# Measuring Performance of Ratio-Exponential-Log Type General Class of Estimators Using Two Auxiliary Variables

Authors: Javid Shabbir; Shakeel Ahmed; Aamir Sanaullah; Ronald Onyango
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a ratio-exponential-log type general class of estimators is proposed in estimating the finite population mean using two auxiliary variables when population parameters of the auxiliary variables are known. From the proposed estimator, some special estimators are identified as members of the proposed general class of estimators. The mean square error (MSE) expressions are obtained up to the first order of approximation. This study finds that the proposed general class of estimators outperforms as compared to the conventional mean estimator, usual ratio estimators, exponential-ratio estimators, log-ratio type estimators, and many other competitor regression type estimators. Four real-life applications are used for efficiency comparison.


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## Body

## 1. Introduction

In survey sampling, ratio, product, exponential-ratio, log-ratio, and regression type estimators are modified or constructed by many researchers to enhance the precision of the estimators under different sampling designs by using the auxiliary variables. These estimators are commonly used by taking the advantage of correlation coefficient between the study variable and the auxiliary variable(s). Some notable work by the authors includes Olkin [1]; Mohanty [2]; Abu-Dayyeh et al. [3]; Koyuncu and Kadilar [4]; Swain [5]; Lu and Yan [6]; Lu et al. [7]; Sanaullah et al. [8]; Lu [9]; Muneer et al. [10]; Shabbir and Gupta [11]; Akingbade and Okafor [12]; Shabbir et al. [13]; Shabbir et al. [14]; Bhushan et al. [15]; Lone et al. [16]; and Kumari and Thaur [17].Consider a finite populationΓ=Γ1,Γ2,…,ΓN of Nunits. A sample of size n units is drawn from a population by using simple random sampling without replacement (SRSWOR). Let yi and xi,zi be the characteristics of the study variable Y and the auxiliary variables X,Z, respectively. Let y¯=n−1∑i=1nyi, and x¯=n−1∑i=1nxi,z¯=n−1∑i=1nzi, respectively, be the sample means corresponding to the population means Y¯=N−1∑i=1Nyi, and X¯=N−1∑i=1Nxi,Z¯=N−1∑i=1Nzi. To obtain the bias and MSE expressions, we define the following error terms: Ξ0=y¯/Y¯−1, and Ξ1=x¯/X¯−1, Ξ2=z¯/Z¯−1, such that EΞi=0,i=0,1,2, EΞ02=ΘCy2, EΞ12=ΘCx2, EΞ22=ΘCz2, EΞ0Ξ1=ΘCyx, EΞ0Ξ2=ΘCyz, and EΞ1Ξ2=ΘCxz where Θ=n−1−N−1, Cy=Y¯−1Sy, Cx=X¯−1Sx, Cz=Z¯−1Sz,Cyx=ρyxCyCx, Cyz=ρyzCyCz, Cxz=ρxzCxCz, ρyx=SySx−1Syx, ρyz=SySz−1Syz, ρxz=SxSz−1SxzSy=N−1−1∑i=1Nyi−Y¯2, Sx=N−1−1∑i=1Nxi−X¯2, Sz=N−1−1∑i=1Nzi−Z¯2, Syx=N−1−1∑i=1Nyi−Y¯xi−X¯,Syz=N−1−1∑i=1Nyi−Y¯zi−Z¯, and Sxz=N−1−1∑i=1Nxi−X¯zi−Z¯.


## 2. Some Existing Estimators

Some existing estimators available in the literature are essential to be discussed here.


### 2.1. Sample Mean Estimator

The usual sample mean estimator and its variance are given as(1)Y¯^0=y¯,(2)VarY¯^0=ΘY¯2Cy2.


### 2.2. Ratio Estimators

The usual ratio estimators when using single and two auxiliary variables are given by(3)iY¯^R1=y¯X¯x¯,iiY¯^R2=y¯Z¯x¯,iiiY¯^R3=y¯X¯x¯Z¯z¯.The MSEs of ratio estimatorsY¯^Rii=1,2,3 to first order of approximation are given by(4)iMSEY¯^R1≅ΘY¯2Cy2+Cx2−2Cyx,(5)iiMSEY¯^R2≅ΘY¯2Cy2+Cz2−2Cyz,(6)iiiMSEY¯^R3≅ΘY¯2Cy2+Cx2+Cz2−2Cyx+Cyz+2Cxz.The ratio estimatorsY¯^Rii=1,2,3 are performing better than Y¯^0under certain conditions.


### 2.3. Exponential-Ratio Estimators

The usual exponential-ratio estimators when using single and two auxiliary variables are given by(7)iY¯^E1=y¯expX¯−x¯X¯+x¯,iiY¯^E2=y¯expZ¯−z¯Z¯+z¯,iiiY¯^E3=y¯expX¯−x¯X¯+x¯expZ¯−z¯Z¯+z¯.The MSEs of exponential-ratio estimatorsY¯^Eii=1,2,3 to first order of approximation are given by(8)iMSEY¯^E1≅ΘY¯2Cy2+14Cx2−Cyx,(9)iiMSEY¯^E2≅ΘY¯2Cy2+14Cz2−Cyz,(10)iiiMSEY¯^E3≅ΘY¯2Cy2+14Cx2+Cz2−Cyx+Cyz+12Cxz.The exponential-ratio estimatorsY¯^Eii=1,2,3 are performing better than Y¯^0 and Y¯^Rii=1,2,3under certain conditions.


### 2.4. Log-Ratio Estimators

Recently many log-type estimators have appeared in the literature in various forms when the logarithmic relationship between the study variable and the auxiliary variables exists.The usual log-ratio estimators when using single and two auxiliary variables are given by(11)iY¯^log1=y¯1+logX¯x¯,iiY¯^log2=y¯1+logZ¯z¯,iiiY¯^log3=y¯1+logX¯x¯1+logZ¯z¯.The MSEs of log-ratio estimatorsY¯^logii=1,2,3 to first order of approximation are given by(12)iMSEY¯^log1≅ΘY¯2Cy2+Cx2−2Cyx,(13)iiMSEY¯^log2≅ΘY¯2Cy2+Cz2−2Cyz,(14)iiiMSEY¯^log3≅ΘY¯2Cy2+Cx2+Cz2−2Cyx+Cyx+2Cxz.The MSEs of log-ratio estimatorsY¯^logii=1,2,3 are exactly equal to the MSEs of ratio estimators Y¯^Rii=1,2,3 but their biases are different (not shown here).


### 2.5. Regression Estimators

The usual regression estimators when using single and two auxiliary variables are given by(15)iY¯^Reg1=y¯+byxX¯−x¯,iiY¯^Reg2=y¯+byzZ¯−z¯,iiiY¯^Reg3=y¯+byxX¯−x¯+byzZ¯−z¯,where byx=syx/sx2 and byz=syz/sz2 are the sample regression coefficients.The MSEs of regression estimatorsY¯^Regii=1,2,3 are given by(16)iMSEY¯^Reg1≅ΘY¯2Cy21−ρyx2,(17)iiMSEY¯^Reg2≅ΘY¯2Cy21−ρyz2,(18)iiiMSEY¯^Reg3≅ΘY¯2Cy21−ρyx2−ρyz2+2ρyxρyzρxz.These regression estimatorsY¯^Regii=1,2,3 are performing better than Y¯^0andY¯^jii=1,2,3;j=R,E,logunder certain conditions.


### 2.6. Some More Regression Type Estimators

Mohanty [2] suggested the following regression-type estimator:(19)Y¯^RegM=y¯+byxX¯−x¯Z¯z¯.The MSE ofY¯^RegM is given by(20)MSEY¯^RegM≅ΘY¯2Cy21−ρyx2+Cz2−2Cyz+2CyCzρyxρxz.Swain [5] introduced the following regression-type estimator(21)Y¯^RegS=y¯+d0X¯−x¯Z¯z¯,where d0is constant.The minimum MSE ofY¯^RegS at optimum value of d0opt=Y¯Cyρyx−Czρxz/X¯Cx is given by(22)MSEY¯^RegSmin≅ΘY¯2Cy2Cy2+Cz2−2Cyz−Cyρyx−Czρxz2.The unbiased regression estimator when using two auxiliary variables is given by(23)Y¯^RegU=y¯+d1X¯−x¯+d2Z¯−z¯,where d1and d2are constants.The minimum MSE ofY¯^RegU at optimum values of d1opt=Syρyx−ρyzρxz/Sx1−ρxz2 and d2opt=Syρyz−ρyxρxz/Sz1−ρxz2 is given by(24)MSEY¯^RegUmin≅ΘY¯2Cy21−Ry.xz2,where Ry⋅xz2=ρyx2+ρyz2−2ρyxρyzρxz/1−ρxz2 is the multiple correlation coefficient.


## 2.1. Sample Mean Estimator

The usual sample mean estimator and its variance are given as(1)Y¯^0=y¯,(2)VarY¯^0=ΘY¯2Cy2.


## 2.2. Ratio Estimators

The usual ratio estimators when using single and two auxiliary variables are given by(3)iY¯^R1=y¯X¯x¯,iiY¯^R2=y¯Z¯x¯,iiiY¯^R3=y¯X¯x¯Z¯z¯.The MSEs of ratio estimatorsY¯^Rii=1,2,3 to first order of approximation are given by(4)iMSEY¯^R1≅ΘY¯2Cy2+Cx2−2Cyx,(5)iiMSEY¯^R2≅ΘY¯2Cy2+Cz2−2Cyz,(6)iiiMSEY¯^R3≅ΘY¯2Cy2+Cx2+Cz2−2Cyx+Cyz+2Cxz.The ratio estimatorsY¯^Rii=1,2,3 are performing better than Y¯^0under certain conditions.


## 2.3. Exponential-Ratio Estimators

The usual exponential-ratio estimators when using single and two auxiliary variables are given by(7)iY¯^E1=y¯expX¯−x¯X¯+x¯,iiY¯^E2=y¯expZ¯−z¯Z¯+z¯,iiiY¯^E3=y¯expX¯−x¯X¯+x¯expZ¯−z¯Z¯+z¯.The MSEs of exponential-ratio estimatorsY¯^Eii=1,2,3 to first order of approximation are given by(8)iMSEY¯^E1≅ΘY¯2Cy2+14Cx2−Cyx,(9)iiMSEY¯^E2≅ΘY¯2Cy2+14Cz2−Cyz,(10)iiiMSEY¯^E3≅ΘY¯2Cy2+14Cx2+Cz2−Cyx+Cyz+12Cxz.The exponential-ratio estimatorsY¯^Eii=1,2,3 are performing better than Y¯^0 and Y¯^Rii=1,2,3under certain conditions.


## 2.4. Log-Ratio Estimators

Recently many log-type estimators have appeared in the literature in various forms when the logarithmic relationship between the study variable and the auxiliary variables exists.The usual log-ratio estimators when using single and two auxiliary variables are given by(11)iY¯^log1=y¯1+logX¯x¯,iiY¯^log2=y¯1+logZ¯z¯,iiiY¯^log3=y¯1+logX¯x¯1+logZ¯z¯.The MSEs of log-ratio estimatorsY¯^logii=1,2,3 to first order of approximation are given by(12)iMSEY¯^log1≅ΘY¯2Cy2+Cx2−2Cyx,(13)iiMSEY¯^log2≅ΘY¯2Cy2+Cz2−2Cyz,(14)iiiMSEY¯^log3≅ΘY¯2Cy2+Cx2+Cz2−2Cyx+Cyx+2Cxz.The MSEs of log-ratio estimatorsY¯^logii=1,2,3 are exactly equal to the MSEs of ratio estimators Y¯^Rii=1,2,3 but their biases are different (not shown here).


## 2.5. Regression Estimators

The usual regression estimators when using single and two auxiliary variables are given by(15)iY¯^Reg1=y¯+byxX¯−x¯,iiY¯^Reg2=y¯+byzZ¯−z¯,iiiY¯^Reg3=y¯+byxX¯−x¯+byzZ¯−z¯,where byx=syx/sx2 and byz=syz/sz2 are the sample regression coefficients.The MSEs of regression estimatorsY¯^Regii=1,2,3 are given by(16)iMSEY¯^Reg1≅ΘY¯2Cy21−ρyx2,(17)iiMSEY¯^Reg2≅ΘY¯2Cy21−ρyz2,(18)iiiMSEY¯^Reg3≅ΘY¯2Cy21−ρyx2−ρyz2+2ρyxρyzρxz.These regression estimatorsY¯^Regii=1,2,3 are performing better than Y¯^0andY¯^jii=1,2,3;j=R,E,logunder certain conditions.


## 2.6. Some More Regression Type Estimators

Mohanty [2] suggested the following regression-type estimator:(19)Y¯^RegM=y¯+byxX¯−x¯Z¯z¯.The MSE ofY¯^RegM is given by(20)MSEY¯^RegM≅ΘY¯2Cy21−ρyx2+Cz2−2Cyz+2CyCzρyxρxz.Swain [5] introduced the following regression-type estimator(21)Y¯^RegS=y¯+d0X¯−x¯Z¯z¯,where d0is constant.The minimum MSE ofY¯^RegS at optimum value of d0opt=Y¯Cyρyx−Czρxz/X¯Cx is given by(22)MSEY¯^RegSmin≅ΘY¯2Cy2Cy2+Cz2−2Cyz−Cyρyx−Czρxz2.The unbiased regression estimator when using two auxiliary variables is given by(23)Y¯^RegU=y¯+d1X¯−x¯+d2Z¯−z¯,where d1and d2are constants.The minimum MSE ofY¯^RegU at optimum values of d1opt=Syρyx−ρyzρxz/Sx1−ρxz2 and d2opt=Syρyz−ρyxρxz/Sz1−ρxz2 is given by(24)MSEY¯^RegUmin≅ΘY¯2Cy21−Ry.xz2,where Ry⋅xz2=ρyx2+ρyz2−2ρyxρyzρxz/1−ρxz2 is the multiple correlation coefficient.


## 3. Proposed General Class of Estimators

We propose a ratio-exponential-log type general class of estimators in estimating the finite population mean using two auxiliary variables when some parameters of the auxiliary variables are known. We also obtain different special estimators as members of the general class of estimators which are useful in different real-life situations. The proposed estimator is the combination of three special estimators including ratio, exponential-ratio, and log-ratio by using the linear transformation as(25)Y¯^Mα1,α2,α3,γ1,γ2,γ3=y¯M1X¯∗x¯∗α1expX¯∗−x¯∗X¯∗+x¯∗α21+logX¯∗x¯∗α3+M2Z¯∗z¯∗γ1expZ¯∗−z¯∗Z¯∗+z¯∗γ21+logZ¯∗z¯∗γ3,where Mii=1,2 are constants, whose values are to be determined; αii=1,2,3 and γii=1,2,3 are scaler quantities; and x¯∗=ax¯+b, X¯∗=aX¯+b, z¯∗=cz¯+d, and Z¯∗=cZ¯+d. Here a,b,c,d are the known population parameters of the auxiliary variables which may be coefficients of variation Cx,Cz, coefficients of kurtosis β2x,β2z and correlation coefficients ρyx,ρyz.Solving (25) in terms of errors to the first order of approximation, we have(26)Y¯^Mα1,α2,α3,γ1,γ2,γ3−Y¯≅Y¯M11+Ξ0−Δ1g1Ξ1−Δ1g1Ξ0Ξ1+Δ2g12Ξ12+M21+Ξ0−Ω1g2Ξ2−Ω1g2Ξ0Ξ2+Δ2g22Ξ22−1,where(27)g1=aX¯+bax¯+b,g2=cZ¯+dcz¯+d,Δ1=α1+12α2+α3,Δ2=α1α3+12α2α3+12α1α1+1+18α2α2+2+12α1α2+12α32,Ω1=γ1+12γ2+γ3,Ω2=γ1γ3+12γ2γ3+12γ1γ1+1+18γ2γ2+2+12γ1γ2+12γ32.The bias ofY¯^Mα1,α2,α3,γ1,γ2,γ3 to the first order of approximation is given by(28)BiasY¯^Mα1,α2,α3,γ1,γ2,γ3≅Y¯M11+ΘΔ2g12Cx2−Δ1g1Cyx+M21+ΘΩ2g22Cz2−Ω1g2Cyz−1.The MSE ofY¯^Mα1,α2,α3,γ1,γ2,γ3 to the first order of approximation is given by(29)MSEY¯^Mα1,α2,α3,γ1,γ2,γ3≅Y¯2EM11+Ξ0−Δ1g1Ξ1−Δ1g1Ξ0Ξ1+Δ2g12Ξ12+M21+Ξ0−Ω1g2Ξ2−Ω1g2Ξ0Ξ2+Δ2g22Ξ22−12.Solving (29), we get(30)MSEY¯^Mα1,α2,α3,γ1,γ2,γ3≅Y¯21+M12Am+M22Bm−2M1Cm−2M2Dm+2M1M2Em,where(31)Am=1+ΘCy2Δ+122Δ2g12Cx2−4Δ1g1Cyx,Bm=1+ΘCy2Ω+122Ω2g22Cz2−4Ω1g2Cyz,Cm=1+ΘΔ2g12Cx2−Δ1g1Cyx,Dm=1+ΘΩ2g22Cz2−Ω1g2Cyz,Em=1+ΘCy2+Δ2g12Cx2+Ω2g22Cz2−2Δ1g1Cyx−2Ω1g2Cyz+Δ1Ω1g1g2Cxz.Solving (30), the optimum values Mii=1,2 are given as(32)M1opt=BmCm−DmEmAmBm−Em2,M2opt=AmDm−CmEmAmBm−Em2.The minimum MSE ofY¯^Mα1,α2,α3,γ1,γ2,γ3 to the first order of approximation is given by(33)MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min≅Y¯21−AmDm2+BmCm2−2CmDmEmAmBm−Em2.Some special estimators as members of the proposed general class of estimators are given by(i)
Puttingα1=γ1=1,α2=α3=γ2=γ3=0 in (25), we get(34)Y¯^M1,0,0,1,0,0=y¯M1X¯∗x¯∗+M2Z¯∗z¯∗.(ii)
Puttingα2=γ2=1,α1=α3=γ1=γ3=0 in (25), we get(35)Y¯^M0,1,0,0,1,0=y¯M1expX¯∗−x¯∗X¯∗+x¯∗+M2expZ¯∗−z¯∗Z¯∗+z¯∗.(iii)
Puttingα3=γ3=1,α1=α2=γ1=γ2=0 in (25), we get(36)Y¯^M0,0,1,0,0,1=y¯M11+logX¯∗x¯∗+M21+logZ¯∗z¯∗.(iv)
Puttingα1=α2=γ2=1,α3=γ1=γ3=0 in (25), we get(37)Y¯^M1,1,0,0,1,0=y¯M1X¯∗x¯∗expX¯∗−x¯∗X¯∗+x¯∗+M2expZ¯∗−z¯∗Z¯∗+z¯∗.(v)
Puttingα1=α3=γ2=1,α2=γ1=γ3=0 in (25), we get(38)Y¯^M1,0,1,0,1,0=y¯M1X¯∗x¯∗1+logX¯∗x¯∗+M2expZ¯∗−z¯∗Z¯∗+z¯∗.(vi)
Puttingα2=α3=γ2=1,α1=γ1=γ3=0 in (25), we get(39)Y¯^M0,1,1,0,1,0=y¯M1expX¯∗−x¯∗X¯∗+x¯∗1+logX¯∗x¯∗+M2expZ¯∗−z¯∗Z¯∗+z¯∗.(vii)
Puttingα1=α2=α3=γ2=1,γ1=γ3=0 in (25), we get(40)Y¯^M1,1,1,0,1,0=y¯M1X¯∗x¯∗expX¯∗−x¯∗X¯∗+x¯∗1+logX¯∗x¯∗+M2expZ¯∗−z¯∗Z¯∗+z¯∗.(viii)
Puttingα1=γ1=γ2=γ3=1,α2=α3=0 in (25), we get(41)Y¯^M1,0,0,1,1,1=y¯M1X¯∗x¯∗+M2Z¯∗z¯∗expZ¯∗−z¯∗Z¯∗+z¯∗1+logZ¯∗z¯∗.(i)
Puttingα2=γ1=γ2=γ3=1,α1=α3=0 in (25), we get(42)Y¯^M0,1,0,1,1,1=y¯M1expX¯∗−x¯∗X¯∗+x¯∗+M2Z¯∗z¯∗expZ¯∗−z¯∗Z¯∗+z¯∗1+logZ¯∗z¯∗.(x)
Puttingα3=γ1=γ2=γ3=1,α1=α2=0 in (25), we get(43)Y¯^M0,0,1,1,1,1=y¯M11+logX¯∗x¯∗+M2Z¯∗z¯∗expZ¯∗−z¯∗Z¯∗+z¯∗1+logZ¯∗z¯∗.Note: We can generate more sub-classes of the proposed general class of estimators by using different combinations.


## 4. Numerical Example

We use the following four real data sets for a numerical study.Population 1 (see [19]):(44)Y=Area under wheat in acres in1974,X=Area under wheat in acres in1971,Z=Area under wheat in acres in1973,N=34,n=20,Y¯=856.412,X¯=208.882,Z¯=199.441,Cy=0.8561,Cx=0.721,Cz=0.753,ρyx=0.449,ρyz=0.443,ρxz=0.980,β2x=2.910,β2z=3.732.Population 2 (see [18]):(45)Y=Output of the factory,X=Number of workers,Z=Fixed capital,N=80,n=20,Y¯=5182.637,X¯=285.125,Z¯=1126.463,Cy=0.354,Cx=0.948,Cz=0.751,ρyx=0.915,ρyz=0.941,ρxz=0.988,β2x=0.698,β2z=1.050.Population 3 (Punjab Development Statistics (2019)):This data is taken from Punjab development of statistics of 36 districts of Punjab, Pakistan during 2018.(46)Y=Number of reported crimes by hurt,X=Number of reported crimes by murdered,Z=Number of reported crimes by kidnapped,N=36,n=10,Y¯=421.9722,X¯=112.3889,Z¯=412.2222,Cy=0.5718,Cx=0.8336,Cz=1.4229,ρyx=0.7786,ρyz=0.6900,ρxz=0.8483,β2x=7.5508,β2z=27.5415.Population 4 (see [19]):This data are based on 69 villages of Doraha development bloc of Punjab, India.(47)Y=Number of tube wells,X=Number of tractors,Z=Net irrigated area in hectares,N=69,n=10,Y¯=135.2609,X¯=21.2319,Z¯=345.7536,Cy=0.8422,Cx=0.7969,Cz=0.8478,ρyx=0.9118,ρyz=0.9224,ρxz=0.9007,β2x=3.7653,β2z=7.2159.The results based on Populations 1–4 are given in Tables1–11. We use the following expression to obtain the percent relative efficiency (PRE) as(48)PRE=VarY¯^0MSE.×100,where .=Y¯^0,Y¯^Ri,Y¯^Ei,Y¯^logi,Y¯^regi,Y¯^regji=1,2,3;j=M,S,U.Table 1
PRE of different estimators with respectY¯^0 when a=c=1,b=d=0.
EstimatorPop.1Pop. 2Pop. 3Pop. 4Y¯^0100.00100.00100.00100.00Y¯^R1,Y¯^log1104.932∗116.936588.911Y¯^R2,Y¯^log2100.926∗∗639.893Y¯^R3,Y¯^log3∗∗∗∗Y¯^E1125.129291.940252.366276.952Y¯^E2124.688776.064120.327307.879Y¯^E3103.718∗∗870.810Y¯^Reg1125.251614.345253.948593.047Y¯^Reg2124.692873.218190.876670.339Y¯^Reg3100.807102.181120.605120.069Y¯^RegM∗∗∗102.214Y¯^RegS121.910883.094∗640.000Y¯^RegU125.349948.307255.867873.966“∗” indicates that the estimators are not efficient as compared to the usual mean estimator.Table 2
PRE of proposed estimator whenα1=γ1=1,α2=α3=γ2=γ3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010107.697186.585132.429886.8151Cx1Cz107.934184.610134.467885.501β2xCxβ2zCz107.780183.288132.707886.865Cxβ2xCzβ2z107.780183.288132.707886.8651ρyx1ρyz107.780183.288132.707886.8651β2x1β2z108.628185.787149.748852.370CxρyxCzρyz107.903185.219134.781883.828ρyxCxρyzCz108.220184.398135.032885.176β2xρyzβ2zρyz107.749183.672132.690886.850ρyxβ2xρyzβ2z109.745185.671154.632846.972Table 3
PRE of proposed estimator whenα2=γ2=1,α1=α3=γ1=γ3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010127.090956.748264.876311.8681Cx1Cz127.073956.817264.643310.282β2xCxβ2zCz127.084956.854264.849311.339Cxβ2xCzβ2z126.980956.775261.317319.8391ρyx1ρyz127.079956.808264.675310.2911β2x1β2z127.011956.783261.624314.469CxρyxCzρyz127.075956.803264.642310.422ρyxCxρyzCz127.051956.824264.568310.257β2xρyzβ2zρyz127.086956.846264.851311.285ρyxβ2xρyzβ2z126.900956.788260.260316.213Table 4
PRE of proposed estimator whenα3=γ3=1,α1=α2=γ1=γ2=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010106.402186.926127.151919.6961Cx1Cz106.646184.941129.254921.254β2xCxβ2zCz106.488183.610127.443920.619Cxβ2xCzβ2z107.724186.467149.441873.7451ρyx1ρyz106.555185.296129.179921.0521β2x1β2z107.357186.124144.723889.198CxρyxCzρyz106.615185.553129.618920.360ρyxCxρyzCz106.941184.727129.828921.127β2xρyzβ2zρyz106.456183.997127.426920.719ρyxβ2xρyzβ2z108.505186.007149.286883.271Table 5
PRE of proposed estimator whenα1=α2=γ2=1,α3=γ1=γ3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010126.769914.223141.181807.8501Cx1Cz126.777914.152141.896794.946β2xCxβ2zCz126.771914.197141.216804.456Cxβ2xCzβ2z126.818913.997150.724732.9601ρyx1ρyz126.774914.109141.569793.1581β2x1β2z126.807914.066153.920747.374CxρyxCzρyz126.775914.048141.497789.553ρyxCxρyzCz126.786914.150142.202793.730β2xρyzβ2zρyz126.770914.155141.204803.975ρyxβ2xρyzβ2z126.849914.058159.313741.821Table 6
PRE of proposed estimator whenα1=α3=γ2=1,α2=γ1=γ3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010126.737912.457139.553845.1471Cx1Cz126.743912.383140.132836.605β2xCxβ2zCz126.739912.423139.573842.896Cxβ2xCzβ2z126.779912.237147.425797.4741ρyx1ρyz126.741912.342139.830835.4521β2x1β2z126.769912.303150.655806.104CxρyxCzρyz126.742914.284139.744833.140ρyxCxρyzCz126.752912.380140.393835.815β2xρyzβ2zρyz126.738914.383139.562842.580ρyxβ2xρyzβ2z126.805912.295155.475802.724Table 7
PRE of proposed estimator whenα2=α3=γ2=1,α1=γ1=γ3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010126.710920.174140.021784.4431Cx1Cz126.715920.076140.691775.612β2xCxβ2zCz126.711920.126140.050782.185Cxβ2xCzβ2z126.743919.891149.088727.2611ρyx1ρyz126.713920.025140.371774.3621β2x1β2z126.735919.976152.300739.153CxρyxCzρyz126.714919.950140.293771.816ρyxCxρyzCz126.721920.073140.983774.759β2xρyzβ2zρyz126.710920.075140.038781.863ρyxβ2xρyzβ2z126.763919.965157.553734.610Table 8
PRE of proposed estimator whenα1=α2=α3=γ2=1,γ1=γ3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010126.768907.889139.833902.5031Cx1Cz126.776907.822140.396890.351β2xCxβ2zCz126.770907.856139.852899.195Cxβ2xCzβ2z126.818907.695147.416843.1791ρyx1ρyz126.772907.787140.103888.7441β2x1β2z126.806907.753150.567852.694CxρyxCzρyz126.774907.736140.019885.566ρyxCxρyzCz126.785907.820140.649889.257β2xρyzβ2zρyz126.769907.821139.842898.731ρyxβ2xρyzβ2z126.850907.746155.212848.913Table 9
PRE of proposed estimator whenα1=γ1=γ2=γ3=1,α2=α3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010123.139145.452159.789669.5271Cx1Cz123.185146.707161.602702.708β2xCxβ2zCz123.155147.296160.040678.643Cxβ2xCzβ2z123.384146.324178.354820.6211ρyx1ρyz123.168146.636161.528707.2181β2x1β2z123.317146.313174.823798.447CxρyxCzρyz123.179146.668161.898716.133ρyxCxρyzCz123.240146.826162.099705.743β2xρyzβ2zρyz123.149147.206160.025679.938ρyxβ2xρyzβ2z123.523146.397178.726807.318Table 10
PRE of proposed estimator whenα2=γ1=γ2=γ3=1,α1=α3=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010127.304483.947272.583956.1651Cx1Cz127.306485.394272.843952.519β2xCxβ2zCz127.304486.023272.627955.357Cxβ2xCzβ2z127.314485.060274.208928.0191ρyx1ρyz127.305485.341272.869952.0321β2x1β2z127.311485.006273.258933.369CxρyxCzρyz127.306485.413272.947952.016ρyxCxρyzCz127.308485.528272.898952.166β2xρyzβ2zρyz127.304485.948272.626955.247ρyxβ2xρyzβ2z127.318485.105272.896931.187Table 11
PRE of proposed estimator whenα3=γ1=γ2=γ3=1,α1=α2=0.
abcdPop. 1Pop. 2Pop. 3Pop. 41010122.150178.926180.747649.6161Cx1Cz122.196180.406181.991682.237β2xCxβ2zCz122.167181.159180.961658.386Cxβ2xCzβ2z122.392179.818192.910797.5031ρyx1ρyz122.179180.286182.139686.6751β2x1β2z122.324179.859188.365776.097CxρyxCzρyz122.191180.278182.524695.431ρyxCxρyzCz122.251180.550182.263685.229β2xρyzβ2zρyz122.161181.016180.955659.670ρyxβ2xρyzβ2z122.529179.955190.082784.693In Table1, we observed the following:(i)
The ratio and log-ratio estimatorsY¯^R1,Y¯^log1 in population 2, Y¯^R2,Y¯^log2 in populations 2 and 3, and Y¯^R3,Y¯^log3 in all four populations are performing poorly as compared to Y¯^0.(ii)
The exponential-ratio estimatorY¯^E3 in Populations 2 and 3 is not performing good.(iii)
Mohanty [2] regression estimator Y¯^RegM in populations 1–3 and Swain [5] estimator Y¯^RegS in Population 3 are not efficient as compared toY¯^0.(iv)
Among all the estimators above discussed, the performance ofY¯^RegU is the best.


## 5. Comparison of Estimators

Now we compare the proposed class of estimators with other existing estimators.Condition 1.
By (2) and (33), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<VarY¯^0 if (49)ΘCy2+O1O2−1>0,where O1=AmDm2+BmCm2−2CmDmEm and O2=AmBm−Em2.Condition 2.
By ((4)–(6)) or ((12)–(14)) and (33), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<MSEY¯^Ri,MSEY¯^logii=1,2,3 if (50)ΘCy2+Cx2−2Cyx+O1O2−1>0,ΘCy2+Cz2−2Cyz+O1O2−1>0,ΘCy2+Cx2+Cz2−2Cyx+Cyz+2Cxz+O1O2−1>0.Condition 3.
By ((8)–(10)) and (33), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<MSEY¯^Eii=1,2,3 if (51)ΘCy2+14Cx2−Cyx+O1O2−1>0,ΘCy2+14Cz2−Cyz+O1O2−1>0,ΘCy2+14Cx2+Cz2−Cyx+Cyz+12Cxz+O1O2−1>0.Condition 4.
By ((16)–(18)) and (33), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<MSEY¯^Regii=1,2,3 if (52)ΘCy21−ρyx2+O1O2−1>0,ΘCy21−ρyz2+O1O2−1>0,ΘCy21−ρyx2−ρyz2−2ρyxρyzρxz+O1O2−1>0.Condition 5.
By (20) and (38), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<MSEY¯^RegM if (53)ΘCy21−ρyx2+Cz2−2Cyz+2CyCzρyxρxz+O1O2−1>0.Condition 6.
By (22) and (38), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<MSEY¯^RegSmin if (54)ΘCy2Cy2+Cz2−2Cyz−Cyρyx−Czρxz2+O1O2−1>0.Condition 7.
By (32) and (38), MSEY¯^Mα1,α2,α3,γ1,γ2,γ3min<MSEY¯^RegUmin if (55)ΘCy21−Ry.xz2+O1O2−1>0.
The proposed class of estimators will perform better when conditions 1–7 are satisfied.


## 6. Conclusion

In this study, we have proposed ratio-exponential-log type generalized class of estimators by combing a ratio, exponential-ratio, and log-ratio type estimators by using the linear transformation for finite population mean in simple random sampling. Expressions for the bias and MSE of proposed general class of estimators are obtained up to the first order of approximation. Four data sets are used for numerical study. Based on Tables1–11, we observe that the proposed sub-classes of general estimators are performing well as compared to their competitor estimators. We have generated 10 sub-classes from the proposed general estimators with different combinations which all are efficient in different situation as compared to SRS. So, the proposed general class of estimators is preferable in further study.



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*Source: 5245621-2021-10-01.xml*