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

Authors: Ramkrishna S. Solanki; Housila P. Singh; Anjana Rathour
Journal: ISRN Probability and Statistics
 (2012)

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## Abstract

This paper suggests a class of estimators for estimating the finite population meanY¯ 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.


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## Body

## 1. 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], Srivastava [6–8], Reddy [9, 10], Gupta [11], Sahai [12], Sahai and Ray [13], Ray and Sahai [14], Srivenkataramana and Tracy [15, 16], Bandyopadhyay [17], Vos [18], Sisodia and Dwivedi [19], Adhvaryu and Gupta [20], Chaubey et al. [21], Mohanty and Sahoo [22], Singh and Shukla [23], Naik and Gupta [24], Bahl and Tuteja [25], Mohanty and Sahoo  [26], Upadhyaya and Singh [27], Singh and Tailor [28], Kadilar and Cingi [29], Singh et al. [30], and others.In this paper, we have envisaged a new class of estimators for population mean of study variabley 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 populationU=(u1,u2,…,uN) of size N from which a sample of size n is drawn according to simple random sampling without replacement (SRSWOR). Let y̅ and x̅ be the sample mean estimators of the population means Y̅ and X,̅ respectively, of the study variable y and the auxiliary variable x. Let Cy(=Sy/Y̅)), Cx(=Sx/X̅), ρ(=Syx/SxSy) (correlation coefficient between the variables y and x)  and  k=(ρCy/Cx), where(1.1)Sy2=(N-1)-1∑i=1N(yi-Y̅)2,Sx2=(N-1)-1∑i=1N(xi-X̅)2,Sxy=(N-1)-1∑i=1N(xi-X̅)(yi-Y̅).The remaining part of the paper is organized as follows. Section2 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.


## 2. Reviewing Estimators

It is very well known that the sample meany̅   is an unbiased estimator of population mean Y̅, and under (SRSWOR), its variance is given by(2.1)Var⁡(y̅)=(1-f)nSy2=(1-f)nY̅2Cy2,
where f=(n/N).The usual ratio and product estimators of population meanY̅ of the study variable y are, respectively, defined as(2.2)y̅R=y̅X̅x̅,y̅P=y̅x̅X̅.Bahl and Tuteja [25] suggested exponential ratio-type and product-type estimators for population mean Y̅, respectively, as(2.3)y̅Re=y̅exp⁡(X̅-x̅X̅+x̅),y̅Pe=y̅exp⁡(x̅-X̅x̅+X̅).Kadilar and Cingi [31] suggested a chain ratio-type estimator for population mean Y̅ as(2.4)y̅CR=y̅(X̅2x̅2).Following the procedure adopted by Kadilar and Cingi [31], one may define a chain product-type estimator for population mean Y̅ as(2.5)y̅CP=y̅(x̅2X̅2).
To the first degree of approximation, the biases and variances of estimators y̅R, y̅p, y̅Re, y̅Pe, y̅CR,  and y̅CP are, respectively, given by(2.6)B(y̅R)=(1-f)nY̅Cx2(1-k),(2.7)B(y̅P)=(1-f)nY̅Cx2k,(2.8)B(y̅Re)=(1-f)nCx2(Y̅8)(3-4k),(2.9)B(y̅Pe)=(1-f)nCx2(Y̅8)(4k-1),(2.10)B(y̅CR)=(1-f)nY̅Cx2(3-2k),(2.11)B(y̅CP)=(1-f)nY̅Cx2(1+2k),(2.12)Var⁡(y̅R)=(1-f)nY̅2[Cy2+Cx2(1-2k)],(2.13)Var⁡(y̅P)=(1-f)nY̅2[Cy2+Cx2(1+2k)],(2.14)Var⁡(y̅Re)=(1-f)nY̅2[Cy2+Cx24(1-4k)],(2.15)Var⁡(y̅Pe)=(1-f)nY̅2[Cy2+Cx24(1+4k)],(2.16)Var⁡(y̅CR)=(1-f)nY̅2[Cy2+4Cx2(1-k)],(2.17)Var⁡(y̅CP)=(1-f)nY̅2[Cy2+4Cx2(1+k)].
From (2.1) and (2.12)–(2.17), we have made some efficiency comparisons between the estimators y̅, y̅R, y̅p, y̅Re, y̅Pe, y̅CR, and  y̅CP, as shown in Table 1.Table 1
Efficiency comparisons between different estimators.
EstimatorMore efficient than estimatorIfy̅y̅Rk≤1/2y̅y̅Pk≥-1/2y̅Ry̅k>1/2y̅Py̅k<-1/2y̅y̅Rek≤1/4y̅y̅Pek≥-1/4y̅Rey̅k>1/4y̅Pey̅k<-1/4y̅y̅CRk≤1y̅y̅CPk≥-1y̅CRy̅k>1y̅CPy̅k<-1y̅CRy̅Rk>3/2y̅CPy̅Pk<-3/2


## 3. Suggested Class of Estimators

We define the following class of estimators for the population meanY̅ as(3.1)t(α,δ)=y̅[2-(x̅X̅)αexp⁡{δ(x̅-X̅)(x̅+X̅)}],
where (α,δ) are suitable chosen scalars.It is to be mentioned that forδ=0, the class of estimators t(α,δ) reduces to the following class of estimators(3.2)t(α,0)=y̅[2-(x̅X̅)α],
which is due to Sahai and Ray [13].While forα=0, it reduces to the new transformed class of estimators defined as(3.3)t(0,δ)=y̅[2-exp⁡{δ(x̅-X̅)(x̅+X̅)}].To obtain the bias and variance of suggested class of estimatorst(α,δ), we write(3.4)y̅=Y̅(1+e0),x̅=X̅(1+e1),such that(3.5)E(e0)=E(e1)=0,E(e02)=(1-f)nCy2,E(e12)=(1-f)nCx2,E(e0e1)=(1-f)nρCyCx=(1-f)nkCx2.
Expressing (3.1) in terms of e’s, we have(3.6)t(α,δ)=Y̅(1+e0)[2-(1+e1)αexp⁡{δe12+e1}]=Y̅(1+e0)[2-(1+e1)αexp⁡{δe12(1+e12)-1}].
We assume that |e1|<1, so that (1+e1)α, exp{(δe1/2)(1+e1/2)-1}, and (1+e1/2)-1 are expandable. Now expanding the right-hand side of (3.6), we have(3.7)t(α,δ)=Y̅(1+e0)[2-{1+αe1+α(α-1)2e12+⋯}×{1+δe12(1+e12)-1+δ2e128(1+e12)-2+⋯}]=Y̅(1+e0)[2-{1+αe1+α(α-1)2e12+⋯}×{1+δe12(1-e12+e124-⋯+δ2e128(1-e1)+⋯)}]=Y̅(1+e0)[2-{1+(2α+δ)2e1+(2α+δ)(2δ+δ-2)8e12+⋯}]=Y̅[{1+e0-(2α+δ)2(e1+e0e1)-(2α+δ)(2δ+δ-2)8(e12+e0e12)-⋯}].
Neglecting the terms of e’s having greater than two in (3.7), we have(3.8)t(α,δ)≅Y̅[1+e0-(2α+δ)2{e1+e0e1-(2δ+δ-2)4e12}]or(3.9)(t(α,δ)-Y̅)≅Y̅[e0-(2α+δ)2{e1+e0e1-(2δ+δ-2)4e12}].
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)B(t(α,δ))=(1-f)nY̅Cx2[-(2α+δ)2{(2δ+δ-2)4+k}].
Squaring both sides of (3.9) and neglecting terms e’s having power greater than two, we have(3.11)(t(α,δ)-Y̅)2≅Y̅[e02+(2α+δ)2{(2δ+δ)4e12-2e0e1}].
Taking expectation of both sides of (3.11), we get the variance of t(α,δ) to the first degree of approximation as(3.12)Var⁡(t(α,δ))=(1-f)nY̅2[Cy2+(2α+δ)4Cx2{(2α+δ)-4k}],
which is minimum when(3.13)(2α+δ)=2k.
Thus, the resulting minimum variance of t(α,δ) is obtained as(3.14)Varmin⁡(t(α,δ))=(1-f)nSy2(1-ρ2).
The minimum variance of t(α,δ) equals to the approximate variance of the usual linear regression estimator defined as(3.15)y̅lr=y̅+β̂(X̅-x̅),
where β̂=(sxy/sx2), sxy=(n-1)-1∑i=1n(xi-x̅)(yi-y̅) and sx2=(n-1)-1∑i=1n(xi-x̅)2.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)Var⁡(t(α,0))=(1-f)nY̅2[Cy2+αCx2(α-2k)],Var⁡(t(0,δ))=(1-f)nY̅2[Cy2+(δ4)Cx2(δ-4k)].
If we assume the value of α is specified by αo (say), then the variances of t(α,0) and t(α,δ) are, respectively, given by(3.17)Var⁡(t(α=αo,0))=(1-f)nY̅2[Cy2+αoCx2(αo-2k)],Var⁡(t(α=αo,δ))=(1-f)nY̅2[Cy2+(2αo+δ)4Cx2{(2αo+δ)-4k}].
If the value of δ is specified δo, then the variances of the estimators t(0,δ) and t(α,δ) are, respectively, given by(3.18)Var⁡(t(0,δ=δo))=(1-f)nY̅2[Cy2+(δo4)Cx2(δo-4k)],Var⁡(t(α,δ=δo))=(1-f)nY̅2[Cy2+(2αo+δ)4Cx2{(2α+δo)-4k}].
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 Y̅ as(3.19)t(1,1)=y̅[2-(x̅X̅)exp⁡(x̅-X̅x̅+X̅)].
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(3.20)B(t(1,1))=-3(1-f)8nY̅Cx2(1+4K),(3.21)Var⁡(t(1,1))=(1-f)nY̅2[Cy2+3Cx24(3-4k)].
Expression (3.20) clearly indicates that the proposed estimator t(1,1) is better than conventional unbiased estimator y̅  if  k>(3/4), a condition which is usually met in survey situations.


## 4. Efficiency Comparisons

From (3.17), we have
(4.1)Var⁡(t(α=αo,δ))-Var⁡(t(α=αo,0))=(1-f)nY̅2Cx2[(2αo+δ)4{(2αo+δ)-4k}-αo(αo-2k)]<0,if[(2αo+δ)2-4(2αo+δ)k-4αo2+8αok]<0,i.e.,ifδ(δ+4αo-4k)<0,i.e.,if{either0<δ<4(k-αo)or4(k-αo)<δ<0,
or equivalently(4.2)min⁡⋅{0,4(k-αo)}<δ<max⁡⋅{0,4(k-αo)}.
From (3.18), we have(4.3)Var⁡(t(α,δ=δo))-Var⁡(t(0,δo))=(1-f)nY̅2Cx2[(2α+δo)24-(2α+δo)k-δo24+δok}]<0,if[α2+αδo-2αk]<0,i.e.,if{either0<α<(2k-δo)or(2k-δo)<α<0,
or equivalently(4.4)min⁡⋅{0,(2k-δo)}<α<max⁡⋅{0,(2k-δo)}.From (2.1), (2.12)–(2.17), and (3.12), we have(i)
(4.5)Var⁡(t(α,δ))-Var⁡(y̅)=(1-f)4nY̅2Cx2[(2α+δ)2-k(2α+δ)]<0,if{either0<(2α+δ)<kork<(2α+δ)<0,
or equivalently(4.6)min⁡⋅(0,k)<(2α+δ)<max⁡⋅(0,k).(ii)
(4.7)Var⁡(t(α,δ))-Var⁡(y̅R)=(1-f)nY̅2Cx2[(2α+δ)4{(2α+δ)-4k}-(1+2k)]<0,if{either2<(2α+δ)<2(2k-1)or2(2k-1)<(2α+δ)<2,
or equivalently(4.8)min⁡⋅{2,2(2k-1)}<(2α+δ)<max⁡⋅{2,2(2k-1)}.(iii)
(4.9)Var⁡(t(α,δ))-Var⁡(y̅p)=(1-f)nY̅2Cx2[(2α+δ)4{(2α+δ)-4k}-(1+2k)]<0,if{either-2<θ<2(2k+1)or2(2k+1)<θ<-2,
or equivalently(4.10)min⁡⋅{-2,2(2k+1)}<(2α+δ)<max⁡⋅{-2,2(2k+1)}.(iv)
(4.11)Var⁡(t(α,δ))-Var⁡(y̅Re)=(1-f)4nY̅2Cx2[(2α+δ){(2α+δ)-4k}-(1-4k)]<0,if{either1<(2α+δ)<(4k-1)or(4k-1)<(2α+δ)<1,
or equivalently(4.12)min⁡⋅{1,(4k-1)}<(2α+δ)<max⁡⋅{1,(4k-1)}.(v)
(4.13)Var⁡(t(α,δ))-Var⁡(y̅Pe)=(1-f)4nY̅2Cx2[(2α+δ){(2α+δ)-4k}-(1+4k)]<0,if{either-1<(2α+δ)<(4k+1)or(4k+1)<(2α+δ)<-1,
or equivalently(4.14)min⁡⋅{-1,(4k+1)}<(2α+δ)<max⁡⋅{-1,(4k+1)}.(vi)
(4.15)Var⁡(t(α,δ))-Var⁡(y̅CR)=(1-f)nY̅2Cx2[(2α+δ)4{(2α+δ)-4k}-4(1-k)]<0,if{either4<(2α+δ)<4(k-1)or4(k-1)<(2α+δ)<4,
or equivalently(4.16)min⁡⋅{4,4(k-1)}<(2α+δ)<max⁡⋅{4,4(k-1)}.(vii)
(4.17)Var⁡(t(α,δ))-Var⁡(y̅CP)=(1-f)nY̅2Cx2[(2α+δ)4{(2α+δ)-4k}-4(1+k)]<0,if{either-4<(2α+δ)<4(k+1)or4(k+1)<(2α+δ)<-4,
or equivalently(4.18)min⁡⋅{-4,4(k+1)}<(2α+δ)<max⁡⋅{-4,4(k+1)}.
From (2.1), (2.12), (2.14), (2.16), and (3.21), we have made some efficiency comparisons between the estimators t(1,1), y̅, y̅R, y̅Re, and  y̅CR, as shown in Table 2.Table 2
Efficiency comparisons between different estimators.
EstimatorMore efficient than estimatorIft(1,1)y̅k>3/4t(1,1)y̅Rk>5/4t(1,1)y̅Rek>1t(1,1)y̅CRk<7/3It is clearly indicated from Table2 that the estimator t(1,1) is more efficient than estimators y̅, y̅R, y̅Re, and y̅CR if(4.19)54<k<73.


## 5. Empirical Study

To judge the merits of the suggested estimatort(1,1) over usual unbiased estimator y̅, usual ratio estimator y̅R, Bahl and Tuteja [25] exponential ratio-type estimator y̅Re and Kadilar and Cingi [31] chain ratio-type estimator  y̅CR we have considered three populations. Descriptions of the populations are given below.Population I (source: Srivastava [7], page 406)
y: fiber per plant in jute fiber crops,x: height,Cy2=0.0568, Cx2=0.00846, ρ=0.7418,  k=1.9221.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),Cy2=17.4724, Cx2=4.0804, ρ=0.82, k=1.6968.Population III (Source: Kadilar and Cingi [33])
y: the level of apple production amount,x: the number of apple trees,Cy2=34.1056, Cx2=14.8225, ρ=0.92, k=1.3955.
We have computed the percent relative efficiencies (PREs) of different estimators(·)y̅, y̅R, y̅Re,  y̅CR, and t(1,1) of the population mean Y̅ with respect to y̅ by using the following formula:(5.1)PRE(⋅,y̅)=Var⁡(y̅)Var⁡(⋅)×100,
and results are summarized in Table 3.
Table3 exhibits that the proposed estimator t(1,1) is more efficient than usual unbiased estimator y̅, usual ratio estimator y̅R, Bahl and Tuteja [25] estimator y̅Re, and Kadilar and Cingi [31] estimator y̅CR 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 y̅CR. 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 3
PREs of different estimators of population meany̅.
EstimatorPRE(·,y̅) for populationIIIIIIy̅ (sample mean)100.00100.00100.00y̅R (ratio estimator)173.46226.77451.25y̅Re (bahl and Tuteja [25])133.16151.03199.15y̅CR (kadilar and Cingi [31])221.83286.48320.12t(1,1) (proposed estimator)224.64297.05631.59



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