Optimality of Ratio-Type Imputation Methods for Estimation of Population Mean Using Higher Order Moment of an Auxiliary Variable

Abstract

This paper introduces some improved ratio methods of imputation using higher-order moment of an auxiliary variable while imputing missing values. It is well-known fact that the optimal ratio estimators attain the MSE of regression estimator, but while using Searls-type transformation this may not always happen. The performance of the proposed imputation methods is investigated relative to the methods proposed by Bhushan et al. (Commun Stat Simul Comput, 2018. https://doi.org/10.1080/03610918.2018.1500595), Mohamed et al. (Commun Stat Simul Comput, 2017. https://doi.org/10.1080/03610918.2016.1208235) and Bhushan and Pandey (J Stat Manag Syst 19(6):755–769, 2016, Commun Stat Theory Methods 47(11):2576–2589, 2018). A comparative study has been carried out, and it has been shown that the proposed methods perform better in comparison with methods proposed by Bhushan et al. (2018), Mohamed et al. (2017) and Bhushan and Pandey (2016, 2018). The theoretical findings are supported by numerical study on two real populations and a simulation study using hypothetical population.

Appendices

Appendix 1

An outline of derivation of Theorem 1. The bias of $T_{rat1}$

$$\begin{aligned} T_{rat1}={\psi _{1}}{\bar{y}}_{r}\left( \dfrac{{\bar{X}}}{{\bar{x}}_{n}}\right) ^{\omega _{1}} \left( \dfrac{S_{x}^{2}}{s_{x\left( n\right) }^{2}}\right) ^{\omega _{2}} \end{aligned}$$

we can write it in the term of $t_{rat1}$, is given by

$$\begin{aligned}&T_{rat1}=\psi _{1}t_{rat1}\\&(T_{rat1}-{\bar{Y}})=(\psi _{1}-1){\bar{Y}}+\psi _{1}(t_{rat1}-{\bar{Y}}) \end{aligned}$$

Taking expectation on both sides

$$\begin{aligned}&E(T_{rat1}-{\bar{Y}})=(\psi _{1}-1){\bar{Y}}+\psi _{1}E(t_{rat1}-{\bar{Y}})\\&Bias\left( T_{rat_{1}}\right) ={\bar{Y}}\left( \psi _{1}-1\right) +{\psi _{1}} Bias(t_{rat_{1}}) \end{aligned}$$

The MSE $T_{rat1}$ is given by

$$\begin{aligned}&MSE\left( T_{rat1}\right) \\&\quad =\left( \psi _{1}-1\right) ^{2}{\bar{Y}}^{2}+\psi _{1}^{2}{\bar{Y}}^{2}\left[ f_{r}C_{y}^2+\omega _{1}f_{n}C_{x}^2+\omega _{2}f_{n}(\lambda _{04}-1)-2\omega _{1}f_{n}{\rho _{yx}}C_{y}C_{x}\right. \\&\qquad -\left. 2\omega _{2}f_{n}C_{y}{\lambda _{12}}+2\omega _{1}\omega _{2}f_{n}C_{x}{\lambda _{03}}\right] +2 \psi _{1}\left( \psi _{1}-1\right) {\bar{Y}}^{2}\left[ -\omega _{1}f_{n}{\rho _{yx}}C_{y}C_{x}-\omega _{2}f_{n}\right. \\&\qquad \left. C_{y}{\lambda _{12}}+\omega _{1}\omega _{2}f_{n}C_{x}{\lambda _{03}}+\frac{\omega _{1}(\omega _{1}+1)}{2}f_{n}C_{x}^2+\frac{\omega _{2}(\omega _{2}+1)}{2}f_{n}(\lambda _{04}-1)\right] \\&MSE\left( T_{rat1}\right) \\&\quad ={\bar{Y}}^{2}\left[ 1+\psi _{1}^2\left\{ 1+f_{r}C_{y}^2+2\omega _{1}^2f_{n}C_{x}^2+2\omega _{2}^2f_{n}(\lambda _{04}-1)-4\omega _{1}f_{n}{\rho _{yx}}C_{y}C_{x}-\right. \right. \\&\qquad \left. \left. 4\omega _{2}f_{n}C_{y}{\lambda _{12}}+4\omega _{1}\omega _{2}f_{n}C_{x}{\lambda _{03}}+\omega _{1}f_{n}C_{x}^2+\omega _{2}f_{n}(\lambda _{04}-1)\right\} -2\psi _{1}\left\{ 1+\right. \right. \\&\qquad \left. \left. \frac{\omega _{1}^2}{2}f_{n}C_{x}^2+\frac{\omega _{2}^2}{2}f_{n}(\lambda _{04}-1)+\frac{\omega _{1}}{2}f_{n}C_{x}^2+\frac{\omega _{2}}{2}f_{n}(\lambda _{04}-1)-\omega _{1}f_{n}{\rho _{yx}}C_{y}C_{x}-\right. \right. \\&\qquad \left. \left. \omega _{2}f_{n}C_{y}{\lambda _{12}}+\omega _{1}\omega _{2}f_{n}C_{x}{\lambda _{03}}\right\} \right] \end{aligned}$$

which can be expressed as

$$\begin{aligned} MSE\left( T_{rat1}\right) =\left[ 1+\psi _{1}^2A_{1}-2\psi _{1}B_{1}\right] \end{aligned}$$

For optimum value of $\psi _{1}$ differentiating the $MSE\left( T_{rat1}\right)$ with respect to $\psi _{1}$ and equating to zero, we get

$$\begin{aligned} \psi _{1opt}=\dfrac{B_{1}}{A_{1}} \end{aligned}$$

substituting the optimum value of $\psi _{1}$ in $MSE\left( T_{rat1}\right)$, we get the minimum MSE

$$\begin{aligned} min.MSE\left( T_{rat1}\right) ={\bar{Y}}^2\left( 1-\frac{B_{1}^2}{A_{1}}\right) \end{aligned}$$

The derivations for other estimators $T_{rat2}$ and $T_{rat3}$ can be done on similar lines. In general, we have

$$\begin{aligned} MSE\left( T_{rati}\right) =\left[ 1+\psi _{i}^2A_{i}-2\psi _{i}B_{i}\right] \end{aligned}$$

The optimum values of scalars involved are tabulated below for ready reference:

$$\begin{aligned} \psi _{iopt}= & {} \dfrac{B_{i}}{A_{i}}; i=1,2,3.\\ A_{1}= & {} 1+f_{r}C_{y}^2+f_{n}\left\{ 2\omega _{1}^2C_{x}^2+2\omega _{2}^2(\lambda _{04}-1)-4\omega _{1}{\rho _{yx}}C_{y}C_{x}-4\omega _{2}C_{y}{\lambda _{12}}+\right. \\&\left. 4\omega _{1}\omega _{2}C_{x}{\lambda _{03}}+\omega _{1}C_{x}^2+\omega _{2}(\lambda _{04}-1)\right\} \\ B_{1}= & {} 1+f_{n}\left\{ \frac{\omega _{1}^2}{2}C_{x}^2+\frac{\omega _{2}^2}{2}(\lambda _{04}-1)+\frac{\omega _{1}}{2}C_{x}^2+\frac{\omega _{2}}{2}(\lambda _{04}-1)-\omega _{1}{\rho _{yx}}C_{y}C_{x}- \right. \\&\left. \omega _{2}C_{y}{\lambda _{12}}+\omega _{1}\omega _{2}C_{x}{\lambda _{03}}\right\} \\ A_{2}= & {} 1+f_{r}\left\{ C_{y}^2+2\omega _{3}^2C_{x}^2+2\omega _{4}^2(\lambda _{04}-1)-4\omega _{3}{\rho _{yx}}C_{y}C_{x}-4\omega _{4}C_{y}{\lambda _{12}}+\right. \\&\left. 4\omega _{3}\omega _{4}C_{x}{\lambda _{03}}+\omega _{3}C_{x}^2+\omega _{4}(\lambda _{04}-1)\right\} \\ B_{2}= & {} 1+f_{r}\left\{ \frac{\omega _{3}^2}{2}C_{x}^2+\frac{\omega _{4}^2}{2}(\lambda _{04}-1)+\frac{\omega _{3}}{2}C_{x}^2+\frac{\omega _{4}}{2}(\lambda _{04}-1)-\omega _{3}{\rho _{yx}}C_{y}C_{x}- \right. \\&\left. \omega _{4}C_{y}{\lambda _{12}}+\omega _{3}\omega _{4}C_{x}{\lambda _{03}}\right\} \\ A_{3}= & {} 1+f_{r}C_{y}^2+f_{rn}\left\{ 2\omega _{5}^2C_{x}^2+2\omega _{6}^2(\lambda _{04}-1)-4\omega _{5}{\rho _{yx}}C_{y}C_{x}-4\omega _{6}C_{y}{\lambda _{12}}+\right. \\&\left. 4\omega _{5}\omega _{6}C_{x}{\lambda _{03}}+\omega _{5}C_{x}^2+\omega _{6}(\lambda _{04}-1)\right\} \\ B_{6}= & {} 1+f_{rn}\left\{ \frac{\omega _{5}^2}{2}C_{x}^2+\frac{\omega _{6}^2}{2}(\lambda _{04}-1)+\frac{\omega _{5}}{2}C_{x}^2+\frac{\omega _{6}}{2}(\lambda _{04}-1)-\omega _{5}{\rho _{yx}}C_{y}C_{x}- \right. \\&\left. \omega _{6}C_{y}{\lambda _{12}}+\omega _{5}\omega _{6}C_{x}{\lambda _{03}}\right\} \end{aligned}$$

where $\omega _{1}=\omega _{3}=\omega _{5}=\dfrac{C_y(\rho (\lambda _{04}-1)-\lambda _{12}\lambda _{03})}{C_x(\lambda _{04}-1-\lambda _{03}^2)}$ and $\omega _{2}=\omega _{4}=\omega _{6}=\dfrac{C_y(\lambda _{12}-\rho _{yx}\lambda _{03})}{(\lambda _{04}-1-\lambda _{03}^2)}$ are used as optimizing value of the constant in this study, which are the optimum values of these scalars when $\psi _{i}=1$ is set in the respective estimators.

Appendix 2

N$<-$ 2000

X$<-$ rep(0,N);

Y$<-$ rep(0,N);

x1$<-$rgamma(N,0.085,1.2)

y1$<-$rgamma(N,0.109,2.0)

X$<-$ 2.4+x1

SX$<-$ sqrt(var(x1))

SY$<-$ sqrt(var(y1))

rho$<-$ 0.6

$Y<- 2.8+(sqrt(1-rho^2)*y1)+(rho*(SY/SX)*x1)$

set.seed(123456789)

$N<- 2000$

$X<- rep(0,N);$

$Y<- rep(0,N);$

$x1<-rgamma(N,0.085,1.2)$

$y1<-rgamma(N,0.109,2.0)$

$X<- 2.4+x1$

$SX<- sqrt(var(x1))$

$SY<- sqrt(var(y1))$

$rho<- 0.6$

$Y<- 2.8+(sqrt(1-rho^2)*y1)+(rho*(SY/SX)*x1)$

$df = data.frame(X, Y)$

$X1<- df$X

X2$<-$ dfY

$Sxy<-cov(X1,X2);Sxy$

$Vx<-var(X1);Vx$

$Vy<-var(X2);Vy$

$Sx<-sd(X1);Sx$

$Sy<-sd(X2);Sy$

$ro<-cor(X1, X2);ro$

$R<-(Ybr/Xbr);R$

$beta<-Sxy/Vx;beta$

$n<-200$

$r<-160$

$fn<-(1/n-1/N)$

$fr<-(1/r-1/N);$

$frn<-(1/r-1/n);$

$Ybr<-mean(X2);Ybr$

$Xbr<-mean(X1);Xbr$

$Cy<-Sy/Ybr;Cy$

$Cx<-Sx/Xbr;Cx$

$Beta<-(ro*(Cy/Cx));Beta$

$mu11<-sum((X2-Ybr)*(X1-Xbr))/(N-1)$

$mu12<-sum((X2-Ybr)*((X1-Xbr)^2))/(N-1)$

$mu02<-sum((X1-Xbr)^2)/(N-1)$

$mu20<-sum((X2-Ybr)^2)/(N-1)$

$mu03<-sum((X1-Xbr)^3)/(N-1)$

$mu04<-sum((X1-Xbr)^4)/(N-1)$

$lambda12<-((mu12)/((sqrt(mu20))*mu02));lambda12$

$lambda03<-((mu03)/((mu02)^(3/2)));lambda03$

$lambda04<-((mu04)/(mu02^2));lambda04$

$beta1<-((Sxy*(lambda04-1))-(Sy*Sx*lambda12*lambda03))/(Vx*(lambda04-1-(lambda03^2)));beta1$

$beta2<-((Sy*Sx*lambda12)-(Sxy*lambda03))/((Sx^3)*(lambda04-1-(lambda03^2)));beta2$

Strategy1

$A1<-1/(1+(fr*(Cy^2))-(fn*(ro^2)*(Cy^2)));$

$B1<-(ro*Sy)/(Sx*(1+(fr*(Cy^2))-(fn*(ro^2)*(Cy^2))));$

$C1=(1+(fr*Cy^2)+(2*(Beta^2)*fn*Cx^2)+((Beta*fn)*((Cx^2)-(4*ro*Cy*Cx))));$

$D1=(1+(((Beta^2)/2)*fn*Cx^2)+(((Beta/2)*fn)*((Cx^2)-(2*ro*Cy*Cx))));D1$

$alpha1=(D1/C1);$

$d1<-(Cy*((ro*(lambda04-1))-(lambda12*lambda03)))/(Cx*(lambda04-1-(lambda03^2)))$

$d2<-(Cy*(lambda12-(ro*lambda03)))/(lambda04-1-(lambda03^2));$

$K1<-(1+(2*(d1^2)*fn*Cx^2)+(2*(d2^2)*fn*(lambda04-1))+(d1*fn*((Cx^2)-(4*ro*Cy*Cx)))$

$+(d2*fn*((lambda04-1)-(4*Cy*lambda12)))+(4*d1*d2*fn*Cx*lambda03));K1 K2<-(1+(((d1^2)/2)*fn*Cx^2)+(((d2^2)/2)*fn*(lambda04-1))+((d1/2)*fn*((Cx^2)-(2*ro*Cy*Cx)))+((d2/2)*fn*((lambda04-1)-(2*Cy*lambda12)))+(d1*d2*fn*Cx*lambda03));K1$

$M1<-(fr*Vy);M1$

$M2<-((fr*Vy)+(fn*(((R^2)*Vx)-(2*R*Sxy))));M2$

$M3<-((fr*Vy)-(fn*Vy*(ro^2)));M3$

$M4<-((Ybr^2*M3)/((Ybr^2)+M3));M4$

$M5<-((fr*Vy)-((fn*Vy)*((ro^2)+(((lambda12-(ro*lambda03))^2)/(lambda04-1-(lambda03^2))))));M5$

$M6<-((Ybr^2*M5)/((Ybr^2)+M5));M6$

$M7<-((Ybr^2)*(1-((D1^2)/C1)));M7$

$M8<-((Ybr^2)*(1-((K1^2)/K1)));M8$

$P1<-100$

$P2<-(M1/M2)*100;P2$

$P3<-(M1/M3)*100;P3$

$P4<-(M1/M4)*100;P4$

$P5<-(M1/M5)*100;P5$

$P6<-(M1/M6)*100;P6$

$P7<-(M1/M7)*100;P7$

$P8<-(M1/M8)*100;P8$

$g1<-((Ybr^2)/((Ybr^2)+(fr*Vy)-((fn*Vy)*((ro^2)+(((lambda12-(ro*lambda03))^2)/(lambda04-1-(lambda03^2)))))));g1$

$g2<-(g1*Sy*((ro*(lambda04-1))-(lambda12*lambda03)))/(Sx*(lambda04-1-(lambda03^2)));g2$

$g3<-(g1*Sy*(lambda12-(ro*lambda03)))/(Vx*(lambda04-1-(lambda03^2)));g3$

Strategy2

$A2<-1/(1+(fr*(Cy^2))-(fr*(ro^2)*(Cy^2)));A2$

$B2<-(ro*Sy)/(Sx*(1+(fr*(Cy^2))-(fr*(ro^2)*(Cy^2))));B2$

$C2=(1+(fr*Cy^2)+(2*(Beta^2)*fr*Cx^2)+((Beta*fr)*((Cx^2)-(4*ro*Cy*Cx))));C2$

$D2=(1+(((Beta^2)/2)*fr*Cx^2)+(((Beta/2)*fr)*((Cx^2)-(2*ro*Cy*Cx))));D2$

$alpha2=(D2/C2);alpha2$

$M21<-(fr*Vy);M21$

$M22<-((fr*Vy)+(fr*(((R^2)*Vx)-(2*R*Sxy))));M22$

$M23<-((fr*Vy)-(fr*Vy*(ro^2)));M23$

$M24<-((Ybr^2*M23)/((Ybr^2)+M23));M24$

$M25<-((fr*Vy)-((fr*Vy)*((ro^2)+(((lambda12-(ro*lambda03))^2)/(lambda04-1-(lambda03^2))))));M25$

$M26<-((Ybr^2*M25)/((Ybr^2)+M25));M26$

$P21<-100$

$P22<-(M21/M22)*100;P22$

$P23<-(M21/M23)*100;P23$

$P24<-(M21/M24)*100;P24$

$P25<-(M21/M25)*100;P25$

$P26<-(M21/M26)*100;P26$

$g4<-((Ybr^2)/((Ybr^2)+(fr*Vy)-((fr*Vy)*((ro^2)+(((lambda12-(ro*lambda03))^2)/(lambda04-1-(lambda03^2)))))));g4$

$g5<-(g4*Sy*((ro*(lambda04-1))-(lambda12*lambda03)))/(Sx*(lambda04-1-(lambda03^2)));g5$

$g6<-(g4*Sy*(lambda12-(ro*lambda03)))/(Vx*(lambda04-1-(lambda03^2)));g6$

Strategy3

$A3<-1/(1+(fr*(Cy^2))-(frn*(ro^2)*(Cy^2)));A3$

$B3<-(ro*Sy)/(Sx*(1+(fr*(Cy^2))-(frn*(ro^2)*(Cy^2))));B3$

$C3=(1+(fr*Cy^2)+(2*(Beta^2)*frn*Cx^2)+((Beta*frn)*((Cx^2)-(4*ro*Cy*Cx))));C3$

$D3=(1+(((Beta^2)/2)*frn*Cx^2)+(((Beta/2)*frn)*((Cx^2)-(2*ro*Cy*Cx))));D3$

$alpha3=(D3/C3);alpha3$

$M31<-(fr*Vy);M31$

$M32<-((fn*Vy)+(frn*(Vy+((R^2)*Vx)-(2*R*Sxy))));M32$

$M33<-((fn*Vy)+(frn*Vy*(1-(ro^2))));M33$

$M34<-((Ybr^2*M33)/((Ybr^2)+M33));M34$

$M35<-((fn*Vy)+((frn*Vy)*(1-(ro^2)-(((lambda12-(ro*lambda03))^2)/(lambda04-1-(lambda03^2))))));M35$

$M36<-((Ybr^2*M35)/((Ybr^2)+M35));M36$

$P31<-100$

$P32<-(M31/M32)*100;P32$

$P33<-(M31/M33)*100;P33$

$P34<-(M31/M34)*100;P34$

$P35<-(M31/M35)*100;P35$

$P36<-(M31/M36)*100;P36$

$g7<-((Ybr^2)/((Ybr^2)+(fr*Vy)-((frn*Vy)*((ro^2)+(((lambda12-(ro*lambda03))^2)/(lambda04-1-(lambda03^2)))))));g7$

$g8<-(g7*Sy*((ro*(lambda04-1))-(lambda12*lambda03)))/(Sx*(lambda04-1-(lambda03^2)));g8$

$g9<-(g7*Sy*(lambda12-(ro*lambda03)))/(Vx*(lambda04-1-(lambda03^2)));g9$

$n<- 200$

$ll<- 200$

$r<- 160$

$S<-rep(0,n)$

$XX<- rep(0,r);$

$YY<- rep(0,r);$

$Xn<- rep(0,n);$

$Yn<- rep(0,n);$

$ZX<-rep(0,ll);$

$M11<- rep(0,ll);$

$M12<- rep(0,ll);$

$M13<- rep(0,ll);$

$M14<- rep(0,ll);$

$M15<- rep(0,ll);$

$M16<-rep(0,ll);$

$M17<-rep(0,ll);$

$M21<- rep(0,ll);$

$M22<- rep(0,ll);$

$M23<- rep(0,ll);$

$M24<- rep(0,ll);$

$M25<- rep(0,ll);$

$M26<-rep(0,ll);$

$M27<-rep(0,ll);$

$M31<- rep(0,ll);$

$M32<- rep(0,ll);$

$M33<- rep(0,ll);$

$M34<- rep(0,ll);$

$M35<- rep(0,ll);$

$M36<-rep(0,ll);$

$M37<-rep(0,ll);$

$ZXM<-rep(0,n);$

$ZYM<- rep(0,n);$

$xyr<-0$

$S<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S11<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S12<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S13<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S14<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S15<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S16<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S17<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S21<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S22<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S23<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S24<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S25<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S26<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S27<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S31<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S32<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S33<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S34<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S35<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S36<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$S37<-matrix(data=0, nrow=ll, ncol=n, byrow=$”T”)

$ZY<-rep(0,ll);$

$W<- seq(1:N)$

for (it in 1:n)

$D<-sample(W, n, replace = FALSE, prob = NULL)$

for (kk in 1:n)

Tn$<-$ D[kk]

Xn[kk]$<-$ X1[Tn]

Yn[kk]$<-$ X2[Tn]

for (j in 1:ll)

T$<-$sample(D, r, replace = FALSE, prob = NULL)

for (k in 1:r)

T1$<-$ T[k]

XX[k]$<-$ X1[T1]

YY[k]$<-$ X2[T1]

XX

YY

ZX[j]$<-$mean(XX);

ZY[j]$<-$mean(YY)

yr$<-$mean(YY);

$M11[j]<-((yr-mean(X2))^2);$

$yrat1<-mean(YY)*mean(X1)/mean(Xn);$

$M12[j]<-((yrat1-mean(X2))^2);$

$yreg1<-mean(YY)+beta*(mean(X1)-mean(Xn));$

$M13[j]<-((yreg1-mean(X2))^2)$

$treg1<-(A1*mean(YY))+(B1*(mean(X1)-mean(Xn)));$

$M14[j]<-((treg1-mean(X2))^2);$

$ynew_ch1<-(mean(YY)+(beta1*(mean(X1)-mean(Xn)))+(beta2*(var(X1)-var(Xn))))$; $M15[j]<-((ynew_ch1-mean(X2))^2);$

$T11<-((g1*mean(YY))+g2*(mean(X1)-mean(Xn))+g3*(var(X1)-var(Xn)));$

$M16[j]<-((T11-mean(X2))^2)$

$Tr1<-(alpha1*mean(YY)*((mean(X1)/mean(Xn))^Beta));$

$M17[j]<-((Tr1-mean(X2))^2);$

$yr<-mean(YY)$;

$M21[j]<-((yr-mean(X2))^2);$

$yrat2<-mean(YY)*mean(X1)/mean(XX)$;

$M22[j]<-((yrat2-mean(X2))^2);$

$yreg2<-mean(YY)+beta*(mean(X1)-mean(XX))$;

$M23[j]<-((yreg2-mean(X2))^2);$

$treg2<-(A2*mean(YY))+(B2*(mean(X1)-mean(XX)))$;

$M24[j]<-((treg2-mean(X2))^2);$

$ynew_ch2<-(mean(YY)+(beta1*(mean(X1)-mean(XX)))+(beta2*(var(X1)-var(XX))));$

$M25[j]<-((ynew_ch2-mean(X2))^2);$

$T21<-((g4*mean(YY))+g5*(mean(X1)-mean(XX))+g6*(var(X1)-var(XX)));$

$M26[j]<-((T21-mean(X2))^2)$

$Tr2<-(alpha2*mean(YY)*((mean(X1)/mean(XX))^Beta));$

$M27[j]<-((Tr2-mean(X2))^2);$

$yr<-mean(YY);$

$M31[j]<-((yr-mean(X2))^2)$;

$yrat3<-mean(YY)*mean(Xn)/mean(XX)$;

$M32[j]<-((yrat3-mean(X2))^2);$

$yreg3<-mean(YY)+beta*(mean(Xn)-mean(XX));$

$M33[j]<-((yreg3-mean(X2))^2);$

$treg3<-(A3*mean(YY))+(B3*(mean(Xn)-mean(XX)));$

$M34[j]<-((treg3-mean(X2))^2);$

$ynew_ch3<-(mean(YY)+(beta1*(mean(Xn)-mean(XX)))+(beta2*(var(Xn)-var(XX))))$; $M35[j]<-((ynew_ch3-mean(X2))^2);$

$T31<-((g7*mean(YY))+g8*(mean(Xn)-mean(XX))+g9*(var(Xn)-var(XX)));$

$M36[j]<-((T31-mean(X2))^2)$

$Tr3<-(alpha3*mean(YY)*((mean(Xn)/mean(XX))^Beta));$

$M37[j]<-((Tr3-mean(X2))^2);$

S[it,]$<-$ ZX

S11[it,]$<-$M11

S12[it,]$<-$M12

S13[it,]$<-$M13

S14[it,]$<-$M14

S15[it,]$<-$M15

S16[it,]$<-$M16

S17[it,]$<-$M17

S21[it,]$<-$M21

S22[it,]$<-$M22

S23[it,]$<-$M23

S24[it,]$<-$M24

S25[it,]$<-$M25

S26[it,]$<-$M26

S27[it,]$<-$M27

S31[it,]$<-$M31

S32[it,]$<-$M32

S33[it,]$<-$M33

S34[it,]$<-$M34

S35[it,]$<-$M35

S36[it,]$<-$M36

S37[it,]$<-$M37

Strategy1

MSE1$<-$(sum(S11))/(n*ll);MSE1

MSE2$<-$(sum(S12))/(n*ll);MSE2

MSE3$<-$(sum(S13))/(n*ll);MSE3

MSE4$<-$(sum(S14))/(n*ll);MSE4

MSE5$<-$(sum(S15))/(n*ll);MSE5

MSE6$<-$(sum(S16))/(n*ll);MSE6

MSE7$<-$(sum(S17))/(n*ll);MSE7

PRE1$<-$100

PRE2$<-$(MSE1/MSE2)*100;PRE2

PRE3$<-$(MSE1/MSE3)*100;PRE3

PRE4$<-$(MSE1/MSE4)*100;PRE4

PRE5$<-$(MSE1/MSE5)*100;PRE5

PRE6$<-$(MSE1/MSE6)*100;PRE6

PRE7$<-$(MSE1/MSE7)*100;PRE7

Strategy2

MSE21$<-$(sum(S21))/(n*ll);MSE21

MSE22$<-$(sum(S22))/(n*ll);MSE22

MSE23$<-$(sum(S23))/(n*ll);MSE23

MSE24$<-$(sum(S24))/(n*ll);MSE24

MSE25$<-$(sum(S25))/(n*ll);MSE25

MSE26$<-$(sum(S26))/(n*ll);MSE26

MSE27$<-$(sum(S27))/(n*ll);MSE27

PRE21$<-$100

PRE22$<-$(MSE21/MSE22)*100;PRE22

PRE23$<-$(MSE21/MSE23)*100;PRE23

PRE24$<-$(MSE21/MSE24)*100;PRE24

PRE25$<-$(MSE21/MSE25)*100;PRE25

PRE26$<-$(MSE21/MSE26)*100;PRE26

PRE27$<-$(MSE21/MSE27)*100;PRE27

Strategy3

MSE31$<-$(sum(S31))/(n*ll);MSE31

MSE32$<-$(sum(S32))/(n*ll);MSE32

MSE33$<-$(sum(S33))/(n*ll);MSE33

MSE34$<-$(sum(S34))/(n*ll);MSE34

MSE35$<-$(sum(S35))/(n*ll);MSE35

MSE36$<-$(sum(S36))/(n*ll);MSE36

MSE37$<-$(sum(S37))/(n*ll);MSE37

PRE31$<-$100

PRE32$<-$(MSE31/MSE32)*100;PRE32

PRE33$<-$(MSE31/MSE33)*100;PRE33

PRE34$<-$(MSE31/MSE34)*100;PRE34

PRE35$<-$(MSE31/MSE35)*100;PRE35

PRE36$<-$(MSE31/MSE36)*100;PRE36

PRE37$<-$(MSE31/MSE37)*100;PRE37