# On Using the Conventional and Nonconventional Measures of the Auxiliary Variable for Mean Estimation

Authors: Javid Shabbir; Sat Gupta; Ronald Onyango
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we propose an improved new class of exponential-ratio-type estimators for estimating the finite population mean using the conventional and the nonconventional measures of the auxiliary variable. Expressions for the bias and MSE are obtained under large sample approximation. Both simulation and numerical studies are conducted to validate the theoretical findings. Use of the conventional and the nonconventional measures of the auxiliary variable is very common in survey research, but we observe that this does not add much value in many of the estimators except for our proposed class of estimators.


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## Body

## 1. Introduction

Many research papers have appeared in the literature where authors have used the conventional and the nonconventional measures of the auxiliary variable to enhance the efficiency of estimators. Recently, Gulzar et al. [1] used the nonconventional measures for population variance. Many other researchers have contributed to this area. Singh et al. [2] suggested a class of linear combinations of exponential-ratio- and product-type estimators for mean estimation. Gupta and Shabbir [3] introduced a class of ratio-in-difference-type estimators using the conventional measures for the population mean. Singh et al. [4] presented an improved family of exponential-ratio-type estimator in simple random sampling for population mean. Haq and Shabbir [5] introduced an improved family of ratio-type estimators in simple and stratified sampling. Yadav and Kadilar [6] proposed an exponential family of ratio-type estimators by using conventional measures for estimating the population mean. Shabbir et al. [7] presented a new family of estimators for finite population mean in simple random sampling. Grover and Kaur [8] suggested a generalized class of exponential-ratio-type exponential estimators using the conventional measures for mean estimation. Kadilar [9] discussed a new exponential-type estimator for mean estimation. Irfan et al. [10] suggested a generalized ratio-exponential-type estimator using the conventional measures. Also, Irfan et al. [11] used both conventional and nonconventional measures in their proposed estimator. Ali et al. [12] used the robust-regression-type estimators for mean estimation of sensitive variables, and Shahzad et al. [13] suggested the L-moments based on calibration variance estimators in their recent work.In our study, we propose a new generalized class of exponential-ratio-type estimators by using the conventional and the nonconventional measures of the auxiliary variable and compare our proposed estimator with several existing estimators.Consider a finite populationΛ=Λ1,Λ2,…,ΛN of N units. A sample of size n units is drawn from this population using simple random sampling without replacement (SRSWOR). Let yi and xi be the observed values of the study variable Y and the auxiliary variable X, respectively. Let y¯=n−1∑i=1nyi and x¯=n−1∑i=1nxi, respectively, be the sample means and Y¯=N−1∑i=1Nyi and X¯=N−1∑i=1Nxi be the corresponding population means. Let Cy=Y¯−1Sy,Cx=X¯−1Sx be the coefficient of variations andSy=N−1−1∑i=1Nyi−Y¯2,Sx=N−1−1∑i=1Nxi−X¯2 be the standard deviations of Y,X, respectively. Let ρyx=SySx−1Syx be the correlation coefficient and Syx=N−1−1∑i=1Nyi−Y¯xi−X¯ be the covariance between the variables indicated by the subscripts. We define the following error terms to obtain the bias and MSE expressions: Θ0=y¯/Y¯−1 and Θ1=x¯/X¯−1 such that EΘi=0,i=0,1, EΘ02=γCy2, EΘ12=γCx2, and EΘ0Θ1=γρyxCyCx=γCyx, where γ=n−1−N−1.Now we discuss some of the conventional and nonconventional measures of the auxiliary variable which are used in our study. Arthur Lyon Bowley (1869–1957), a British Statistician, introduced a term skewnessβ1x based on median and the two quartiles, and kurtosis β2x was originated by Karl Pearson (1857–1936). Antonine Augustin Cournot was the first to use the term median MD in 1843. The midrange MR, i.e., MR=X1+XN/2 was introduced by Robert K. Merton. Tukey [14] used the idea of Trimean (TM) TM=Q1+2Q2+Q3/4, where Q1,Q2,Q3 are the first, second, and third quartiles of the auxiliary variable which were discussed by Tukey [15]. The term quartile deviation QD=Q3x−Q1x/2 was used in “Proceeding of the Royal Statistical Society of London” in late 19th century. Hodge and Lehmann [16] used the measure HL: HL=medianXj+Xk/2,1≤j≤k≤N, for estimation of location based on ranks.


## 2. Some Existing Estimators

We discuss the following mean estimators that exist in the literature:(i)
The sample mean estimator isY¯^0=y¯, and its variance is given by(1)VarY¯^0=γY¯2Cy2.(ii)
The usual ratio estimator proposed by Cochran [17] when the regression line Y on X passes through origin is given by(2)Y¯^R=y¯X¯x¯,whereX¯ is the known population mean of the auxiliary variable X. The performance of ratio estimator is better as compared to the usual mean estimator y¯ when ρyx>0.5Cx/Cy.The bias and MSE, respectively, ofY¯^R are given by(3)BY¯^R≈γY¯Cx2−Cyx,(4)MSEY¯^R≈γY¯2Cy2+Cx2−2Cyx.(iii)
Bahl and Tuteja [18] suggested the exponential-ratio-type estimator given by(5)Y¯^EP=y¯expX¯−x¯X¯+x¯.The bias and MSE, respectively, ofY¯^EP are given by(6)BY¯^EP≈γY¯38Cx2−12Cyx,(7)MSEY¯^EP≈γY¯2Cy2+14Cx2−Cyx.The exponential ratio estimatorY¯^EP is superior to the usual mean estimator Y¯^0 and ratio estimator Y¯^R if ρyx>0.25Cx/Cy and ρyx<0.75Cx/Cy, respectively.(iv)
The regression or difference estimator [19, 20] is given by(8)Y¯^D=y¯+wX¯−x¯,wherew is a constant.The minimum MSE ofY¯^D at wopt=Y¯ρyxCy/X¯Cx is given by(9)MSEY¯^Dmin≈γY¯2Cy21−ρyx2.The difference estimator always performs better than usual sample mean estimator, ratio estimator, and exponential ratio estimator ifρyx2>0,Cx−Cyρyx2>0, and 0.5Cx−Cyρyx2>0, respectively.(v)
Singh et al.’s estimator [2] is given by(10)Y¯^S1=y¯sexpX¯−x¯X¯+x¯+1−sexpx¯−X¯x¯+X¯,wheres is a constant.The minimum MSE ofY¯^S1 at sopt=0.5+ρyxCy/Cx is given by(11)MSEY¯^S1min≈γY¯2Cy21−ρyx2=MSEY¯^Dmin.(vi)
Singh et al. [4] suggested the following estimator:(12)Y¯^iS2=y¯expuX¯−x¯uX¯−x¯+2v,whereu and v are the functions of known population parameters of the auxiliary variable. Some members of the family of estimators Y¯^iS2i=1,2,…,12 are given in Table 1.The bias and MSE, respectively, ofY¯^S2 are given by(13)BY¯^S2≈γY¯38φi2Cx2−12φiCyx,(14)MSEY¯^S2≈γY¯2Cy2+14φi2Cx2−φiCyx,whereφi=uX¯/uX¯+vi=1,2,…,12. We can get different values of φii=1,2,…,12 by using different values of u,v, i.e., u=1,v=0, u=1,v=Cx, u=Cx,v=ρyx, u=β1x,v=ρyx, u=β2x,v=β1x, u=1,v=ρyx, u=QD,v=TM, u=QD,v=MR, u=HL,v=TM, u=1,v=QD, u=ρyx,v=QD, u=ρyx,v=β2x resulting in φ1=1,φ2=X¯/X¯+Cx,φ3=CxX¯/X¯+ρyx,φ4=β1xX¯/X¯+ρyx,φ5=β2xX¯/X¯+β1x,φ6=X¯/X¯+ρyx,φ7=QDX¯/X¯+TM,φ8=QDX¯/X¯+MR,φ9=HLX¯/X¯+TM,φ10=X¯/X¯+QD,φ11=ρyxX¯/X¯+QD,φ12=ρyxX¯/X¯+β2x, respectively.(vii)
Yadav and Kadilar [6] suggested the following estimator:(15)Y¯^YK=ky¯expuX¯−x¯uX¯−x¯+2v,wherek is a constant. Some members of the family of estimators Y¯^iYKi=1,2,…,12 are given in Table 1.The bias and minimum MSE, respectively, ofY¯^YK are given by(16)BY¯^YK≈k−1Y¯+kY¯γ38φi2Cx2−12φiCyx,(17)MSEY¯^YKmin≈Y¯21−Yk2i2Yk1i,whereYk1i=1+γCy2+φi2Cx2−2φiCyx and Yk2i=1+γ3/8φi2Cx2−1/2φiCyx.The optimum value ofk is given by kopt=Yk2i/Yk1i.(viii)
Kadilar [9] suggested the following exponential-ratio-type estimator:(18)Y¯^K=y¯X¯x¯mexpX¯−x¯X¯+x¯,wherem is a constant.The bias and MSE, respectively, ofY¯^K at mopt=0.5−ρyxCy/Cx are given by(19)BY¯^K≈Y¯γm−0.5Cyx+Cx2m22−m+38,(20)MSEY¯^Kmin≈γY¯2Cy21−ρyx2=MSEY¯^Dmin.(ix)
Grover and Kaur’s estimator [8] is given by(21)Y¯^GK=g1y¯+g2X¯−x¯expuX¯−x¯uX¯+x¯+2v,whereg1 and g2 are constants. Some members of the family of estimators Y¯^iGKi=1,2,…,12 are given in Table 1.The bias and minimum MSE, respectively, ofY¯^GK at g1opt=BgCg−DgEg+Bg/AgBg−Eg2+Bg and g2opt=Y¯/X¯AgDg−CgEg+Dg−Eg/AgBg−Eg2+Bg are given by(22)BY¯^GK≈g1−1Y¯+g1Y¯Cg+g2X¯Dg,(23)MSEY¯^GKmin≈Y¯21−AgDg2+BgCg2−2CgDgEg+2BgCg+Dg2−2DgEg+BgAgBg−Eg2+Bg,whereAg=γCy2+φi2Cx2−2φiCyx, Bg=γCx2, Cg=γ3/8φi2Cx2−1/2φiCyx, Dg=γφiCx2/2, and Eg=γφiCx2−Cyx.(x)
Recently, Irfan et al. [10] suggested the following exponential-type estimator:(24)Y¯^Ia=I1ay¯uX¯+vux¯+v+I2aX¯−x¯expuX¯−x¯uX¯+x¯+2v,whereI1a and I2a are constants. Some members of the family of estimators Y¯^iIai=1,2,…,12 are given in Table 2.The bias and minimum MSE, respectively, ofY¯^Ia at I1aopt=BaCa−DaEa+Ba/AaBa−Ea2+Ba and I2aopt=Y¯/X¯AaDa−CaEa+Da−Ea/AaBa−Ea2+Ba are given by(25)BY¯^Ia≈I1a−1Y¯+I1aY¯Ca+I2aX¯Da,(26)MSEY¯^Iamin≈Y¯21−AaDa2+BaCa2−2CaDaEa+2BaCa+Da2−2DaEa+BaAaBa−Ea2+Ba,whereAa=γCy2+3φi2Cx2−4φiCyx, Ba=γCx2, Ca=γφi2Cx2−φiCyx, Da=γφiCx2/2, and Ea=γ3/2φiCx2−Cyx.(xi)
Shabbir et al. [7] suggested the following transformed exponential-ratio-difference-type estimator:(27)Y¯^SH=s1y¯+s2X¯−x¯+Ω∗expuX¯−x¯uX¯+x¯+2v,whereΩ∗=y¯/2expuX¯−x¯/uX¯+x¯+2v+expux¯−X¯/ux¯+X¯+2v and s1 and s2 are constants. The members of the family of estimators Y¯^iSHi=1,2,…,12 are given in Table 2.The bias and minimum MSE, respectively, ofY¯^SH, at s1opt=DsEs−BsCs/AsBs−Es2 and s2opt=Y¯/X¯CsEs−AsDs/AsBs−Es2, are given by(28)BY¯^SH≈s1Y¯Ds+s2X¯Bs12,(29)MSEY¯^SHmin≈Y¯2γCy2+0.25φi2Cx2−φiCyx−AsDs2+BsCs2−2CsDsEsAsBs−Es2,whereAs=1+γCy2+φi2Cx2−2φiCyx,Bs=γCx2,Cs=γCy2+3/4φi2Cx2−3/2φiCyx,Ds=γ0.5φiCx2−Cyx, and Es=γφiCx2−Cyx.(xii)
On the lines of Shabbir et al. [7], Irfan et al. [11] suggested an estimator, which is given by(30)Y¯^iIb=I1by¯+I2bX¯−x¯+ΩexpuX¯−x¯uX¯+x¯−2v,whereΩ=y¯/2expX¯−x¯/X¯+x¯+expx¯−X¯/x¯+X¯ and I1b and I2b are constants. The members of the family of estimators Y¯^iIbi=1,2,…,12 are given in Table 3.Table 1
Members of Singh et al. [4], Yadav and Kadilar [6], and Grover and Kaur [8].
uvSingh et al. [4]Yadav and Kadilar [6]Grover and Kaur [8]10Y¯^1S2=y¯exp1Y¯^1YK=ky¯exp1Y¯1GK=g1y¯+g2X¯−x¯exp11CxY¯^2S2=y¯exp2Y¯^2YK=ky¯exp2Y¯2GK=g1y¯+g2X¯−x¯exp2CxρyxY¯^3S2=y¯exp3Y¯^3YK=ky¯exp3Y¯3GK=g1y¯+g2X¯−x¯exp3β1xρyxY¯^4S2=y¯exp4Y¯^4YK=ky¯exp4Y¯4GK=g1y¯+g2X¯−x¯exp4β2xβ1xY¯^5S2=y¯exp5Y¯^5YK=ky¯exp5Y¯5GK=g1y¯+g2X¯−x¯exp51ρyxY¯^6S2=y¯exp6Y¯^6YK=ky¯exp6Y¯6GK=g1y¯+g2X¯−x¯exp6QDTMY¯^7S2=y¯exp7Y¯^7YK=ky¯exp7Y¯7GK=g1y¯+g2X¯−x¯exp7QDMRY¯^8S2=y¯exp8Y¯^8YK=ky¯exp8Y¯8GK=g1y¯+g2X¯−x¯exp8HLTMY¯^9S2=y¯exp9Y¯^9YK=ky¯exp9Y¯9GK=g1y¯+g2X¯−x¯exp91QDY¯^10S2=y¯exp10Y¯^10YK=ky¯exp10Y¯10GK=g1y¯+g2X¯−x¯exp10ρyxQDY¯^11S2=y¯exp11Y¯^11YK=ky¯exp11Y¯11GK=g1y¯+g2X¯−x¯exp11ρyxβ2xY¯^12S2=y¯exp12Y¯^12YK=ky¯exp12Y¯12GK=g1y¯+g2X¯−x¯exp121=X¯−x¯/X¯+x¯,2=X¯−x¯/X¯+x¯+2Cx,3=CxX¯−x¯/CxX¯+x¯+2ρyx,4=β1xX¯−x¯/β1xX¯+x¯+2ρyx,5=β2xX¯−x¯/β2xX¯+x¯+2β1x,6=X¯−x¯/X¯+x¯+2ρyx,7=QDX¯−x¯/QDX¯+x¯+2TM,8=QDX¯−x¯/QDX¯+x¯+2MR,9=HLX¯−x¯/HLX¯+x¯+2TM,10=X¯−x¯/X¯+x¯+2QD,11=ρyxX¯−x¯/ρyxX¯+x¯+2QD, and 12=ρyxX¯−x¯/ρyxX¯+x¯+2β2x.Table 2
Members of Irfan et al. [10] and Shabbir et al. [7].
uvIrfan et al. [10]Shabbir et al. [7]10Y¯^1Ia=I1ay¯1∗+I2aX¯−x¯exp1Y¯^1SH=s1y¯+s2X¯−x¯+Ω1∗exp11CxY¯^2Ia=I1ay¯2∗+I2aX¯−x¯exp2Y¯^2SH=s1y¯+s2X¯−x¯+Ω2∗exp2CxρyxY¯^3Ia=I1ay¯3∗+I2aX¯−x¯exp3Y¯^3SH=s1y¯+s2X¯−x¯+Ω3∗exp3β1xρyxY¯^4Ia=I1ay¯4∗+I2aX¯−x¯exp4Y¯^4SH=s1y¯+s2X¯−x¯+Ω4∗exp4β2xβ1xY¯^5Ia=I1ay¯5∗+I2aX¯−x¯exp5Y¯^5SH=s1y¯+s2X¯−x¯+Ω5∗exp51ρyxY¯^6Ia=I1ay¯6∗+I2aX¯−x¯exp6Y¯^6SH=s1y¯+s2X¯−x¯+Ω6∗exp6QDTMY¯^7Ia=I1ay¯7∗+I2aX¯−x¯exp7Y¯^7SH=s1y¯+s2X¯−x¯+Ω7∗exp7QDMRY¯^8Ia=I1ay¯8∗+I2aX¯−x¯exp8Y¯^8SH=s1y¯+s2X¯−x¯+Ω8∗exp8HLTMY¯^9Ia=I1ay¯9∗+I2aX¯−x¯exp9Y¯^9SH=s1y¯+s2X¯−x¯+Ω9∗exp91QDY¯^10Ia=I1ay¯10∗+I2aX¯−x¯exp10Y¯^10SH=s1y¯+s2X¯−x¯+Ω10∗exp10ρyxQDY¯^11Ia=I1ay¯11∗+I2aX¯−x¯exp11Y¯^11SH=s1y¯+s2X¯−x¯+Ω11∗exp11ρyxβ2xY¯^12Ia=I1ay¯12∗+I2aX¯−x¯exp12Y¯^12SH=s1y¯+s2X¯−x¯+Ω12∗exp121∗=X¯/x¯,2∗=X¯/x¯+Cx,3∗=CxX¯/Cxx¯+ρyx,4∗=β1xX¯/β1xx¯+ρyx,5∗=β2xX¯/β2xx¯+β1x,6∗=X¯/x¯+ρyx,7∗=QDX¯/QDx¯+TM,8∗=QDX¯/QDx¯+MR,9∗=HLX¯/HLx¯+TM,10∗=X¯/x¯+QD,11∗=ρyxX¯/ρyxx¯+QD,12∗=ρyx/ρyxx¯+β2x,Ω1∗=exp1+exp1++,Ω2∗=exp2+exp2++,Ω3∗=exp3+exp3++,Ω4∗=exp4+exp4++,Ω5∗=exp5+exp5++,Ω6∗=exp6+exp6++,Ω7∗=exp7+exp7++,Ω7∗=exp7+exp7++,Ω8∗=exp8+exp8++,Ω9∗=exp9+exp9++,Ω10∗=exp10+exp10++,Ω11∗=exp11+exp11++,Ω12∗=exp12+exp12++, where 1++=x¯−X¯/x¯+X¯,2++=x¯−X¯/x¯+X¯+2Cx,3++=Cxx¯−X¯/Cxx¯+X¯+2ρyx,4++=β1xx¯−X¯/β1xx¯+X¯+2ρyx,5++=β2xx¯−X¯/β2xx¯+X¯+2β1x,6++=x¯−X¯/x¯+X¯+2ρyx,7++=QDx¯−X¯/QDx¯+X¯+2TM,8++=QDx¯−X¯/QDx¯+X¯+2MR,9++=HLx¯−X¯/HLx¯+X¯+2TM,10++=x¯−X¯/x¯+X¯+2QD,11++=ρyxx¯−X¯/ρyxx¯+X¯+2QD, and 11++=ρyxx¯−X¯/ρyxx¯+X¯+2β2x.The bias and minimum MSE, respectively, ofY¯^Ib, at I1bopt=DbEb−BbCb/AbBb−Eb2 and I2bopt=Y¯/X¯CbEb−AbDb/AbBb−Eb2, are given by(31)BY¯^Ib≈γY¯1+3ψi2Cx28−12ψiCyx+IbY¯γ38ψi2Cx2−12ψiCyx+I2bX¯Bs12,(32)MSEY¯^Ibmin≈Y¯2γCy2+0.25ψi2Cx2−ψiCyx−AbDb2+BbCb2−2CbDbEbAbBb−Eb2,where(33)Bb=γCx2,Cb=γCy2+5ψi2+18Cx2−32ψiCyx,Db=γ0.5ψiCx2−Cyx,Eb=γψiCx2−Cyx,ψi=uX¯uX¯−vCx,ψ1=1,ψ2=X¯X¯−Cx,ψ3=CxX¯X¯−ρyx,ψ4=β1xX¯X¯−ρyx,ψ5=β2xX¯X¯−β1x,ψ6=X¯X¯−ρyx,ψ7=QDX¯X¯−TM,ψ8=QDX¯X¯−MR,ψ9=HLX¯X¯−TM,ψ10=X¯X¯−QD,ψ11=ρyxX¯X¯−QD,ψ12=ρyxX¯X¯−β2x.Table 3
Members of Irfan et al. [11] and proposed class of estimators.
uvIrfan et al. [11]Proposed class of estimators10Y¯^1Ib=I1by¯+I2bX¯−x¯+Ωexp1+Y¯^1P=T1y¯+T2exp11CxY¯^2Ib=I1by¯+I2bX¯−x¯+Ωexp2+Y¯^2P=T1y¯+T2exp2CxρyxY¯^3Ib=I1by¯+I2bX¯−x¯+Ωexp3+Y¯^3P=T1y¯+T2exp3β1xρyxY¯^4Ib=I1by¯+I2bX¯−x¯+Ωexp4+Y¯^4P=T1y¯+T2exp4β2xβ1xY¯^5Ib=I1by¯+I2bX¯−x¯+Ωexp5+Y¯^5P=T1y¯+T2exp51ρyxY¯^6Ib=I1by¯+I2bX¯−x¯+Ωexp6+Y¯^6P=T1y¯+T2exp6QDTMY¯^7Ib=I1by¯+I2bX¯−x¯+Ωexp7+Y¯^7P=T1y¯+T2exp7QDMRY¯^8Ib=I1by¯+I2bX¯−x¯+Ωexp8+Y¯^8P=T1y¯+T2exp8HLTMY¯^9Ib=I1by¯+I2bX¯−x¯+Ωexp9+Y¯^9P=T1y¯+T2exp91QDY¯^10Ib=I1by¯+I2bX¯−x¯+Ωexp10+Y¯^10P=T1y¯+T2exp10ρyxQDY¯^11Ib=I1by¯+I2bX¯−x¯+Ωexp11+Y¯^11P=T1y¯+T2exp11ρyxβ2xY¯^12Ib=I1by¯+I2bX¯−x¯+Ωexp12+Y¯^12P=T1y¯+T2exp121+=X¯/x¯,2+=X¯/x¯−Cx,3+=CxX¯/Cxx¯−ρyx,4+=β1xX¯/β1xx¯−ρyx,5+=β2xX¯/β2xx¯−β1x,6+=X¯/x¯−ρyx,7+=QDX¯/QDx¯−TM,8+=QDX¯/QDx¯−MR,9+=HLX¯/HLx¯−TM,10+=X¯/x¯−QD,11+=ρyxX¯/ρyxx¯−QD, and 12+=ρyx/ρyxx¯−β2x.


## 3. Proposed Estimator

We propose a fairly simple class of exponential-ratio-type estimators using the conventional and nonconventional measures as given below:(34)Y¯^P=T1y¯+T2expuX¯−x¯uX¯+x¯+2v,where T1 and T2 are constants and u and v are the known conventional and nonconventional measures of the auxiliary variable. Various members of the family of estimators Y¯^iPi=1,2,…,12 are given in Table 3. The purpose of constructing this new class of estimators is to see its behavior when using both conventional and nonconventional measures.RewritingY¯^P in terms of errors up to first order of approximation, we have(35)Y¯^P−Y¯≈T1−1Y¯+T1Y¯Θ0−12φiΘ1−12φiΘ0Θ1+38φi2Θ12+T21−12φiΘ1+38φi2Θ12.The bias ofY¯^P to first degree of approximation is given by(36)BY¯^P≈T1−1Y¯+T1Y¯Ci+T2Di.Taking square and then expectation in (35), the MSE of Y¯^P to first degree of approximation is given by(37)MSEY¯^P≈T1−12Y¯2+T12Y¯2Ai+T22Bi−2T1Y¯2Ci−2T2Y¯Di+2T1T2Y¯Ei,where Ai=γCy2+φi2Cx2−2φiCyx,Bi=1+γφi2Cx2,Ci=γ3/8φi2Cx2−1/2φiCyx,Di=1+3/8γφi2Cx2, and Ei=1+γφi2Cx2−φiCyx.The optimum values ofT1 and T2 are T1opt=BiCi−DiEi+Bi/AiBi−Ei2+Bi and T2opt=Y¯AiDi−CiEi+Di−Ei/AiBi−Ei2+Bi.Substituting the optimum values in (37), we can get minimum MSE of Y¯^P which is given by(38)MSEY¯^Pmin≈Y¯21−AiDi2+BiCi2−2CiDiEi+2BiCi+Di2−2DiEi+BiAiBi−Ei2+Bi.


## 4. Comparison of Estimators

Now we compare the proposed class of estimators with other existing estimators discussed here.Condition 1.
By (1) and (37), MSEY¯^Pmin<VarY¯^0 if γCy2−1+Δ1/Δ2>0, where Δ1=AiDi2+BiCi2−2CiDiEi+2BiCi+Di2−2DiEi+Bi and Δ2=AiBi−Ei2+Bi.Condition 2.
By (4) and (37), MSEY¯^Pmin<MSEY¯^R if(39)γCy2+Cx2−2Cyx−1+Δ1Δ2>0.Condition 3.
By (7) and (38), MSEY¯^Pmin<MSEY¯^EP if(40)γCy2+14Cx2−Cyx−1+Δ1Δ2>0.Condition 4.
By (9), (11), (20), and (38), MSEY¯^Pmin<MSEY¯^Dmin,MSEY¯^S1min,MSEY¯^Kmin if(41)γCy21−ρyx2−1+Δ1Δ2>0.Condition 5.
By (14) and (38), MSEY¯^Pmin<MSEY¯^S2 if(42)γCy2+φi2Cx2−2φiCyx−1+Δ1Δ2>0.Condition 6.
By (17) and (38), MSEY¯^Pmin<MSEY¯^YKmin if(43)−Yk2i2Yk1i+Δ1Δ2>0.Condition 7.
By (23) and (38), MSEY¯^Pmin<MSEY¯^GKmin if(44)Δ1Δ2−Δ1gkΔ2gk>0,where Δ1gk=AgDg2+BgCg2−2CgDgEg+2BgCg+Dg2−2DgEg+Bg and Δ2gk=AgBg−Eg2+Bg.Condition 8.
By (26) and (38), MSEY¯^Pmin<MSEY¯^Iamin if(45)Δ1Δ2−Δ1IaΔ2Ia>0,where Δ1Ia=AaDa2+BaCa2−2CaDaEa+2BaCa+Da2−2DaEa+Ba and Δ2Ib=AgBg−Eg2+Bg.Condition 9.
By (29) and (38), MSEY¯^Pmin<MSEY¯^SHmin if(46)Δ1Δ2−δ1Jδ2J+δ3J>0,where Δ1J=AsDs2+BsCs2−2CsDsEs and Δ2J=AsBs−Es2+Bs, δ3J=Cy2+0.25φi2Cx2−φiCyx.Condition 10.
By (32) and (38), MSEY¯^Pmin<MSEY¯^Ibmin if(47)Δ1Δ2−Δ1IbΔ2Ib+Δ3Ib>0,where Δ1Ib=AbDb2+BbCb2−2CbDbEb and Δ2Ib=AbBb−Eb2+Bb, Δ3Ib=Cy2+0.25ψi2Cx2−ψiCyx.


## 5. Numerical Examples

Both simulation and numerical studies are conducted to observe the performances of different estimators.


### 5.1. Simulation Study

In this section, a simulation study is conducted to assess the performances of all estimators considered here. We consider two finite populations of size 1000 generated from a bivariate normal distribution with the same theoretical means ofY,X as μ=5,5 but different covariance matrices as given below.Population 1:(48)Σ=91.91.94.Population 2:(49)Σ=61.11.12.For each population, we consider a sample of sizes 50 and 100. The following steps are performed to carry out the simulation study.Step 1. Select a SRSWOR of size n from a population of size N.Step 2. Use a sample data from Step 1 to find the MSE values of all the estimators.Step 3. Steps 1 and 2 are repeated 10,000 times.Step 4. Obtain 10,000 values for MSEs.Step 5. Average of 10,000 values obtained in Step 4 represents the simulated MSE of each estimator.The simulated MSEs based on Populations 1 and 2 for sample sizes 50 and 100 are given in Table4.Table 4
MSE values under simulation for Populations 1 and 2 when usingu=1,v=0.
EstimatorPopulation 1Population 25010050100Y¯^00.1870690.0872320.129170.055353Y¯^R0.1813630.0832890.119170.053420Y¯^EP0.1655960.0764920.1112210.051525Y¯^D0.1646500.0762730.1103070.051128Y¯^S10.1654810.0763730.1106460.051170Y¯^S20.1655960.0762140.1109020.051525Y¯^YK0.1647040.0769140.1112210.051409Y¯^K0.1663450.0769240.1133350.052623Y¯^GK0.1644600.0760740.1102250.051036Y¯^Ia0.1656490.0762540.1106440.051100Y¯^SH0.1645210.0760720.1102450.051034Y¯^Ib0.1645210.0760720.1102450.051034Y¯^P0.0161010.0076280.0980410.003814From Table4, we observed that as we increase the sample size, the MSE values decrease in both populations which are on expected lines. Irfan et al.’s estimator [11] becomes Shabbir et al.’s estimator [7] when u=1,v=0, so MSE values are the same. The proposed estimator Y¯^P shows the least MSE values as compared to all other considered estimators.


### 5.2. Real Datasets

We use the following 7 real datasets for a numerical study.Population 1 (source: Singh and Chaudhary [21]):Y=area under wheat crop in acres during 1974 in 34 villages.X=area under wheat crop in acres during 1971 in 34 villages.The summary statistics areN=34,n=20,Y¯=856.4117,X¯=208.8823,Cy=0.8561,Cx=0.7205,ρyx=0.4491,β1x=0.9782,β2x=0.0978,QD=80.25,TM=162.25,MR=284.5, and HL=190.0.Population 2 (source: Cochran [19]):Y=number of inhabitants (in 1000’s) in 1930.X=number of inhabitants (in 1000’s) in 1920.The summary statistics areN=49,n=12,Y¯=127.7959,X¯=103.1429,Cy=0.9634,Cx=1.0122,ρyx=0.9817,β1x=2.2553,β2x=5.1412,QD=38.50,TM=72.75,MR=254.5, and HL=77.25.Population 3 (source: Singh and Mangat [22]):Y=number of tube wells.X=net irrigated area in hectares for 69 villages.The summary statistics areN=69,n=15,Y¯=135.2608,X¯=345.7536,Cy=0.8422,Cx=0.8479,ρyx=0.9224,β1x=2.3808,β2x=7.2159,QD=138.0,TM=274.5,MR=900, and HL=277.0.Population 4 (source: Singh and Mangat [22]):Y=average duration of sleep in hours.X=age of a person.The summary statistics areN=30,n=10,Y¯=6.3767,X¯=66.9333,Cy=0.1633,Cx=0.1436,ρyx=−0.8669,β1x=0.2984,β2x=−0.7686,QD=7.25,TM=66.5,MR=69.0, and HL=67.0.Population 5 (source: Gujarati [23, p. 433]):Y=average miles per gallons.X=top speed miles per hour of 81 cars.The summary statistics areN=81,n=18,Y¯=33.8346,X¯=112.457,Cy=0.2972,Cx=0.1256,ρyx=−0.6907,β1x=1.9016,β2x=4.1454,QD=5.0,TM=109.5,MR=127.5, and HL=108.75.Population 6 (source: Singh and Chaudhary [21]):Y=area under wheat crop in acres during 1974 in 34 villages.X=area under wheat crop in acres during 1973 in 34 villages.The summary statistics areN=34,n=10,Y¯=856.4117,X¯=199.4412,Cy=0.8561,Cx=0.7531,ρyx=0.4453,β1x=1.1823,β2x=1.0445,QD=89.375,TM=165.562,MR=320.0, and HL=184.0.Population 7 (source: Gujarati [23, p. 433]):Y=average miles per gallons.X=cubic feet of cab space of 81 cars.The summary statistics areN=81,n=18,Y¯=33.8346,X¯=98.7654,Cy=0.2972,Cx=0.2258,ρyx=−0.3683,β1x=0.5902,β2x=0.9202,QD=12.0,TM=101.0,MR=105.0, and HL=98.0.The results based on Populations 1–7 are given in Tables5–16, where we use the following expression to obtain the percent relative efficiency (PRE):(50)PRE=VarY¯^0MSEY¯^jor MSEY¯^jmin×100,where j=0,R,EP,D=S1,K,S2,GK,YK,Ia,SH,Ib,P.Table 5
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=0.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^R105.00437.85639.8130.3256.74100.9746.80Y¯^EP125.14408.93307.9351.1374.82124.7370.22Y¯^D,Y¯^S1,Y¯^K125.262757.47670.34402.44191.23124.73115.69Y¯^S2125.13408.93307.9351.1374.82124.7370.22Y¯^GK127.132898.92683.17402.78191.64131.30116.08Y¯^YK126.95410.92309.9651.4575.33131.1970.74Y¯^Ia127.472781.24676.51405.43192.03132.77116.44Y¯^SH127.533269.75698.26403.00191.68132.89116.15Y¯^Ib127.533269.75698.26403.00191.68132.89116.15Y¯^P717.8811691.202755.202084.594286.18681.20803.92Table 6
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=Cx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.12401.24306.9551.2074.84124.7370.22Y¯^GK127.132894.41683.11402.78191.64131.29116.08Y¯^YK126.93403.21308.9751.5275.36131.1770.80Y¯^Ia127.462777.65676.42405.41192.03132.73116.44Y¯^SH127.533249.28698.08403.00191.68132.90116.15Y¯^Ib127.533280.52698.35403.00191.68132.98116.15Y¯^P722.7711882.802768.172093.544295.75686.07807.59Table 7
PRE of different estimators w.r.tY¯^0 for all populations using u=Cx,v=ρyx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13401.55306.6748.2773.73124.7369.75Y¯^GK127.132894.59683.09402.82191.65131.29116.09Y¯^YK126.94403.52308.7048.6174.25131.1770.27Y¯^Ia127.462777.79676.40406.17192.07132.74116.45Y¯^SH127.533250.12698.03403.11191.69132.87116.15Y¯^Ib127.533280.06698.37402.96191.68132.98116.14Y¯^P722.1111874.742771.851726.023877.55685.02777.66Table 8
PRE of different estimators w.r.tY¯^0 for all populations using u=β1x,v=ρyx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13405.58307.4749.8074.75124.7370.39Y¯^GK127.132896.96683.14402.80191.64131.29116.08Y¯^YK126.94407.56309.5150.1275.26131.1870.91Y¯^Ia127.462779.64676.47405.75192.03132.75116.43Y¯^SH127.533260.80698.18403.05191.68132.88116.14Y¯^Ib127.533274.32698.30402.98191.68132.89116.15Y¯^P720.9911773.342761.121907.814258.56683.63814.09Table 9
PRE of different estimators w.r.tY¯^0 for all populations using u=β2x,v=β1x.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2124.91405.55307.5450.9674.91124.7270.03Y¯^GK127.102896.94683.14402.79191.64131.28116.08Y¯^YK126.66407.54309.5751.2875.42131.1670.56Y¯^Ia127.362779.63676.47405.47192.03132.71116.44Y¯^SH127.463260.73698.19403.01191.68132.85116.15Y¯^Ib127.573274.36698.29403.00191.68132.91116.15Y¯^P787.2411773.982760.242060.524321.19688.53793.55Table 10
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=ρyx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13401.47306.8650.7474.69124.7370.11Y¯^GK127.132894.54683.10402.79191.64131.29116.08Y¯^YK126.94403.43308.8951.0675.20131.1870.73Y¯^Ia127.462777.75676.41405.52191.04132.74116.44Y¯^SH127.533249.88698.06403.02191.68132.87116.15Y¯^Ib127.533280.19698.36403.00191.68132.90116.15Y¯^P720.9311876.992769.312031.024233.73684.08797.96Table 11
PRE of different estimators w.r.tY¯^0 for all populations using u=QD,v=TM.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.10394.85305.6554.9478.39124.7272.44Y¯^GK127.122890.67683.03402.74191.63131.27116.07Y¯^YK126.90396.80307.6655.2578.88131.1372.94Y¯^Ia127.452774.88676.31405.69191.93132.68116.38Y¯^SH127.513232.49697.84402.90191.66132.83116.13Y¯^Ib127.543290.27698.47403.09191.70132.92116.16Y¯^P731.6412050.592785.682694.246116.69693.22946.39Table 12
PRE of different estimators w.r.tY¯^0 for all populations using u=QD,v=MR.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.07364.40300.6255.0878.88124.7072.44Y¯^GK127.122872.92682.73402.74191.63131.25116.07Y¯^YK126.86366.28302.5955.3879.37131.0772.94Y¯^Ia127.432764.35675.88404.67191.92132.60116.38Y¯^SH127.503154.95696.93402.89191.66132.77116.13Y¯^Ib127.553250.10698.97403.09191.70132.95116.16Y¯^P742.0912980.432855.802718.686448.68704.53952.27Table 13
PRE of different estimators w.r.tY¯^0 for all populations using u=HL,v=TM.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.12401.76306.7851.5775.01124.7270.51Y¯^GK127.132894.72683.10402.78191.64131.29116.08Y¯^YK126.93403.73308.8151.8975.52131.1771.03Y¯^Ia127.462777.89676.41405.33191.03132.72116.43Y¯^SH127.523250.67698.05403.01191.68132.90116.15Y¯^Ib127.533279.76698.36403.01191.68132.90116.15Y¯^P723.6811869.392770.362146.796463.21687.02820.74Table 14
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=QD.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2122.04252.99306.7854.1975.73121.6973.30Y¯^GK126.962811.58683.10402.75191.64130.55116.07Y¯^YK123.56255.14308.8154.5076.24126.9073.79Y¯^Ia126.962767.31676.41404.82192.00130.59116.35Y¯^SH127.152913.59683.43402.91191.67131.23116.12Y¯^Ib128.274958.19742.29403.07191.68137.32116.17Y¯^P1365.8220418.935286.472559.994675.491388.081010.57Table 15
PRE of different estimators w.r.tY¯^0 for all populations using u=ρyx,v=QD.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2117.96251.52210.4047.0873.37117.2657.51Y¯^GK126.872810.84677.78402.84191.65130.24116.16Y¯^YK119.38253.68212.2647.8373.89122.1358.15Y¯^Ia126.782767.73671.21406.54192.08129.90117.01Y¯^SH126.982910.95682.86403.16191.69131.58116.32Y¯^Ib245.875081.53753.70402.95191.67—116.12Y¯^P2445.8520609.075537.931596.883752.542631.19362.22Table 16
PRE of different estimators w.r.tY¯^0 for all populations using u=ρyx,v=β2x.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13372.64299.2151.5373.63124.7269.49Y¯^GK127.132877.70682.65402.78191.65131.27116.09Y¯^YK126.94374.53301.1851.8574.15131.1270.02Y¯^Ia127.472767.73675.76405.34192.07132.66116.46Y¯^SH127.533175.50696.68402.99191.69132.81116.15Y¯^Ib127.533331.29699.12403.01191.68132.93116.14Y¯^P719.3612705.242876.092140.113841.23696.43763.87The results based on 7 real datasets are given in Tables5–16.In Tables5–16, PRE values are given based on summary statistics of seven real datasets to observe the performances of all estimators. One can see that the estimators Y¯^j,j=D,S1,K,Y¯^j,j=SH,1b, and Y¯^j,j=EP,S2 are equally efficient. Also, some estimators Y¯^j,j=R,EP,S2,YK show very poor performances in Populations 4, 5, and 7 because of negative correlations. In Tables 5–16, all of the estimators have very large PRE for Population 2 due to the highest value of ρyx=0.9817. Also, in all cases, the proposed estimator shows the best performance.


## 5.1. Simulation Study

In this section, a simulation study is conducted to assess the performances of all estimators considered here. We consider two finite populations of size 1000 generated from a bivariate normal distribution with the same theoretical means ofY,X as μ=5,5 but different covariance matrices as given below.Population 1:(48)Σ=91.91.94.Population 2:(49)Σ=61.11.12.For each population, we consider a sample of sizes 50 and 100. The following steps are performed to carry out the simulation study.Step 1. Select a SRSWOR of size n from a population of size N.Step 2. Use a sample data from Step 1 to find the MSE values of all the estimators.Step 3. Steps 1 and 2 are repeated 10,000 times.Step 4. Obtain 10,000 values for MSEs.Step 5. Average of 10,000 values obtained in Step 4 represents the simulated MSE of each estimator.The simulated MSEs based on Populations 1 and 2 for sample sizes 50 and 100 are given in Table4.Table 4
MSE values under simulation for Populations 1 and 2 when usingu=1,v=0.
EstimatorPopulation 1Population 25010050100Y¯^00.1870690.0872320.129170.055353Y¯^R0.1813630.0832890.119170.053420Y¯^EP0.1655960.0764920.1112210.051525Y¯^D0.1646500.0762730.1103070.051128Y¯^S10.1654810.0763730.1106460.051170Y¯^S20.1655960.0762140.1109020.051525Y¯^YK0.1647040.0769140.1112210.051409Y¯^K0.1663450.0769240.1133350.052623Y¯^GK0.1644600.0760740.1102250.051036Y¯^Ia0.1656490.0762540.1106440.051100Y¯^SH0.1645210.0760720.1102450.051034Y¯^Ib0.1645210.0760720.1102450.051034Y¯^P0.0161010.0076280.0980410.003814From Table4, we observed that as we increase the sample size, the MSE values decrease in both populations which are on expected lines. Irfan et al.’s estimator [11] becomes Shabbir et al.’s estimator [7] when u=1,v=0, so MSE values are the same. The proposed estimator Y¯^P shows the least MSE values as compared to all other considered estimators.


## 5.2. Real Datasets

We use the following 7 real datasets for a numerical study.Population 1 (source: Singh and Chaudhary [21]):Y=area under wheat crop in acres during 1974 in 34 villages.X=area under wheat crop in acres during 1971 in 34 villages.The summary statistics areN=34,n=20,Y¯=856.4117,X¯=208.8823,Cy=0.8561,Cx=0.7205,ρyx=0.4491,β1x=0.9782,β2x=0.0978,QD=80.25,TM=162.25,MR=284.5, and HL=190.0.Population 2 (source: Cochran [19]):Y=number of inhabitants (in 1000’s) in 1930.X=number of inhabitants (in 1000’s) in 1920.The summary statistics areN=49,n=12,Y¯=127.7959,X¯=103.1429,Cy=0.9634,Cx=1.0122,ρyx=0.9817,β1x=2.2553,β2x=5.1412,QD=38.50,TM=72.75,MR=254.5, and HL=77.25.Population 3 (source: Singh and Mangat [22]):Y=number of tube wells.X=net irrigated area in hectares for 69 villages.The summary statistics areN=69,n=15,Y¯=135.2608,X¯=345.7536,Cy=0.8422,Cx=0.8479,ρyx=0.9224,β1x=2.3808,β2x=7.2159,QD=138.0,TM=274.5,MR=900, and HL=277.0.Population 4 (source: Singh and Mangat [22]):Y=average duration of sleep in hours.X=age of a person.The summary statistics areN=30,n=10,Y¯=6.3767,X¯=66.9333,Cy=0.1633,Cx=0.1436,ρyx=−0.8669,β1x=0.2984,β2x=−0.7686,QD=7.25,TM=66.5,MR=69.0, and HL=67.0.Population 5 (source: Gujarati [23, p. 433]):Y=average miles per gallons.X=top speed miles per hour of 81 cars.The summary statistics areN=81,n=18,Y¯=33.8346,X¯=112.457,Cy=0.2972,Cx=0.1256,ρyx=−0.6907,β1x=1.9016,β2x=4.1454,QD=5.0,TM=109.5,MR=127.5, and HL=108.75.Population 6 (source: Singh and Chaudhary [21]):Y=area under wheat crop in acres during 1974 in 34 villages.X=area under wheat crop in acres during 1973 in 34 villages.The summary statistics areN=34,n=10,Y¯=856.4117,X¯=199.4412,Cy=0.8561,Cx=0.7531,ρyx=0.4453,β1x=1.1823,β2x=1.0445,QD=89.375,TM=165.562,MR=320.0, and HL=184.0.Population 7 (source: Gujarati [23, p. 433]):Y=average miles per gallons.X=cubic feet of cab space of 81 cars.The summary statistics areN=81,n=18,Y¯=33.8346,X¯=98.7654,Cy=0.2972,Cx=0.2258,ρyx=−0.3683,β1x=0.5902,β2x=0.9202,QD=12.0,TM=101.0,MR=105.0, and HL=98.0.The results based on Populations 1–7 are given in Tables5–16, where we use the following expression to obtain the percent relative efficiency (PRE):(50)PRE=VarY¯^0MSEY¯^jor MSEY¯^jmin×100,where j=0,R,EP,D=S1,K,S2,GK,YK,Ia,SH,Ib,P.Table 5
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=0.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^R105.00437.85639.8130.3256.74100.9746.80Y¯^EP125.14408.93307.9351.1374.82124.7370.22Y¯^D,Y¯^S1,Y¯^K125.262757.47670.34402.44191.23124.73115.69Y¯^S2125.13408.93307.9351.1374.82124.7370.22Y¯^GK127.132898.92683.17402.78191.64131.30116.08Y¯^YK126.95410.92309.9651.4575.33131.1970.74Y¯^Ia127.472781.24676.51405.43192.03132.77116.44Y¯^SH127.533269.75698.26403.00191.68132.89116.15Y¯^Ib127.533269.75698.26403.00191.68132.89116.15Y¯^P717.8811691.202755.202084.594286.18681.20803.92Table 6
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=Cx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.12401.24306.9551.2074.84124.7370.22Y¯^GK127.132894.41683.11402.78191.64131.29116.08Y¯^YK126.93403.21308.9751.5275.36131.1770.80Y¯^Ia127.462777.65676.42405.41192.03132.73116.44Y¯^SH127.533249.28698.08403.00191.68132.90116.15Y¯^Ib127.533280.52698.35403.00191.68132.98116.15Y¯^P722.7711882.802768.172093.544295.75686.07807.59Table 7
PRE of different estimators w.r.tY¯^0 for all populations using u=Cx,v=ρyx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13401.55306.6748.2773.73124.7369.75Y¯^GK127.132894.59683.09402.82191.65131.29116.09Y¯^YK126.94403.52308.7048.6174.25131.1770.27Y¯^Ia127.462777.79676.40406.17192.07132.74116.45Y¯^SH127.533250.12698.03403.11191.69132.87116.15Y¯^Ib127.533280.06698.37402.96191.68132.98116.14Y¯^P722.1111874.742771.851726.023877.55685.02777.66Table 8
PRE of different estimators w.r.tY¯^0 for all populations using u=β1x,v=ρyx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13405.58307.4749.8074.75124.7370.39Y¯^GK127.132896.96683.14402.80191.64131.29116.08Y¯^YK126.94407.56309.5150.1275.26131.1870.91Y¯^Ia127.462779.64676.47405.75192.03132.75116.43Y¯^SH127.533260.80698.18403.05191.68132.88116.14Y¯^Ib127.533274.32698.30402.98191.68132.89116.15Y¯^P720.9911773.342761.121907.814258.56683.63814.09Table 9
PRE of different estimators w.r.tY¯^0 for all populations using u=β2x,v=β1x.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2124.91405.55307.5450.9674.91124.7270.03Y¯^GK127.102896.94683.14402.79191.64131.28116.08Y¯^YK126.66407.54309.5751.2875.42131.1670.56Y¯^Ia127.362779.63676.47405.47192.03132.71116.44Y¯^SH127.463260.73698.19403.01191.68132.85116.15Y¯^Ib127.573274.36698.29403.00191.68132.91116.15Y¯^P787.2411773.982760.242060.524321.19688.53793.55Table 10
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=ρyx.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13401.47306.8650.7474.69124.7370.11Y¯^GK127.132894.54683.10402.79191.64131.29116.08Y¯^YK126.94403.43308.8951.0675.20131.1870.73Y¯^Ia127.462777.75676.41405.52191.04132.74116.44Y¯^SH127.533249.88698.06403.02191.68132.87116.15Y¯^Ib127.533280.19698.36403.00191.68132.90116.15Y¯^P720.9311876.992769.312031.024233.73684.08797.96Table 11
PRE of different estimators w.r.tY¯^0 for all populations using u=QD,v=TM.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.10394.85305.6554.9478.39124.7272.44Y¯^GK127.122890.67683.03402.74191.63131.27116.07Y¯^YK126.90396.80307.6655.2578.88131.1372.94Y¯^Ia127.452774.88676.31405.69191.93132.68116.38Y¯^SH127.513232.49697.84402.90191.66132.83116.13Y¯^Ib127.543290.27698.47403.09191.70132.92116.16Y¯^P731.6412050.592785.682694.246116.69693.22946.39Table 12
PRE of different estimators w.r.tY¯^0 for all populations using u=QD,v=MR.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.07364.40300.6255.0878.88124.7072.44Y¯^GK127.122872.92682.73402.74191.63131.25116.07Y¯^YK126.86366.28302.5955.3879.37131.0772.94Y¯^Ia127.432764.35675.88404.67191.92132.60116.38Y¯^SH127.503154.95696.93402.89191.66132.77116.13Y¯^Ib127.553250.10698.97403.09191.70132.95116.16Y¯^P742.0912980.432855.802718.686448.68704.53952.27Table 13
PRE of different estimators w.r.tY¯^0 for all populations using u=HL,v=TM.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.12401.76306.7851.5775.01124.7270.51Y¯^GK127.132894.72683.10402.78191.64131.29116.08Y¯^YK126.93403.73308.8151.8975.52131.1771.03Y¯^Ia127.462777.89676.41405.33191.03132.72116.43Y¯^SH127.523250.67698.05403.01191.68132.90116.15Y¯^Ib127.533279.76698.36403.01191.68132.90116.15Y¯^P723.6811869.392770.362146.796463.21687.02820.74Table 14
PRE of different estimators w.r.tY¯^0 for all populations using u=1,v=QD.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2122.04252.99306.7854.1975.73121.6973.30Y¯^GK126.962811.58683.10402.75191.64130.55116.07Y¯^YK123.56255.14308.8154.5076.24126.9073.79Y¯^Ia126.962767.31676.41404.82192.00130.59116.35Y¯^SH127.152913.59683.43402.91191.67131.23116.12Y¯^Ib128.274958.19742.29403.07191.68137.32116.17Y¯^P1365.8220418.935286.472559.994675.491388.081010.57Table 15
PRE of different estimators w.r.tY¯^0 for all populations using u=ρyx,v=QD.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2117.96251.52210.4047.0873.37117.2657.51Y¯^GK126.872810.84677.78402.84191.65130.24116.16Y¯^YK119.38253.68212.2647.8373.89122.1358.15Y¯^Ia126.782767.73671.21406.54192.08129.90117.01Y¯^SH126.982910.95682.86403.16191.69131.58116.32Y¯^Ib245.875081.53753.70402.95191.67—116.12Y¯^P2445.8520609.075537.931596.883752.542631.19362.22Table 16
PRE of different estimators w.r.tY¯^0 for all populations using u=ρyx,v=β2x.
EstimatorPop. 1Pop. 2Pop. 3Pop. 4Pop. 5Pop. 6Pop. 7Y¯^0100.00100.00100.00100.00100.00100.00100.00Y¯^S2125.13372.64299.2151.5373.63124.7269.49Y¯^GK127.132877.70682.65402.78191.65131.27116.09Y¯^YK126.94374.53301.1851.8574.15131.1270.02Y¯^Ia127.472767.73675.76405.34192.07132.66116.46Y¯^SH127.533175.50696.68402.99191.69132.81116.15Y¯^Ib127.533331.29699.12403.01191.68132.93116.14Y¯^P719.3612705.242876.092140.113841.23696.43763.87The results based on 7 real datasets are given in Tables5–16.In Tables5–16, PRE values are given based on summary statistics of seven real datasets to observe the performances of all estimators. One can see that the estimators Y¯^j,j=D,S1,K,Y¯^j,j=SH,1b, and Y¯^j,j=EP,S2 are equally efficient. Also, some estimators Y¯^j,j=R,EP,S2,YK show very poor performances in Populations 4, 5, and 7 because of negative correlations. In Tables 5–16, all of the estimators have very large PRE for Population 2 due to the highest value of ρyx=0.9817. Also, in all cases, the proposed estimator shows the best performance.


## 6. Conclusion

In this study, we have proposed a general class of exponential-ratio-type estimators for finite population mean in simple random sampling using the conventional and the nonconventional measures. Expressions for biases and MSEs are obtained up to first order of approximation. Two datasets are used for simulation study and seven real datasets are used for efficiency comparisons. The simulated results given in Table4 show that the proposed estimator has the least MSE values as compared to other competitive estimators. Estimators in Table 5 do not use the conventional and nonconventional measures, but such measures are used in Tables 6–16. Based on the results in Tables 4–16, the proposed class of estimators performs substantially better than all other estimators considered here. In Table 5, the estimators Y¯^j,j=R,EP,S2,YK perform very poorly for Populations 4, 5, and 7 because of negative correlation. Estimators Y¯^j,j=S2,YK in Tables 6–16 also performed poorly for Populations 4, 5, and 7 due to negative correlation. We observed that the efficiency of the estimators Y¯^j,j=S2,GK,YK,Ia,Ib (see Tables 6–16) in some situations does not increase much as compared to other estimators. The PRE of Irfan et al.’s estimator [11] was undefined in Table 15 under Population 6. We conclude that use of conventional and nonconventional measures does not play a very major role in increasing the efficiency of existing mean estimators except in some cases. However, we observed a significant gain for our proposed estimator with both the conventional and the nonconventional measures.



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*Source: 3741620-2021-06-29.xml*