposed.Expressions for the bias and MSE of the proposed estimator are obtained upto first order of approximation.
The conventional difference estimator ( ŶD ) of Ȳ , is given by where w 1 is a constant.
From ( 8), ( 12), and ( 14), a difference-cum-exponential type estimator ( Ŷ * D ) of Ȳ may be given by where is the average of exponential ratio and exponential product estimators ŶBT,R and ŶBT,P respectively.
By combining the ideas in ( 20) and ( 21), a modified difference-cum-exponential type estimator of Ȳ , is given by where w 1 and w 2 are unknown constants to be determined later.
Rewriting Ŷ * P as Solving Ŷ * P in terms of δ i (i = 0, 1), to first order of approximation, we can write Taking expectation on both sides of (23), we get the bias of Ŷ * P , given by Revista Colombiana de Estadística 37 (2014) 197-209 Squaring both sides of (23) and using first order of approximation, we get Taking expectation on both sides of ( 25), the MSE of Ŷ * P , to first order of approximation, is given by Partially differentiating ( 26) with respect to w 1 and w 2 , we get Setting ∂M SE( Ŷ * P ) ∂w2 =0 for i = 0, 1, the optimum values of w 1 and w 2 are given by , respectively.
Substituting the optimum values of w 1 and w 2 in (26), we can obtain the minimum MSE of Ŷ * P , as given by After some simplifications, ( 27) can be written as where Note that both quantities, T 1 and T 2 , are always positive.

Empirical Study
In this section, we consider 10 real data sets to numerically evaluate the performances of the proposed and the existing estimators considered here.
Population 1: [Source: Cochran (1977), pp. 196] Let y be the peach production in bushels in an orchard and x be the number of peach trees in the orchard Revista Colombiana de Estadística 37 (2014) 197-209 In Table 1, the MSE values and percent relative efficiencies (PREs) of all the estimators considered here are reported based on Populations 1-10.
2. The product estimator ( ŶP ) performs better than ȳ in Population 5 because the condition ρ yx < − Cx 2Cy is satisfied.
5. It is also observed that, regardless of positive or negative correlation between the study and the auxiliary variable, the estimators, ŶReg , ŶR,Reg , ŶGK and Ŷ * P , always perform better than the unbiased sample mean, ratio and product estimators considered here in all populations.Among all competitive estimators, the proposed estimator ( Ŷ * P ) is preferable.

Conclusion
In this paper, we have suggested an improved difference-cum-exponential type estimator of the finite population mean in simple random sampling using information on a single auxiliary variable.Expressions for the bias and MSE of the proposed estimator are obtained under first order of approximation.Based on both the theoretical and numerical comparisons, we showed that the proposed estimator always performs better than the sample mean estimator, traditional ratio and product estimators, linear regression estimator, Bahl & Tuteja (1991) estimators, Rao (1991) estimator, andGrover &Kaur (2011) estimator.Hence, we recommend the use of the proposed estimator for a more efficient estimation of the finite population mean in simple random sampling.
, be the corresponding population means.We assume that the mean of the auxiliary variable ( X) is known.Let s 2 i − X) 2 , be the corresponding population variances.Let ρ yx be the correlation coefficient between y and x.Finally let C y = Sy Ȳ and C x = Sx

Table 1 :
MSE values and PREs of different estimators with respect to ȳ.