Appendix 1
Outline of Derivation of Theorem 3.1. The MSE of \(T_{k}\) and \( T_{k_{mult}}\), \(k=1,2,3\), is given by
$$\begin{aligned} \min MSE\left( T_{k}\right)= & {} \dfrac{{\bar{Y}}^{2}MSE\left( t_{k}\right) }{ {\bar{Y}}^{2}+MSE\left( t_{k}\right) }\hbox { and }\min MSE\left( T_{k_{mult}}\right) =\dfrac{{\bar{Y}}^{2}MSE\left( t_{k_{mult}}\right) }{ {\bar{Y}}^{2}+MSE\left( t_{k_{mult}}\right) }\\ \gamma _{1}= & {} \frac{{\bar{Y}}^{2}}{{\bar{Y}}^{2}+S_{y}^{2}\left\{ \left( \frac{1}{ r}-\frac{1}{n}\right) +\left( \frac{1}{n}-\frac{1}{N}\right) \left( 1-R_{y.zx}^{2}\right) \right\} }, d_{1}=\gamma _{1}\frac{S_{y}\left( \rho _{yx}-\rho _{yz}\rho _{zx}\right) }{S_{x}\left( 1-\rho _{zx}^{2}\right) },\\ d_{2}= & {} \gamma _{1}\frac{S_{y}\left( \rho _{yz}-\rho _{yx}\rho _{zx}\right) }{ S_{z}\left( 1-\rho _{zx}^{2}\right) }\\ \gamma _{2}= & {} \frac{{\bar{Y}}^{2}}{{\bar{Y}}^{2}+S_{y}^{2}\left\{ \left( \frac{1}{r}-\frac{1}{N}\right) \left( 1-R_{y.zx}^{2}\right) \right\} }, d_{3}=\gamma _{7}\frac{S_{y}\left( \rho _{yx}-\rho _{yz}\rho _{zx}\right) }{ S_{x}\left( 1-\rho _{zx}^{2}\right) }, \\ d_{4}= & {} \gamma _{7}\frac{S_{y}\left( \rho _{yz}-\rho _{yx}\rho _{zx}\right) }{S_{z}\left( 1-\rho _{zx}^{2}\right) } \\ \gamma _{3}= & {} \frac{{\bar{Y}}^{2}}{{\bar{Y}}^{2}+S_{y}^{2}\left\{ \left( \frac{1}{ n}-\frac{1}{N}\right) +\left( \frac{1}{r}-\frac{1}{n}\right) \left( 1-R_{y.zx}^{2}\right) \right\} }, d_{5}=\gamma _{13}\frac{S_{y}\left( \rho _{yx}-\rho _{yz}\rho _{zx}\right) }{S_{x}\left( 1-\rho _{zx}^{2}\right) }\, , \\ d_{6}= & {} \gamma _{13}\frac{S_{y}\left( \rho _{yz}-\rho _{yx}\rho _{zx}\right) }{S_{z}\left( 1-\rho _{zx}^{2}\right) }\\ \gamma _{10}= & {} \frac{{\bar{Y}}^{2}}{{\bar{Y}}^{2}+S_{y}^{2}\left\{ \left( \frac{1}{r}-\frac{1}{n}\right) +\left( \frac{1}{n}-\frac{1}{N}\right) \left( 1-R_{y.x_{1},x_{2},\ldots ,x_{p}}^{2}\right) \right\} }, \\ d_{j}= & {} \gamma _{4}\frac{S_{y}\left( \rho _{{yx}_j}-\rho _{yx_{i}}\rho _{x_{i}x_{j}}\right) }{ S_{x_{j}}\left( 1-\rho _{x_{i}x_{j}}^{2}\right) } \ (i\ne j=1,2,\ldots ,p)\\ \gamma _{11}= & {} \frac{{\bar{Y}}^{2}}{{\bar{Y}}^{2}+S_{y}^{2}\left\{ \left( \frac{1 }{r}-\frac{1}{N}\right) \left( 1-R_{y.x_{1},x_{2},\ldots ,x_{p}}^{2}\right) \right\} },\\ d_{j}= & {} \gamma _{10}\frac{S_{y}\left( \rho _{{yx}_j}-\rho _{yx_{i}}\rho _{x_{i}x_{j}}\right) }{S_{x_{j}}\left( 1-\rho _{x_{i}x_{j}}^{2}\right) } \ \ \ \ \ \ \ \ \ (i\ne j=1,2,\ldots ,p)\\ \gamma _{12}= & {} \frac{{\bar{Y}}^{2}}{{\bar{Y}}^{2}+S_{y}^{2}\left\{ \left( \frac{1 }{n}-\frac{1}{N}\right) +\left( \frac{1}{r}-\frac{1}{n}\right) \left( 1-R_{y.x_{1},x_{2},\ldots ,x_{p}}^{2}\right) \right\} },\\ d_{j}= & {} \gamma _{16}\frac{ S_{y}\left( \rho _{{yx}_j}-\rho _{yx_{i}}\rho _{x_{i}x_{j}}\right) }{ S_{x_{j}}\left( 1-\rho _{x_{i}x_{j}}^{2}\right) }\quad (i\ne j=1,2,\ldots ,p) \end{aligned}$$
Outline of Derivation of Theorem 3.2. The MSE of \(T_{sr_{j}}\), \(j=1,2,\ldots ,6\), is given by
$$\begin{aligned} MSE\left( T_{4}\right)= & {} {\bar{Y}}^{2}\left[ 1+\gamma _{4}^{2}\left\{ 1+\left( \frac{1}{r}-\frac{1}{N}\right) C_{y}^{2}+\left( \frac{1}{n}-\frac{1}{N}\right) 2K_{1}^{2}C_{x}^{2}+2K_{2}^{2}C_{z}^{2}+K_{1}\left( C_{x}^{2}\right. \right. \right. \\&\left. \left. -4\rho _{yx}C_{y}C_{x}\right) +K_{2}\left( C_{z}^{2}-4\rho _{yz}C_{y}C_{z}\right) +4K_{1}K_{2}\rho _{zx}C_{z}C_{x}\right\} \\&-2\gamma _{4}\left\{ 1+\left( \frac{1}{n}-\frac{1}{N}\right) \right. \ \\&\left. \left. \left\{ \ \ \frac{K_{1}^{2}}{2}C_{x}^{2}+\dfrac{K_{2}^{2}}{2}C_{z}^{2}+\dfrac{K_{1}}{2}\left( C_{x}^{2}-2\rho _{yx}C_{y}C_{x}\right) +\dfrac{K_{2}}{2}\left( C_{z}^{2}-2\rho _{yz}C_{y}C_{z}\right) \right. \right. \right. \\&\left. \left. \left. +K_{1}K_{2}\rho _{zx}C_{z}C_{x}\right\} \right\} \right] \\ \end{aligned}$$
which can be expressed as
$$\begin{aligned} MSE\left( T_{4}\right) ={\bar{Y}}^{2}\left[ 1+\gamma _{4}^{2}A_{1}-2\gamma _{4}B_{1}\right] \end{aligned}$$
For optimum value of \(\gamma _{4}\) differentiating the \(MSE\left( T_{4}\right) \) with respect to and equating to zero we get
$$\begin{aligned} \gamma _{4opt}=\dfrac{B_{1}}{A_{1}} \end{aligned}$$
substituting the optimum value of \(\gamma _{4}\) in \(MSE\left( T_{4}\right) \) we get minimum MSE
$$\begin{aligned} MSE\left( T_{4}\right) ={\bar{Y}}^{2}\left( 1-\dfrac{B_{1}^{2}}{A_{1}} \right) \end{aligned}$$
The derivation of other estimators \(T_{j}\left( i=4,5,\ldots ,9\right) \) can be done on similar lines. In general, we have
$$\begin{aligned} MSE(T_{j})={\bar{Y}}^{2}\left[ 1+\gamma _{j}^{2}A_{j}-2\gamma _{j}B_{j} \right] \end{aligned}$$
The optimum values of scalars involved are tabulated below for ready reference:
$$\begin{aligned} \gamma _{jopt}= & {} \dfrac{B_{j}}{A_{j}}\ ,\ i=4,5,\ldots ,9.\\ A_{1}= & {} 1+f_{r}C_{y}^{2}+f_{n}\left\{ \begin{array}{c} 2K_{1}^{2}C_{x}^{2}+2K_{2}^{2}C_{z}^{2}+K_{1}\left( C_{x}^{2}-4\rho _{yx}C_{y}C_{x}\right) +K_{2}\left( C_{z}^{2}-4\rho _{yz}C_{y}C_{z}\right) \\ +4K_{1}K_{2}\rho _{zx}C_{z}C_{x} \end{array} \right\} \\ B_{1}= & {} 1+f_{n}\left\{ \dfrac{K_{1}^{2}}{2}C_{x}^{2}+\dfrac{K_{2}^{2}}{2} C_{z}^{2}+\dfrac{K_{1}}{2}\left( C_{x}^{2}-2\rho _{yx}C_{y}C_{x}\right) \right. \\&\left. + \dfrac{K_{2}}{2}\left( C_{z}^{2}-2\rho _{yz}C_{y}C_{z}\right) +K_{1}K_{2}\rho _{zx}C_{z}C_{x}\right\} \\ A_{2}= & {} 1+f_{r}C_{y}^{2}+f_{n}\left\{ 3\delta _{1}^{2}C_{x}^{2}+3\delta _{2}^{2}C_{z}^{2}-4\delta _{1}\rho _{yx}C_{y}C_{x}-4\delta _{2}\rho _{yz}C_{y}C_{z}+4\delta _{1}\delta _{2}\rho _{zx}C_{z}C_{x}\right\} \\ B_{2}= & {} 1+f_{n}\left\{ \delta _{1}^{2}C_{x}^{2}+\delta _{2}^{2}C_{z}^{2}-\delta _{1}\rho _{yx}C_{y}C_{x}-\delta _{2}\rho _{yz}C_{y}C_{z}+\delta _{1}\delta _{2}\rho _{zx}C_{z}C_{x}\right\} \\ A_{1mult}= & {} 1+f_{r}C_{y}^{2}+f_{n}\left\{ 2\sum \limits _{j=1}^{p}K_{j}^{2}C_{j}^{2}+\sum \limits _{j=1}^{p}K_{j}\left( C_{x_{j}}^{2}-4\rho _{yx_{j}}C_{y}C_{x_{j}}\right) \right. \\&\left. +4\sum \sum \limits _{i>j}^{p}K_{i}K_{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ B_{1mult}= & {} 1+f_{n}\left\{ \dfrac{1}{2}\sum \limits _{j=1}^{p}K_{j}^{2}C_{x_{j}}^{2}+\dfrac{1}{2}\sum \limits _{j=1}^{p}K_{j}\left( C_{x_{j}}^{2}-2\rho _{yx_{j}}C_{y}C_{x_{j}}\right) \right. \\&\left. +\sum \sum \limits _{i>j}^{p}K_{i}K_{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ A_{2mult}= & {} 1+f_{r}C_{y}^{2}+f_{n}\left\{ 3\sum \limits _{j=1}^{p}\delta _{j}^{2}C_{x_{j}}^{2}-4\sum \limits _{j=1}^{p}\delta _{j}\rho _{yx_{j}}C_{y_{j}}C_{x_{j}}+4\sum \sum \limits _{i>j}^{p}\delta _{i}\delta _{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ B_{2mult}= & {} 1+f_{n}\left\{ \sum \limits _{j=1}^{p}\delta _{j}^{2}C_{x_{j}}^{2}-\sum \limits _{j=1}^{p}\delta _{j}\rho _{yx_{j}}C_{y_{j}}C_{x_{j}}+\sum \sum \limits _{i>j}^{p}\delta _{i}\delta _{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ A_{3}= & {} 1+f_{r}\left\{ \begin{array}{c} C_{y}^{2}+2K_{3}^{2}C_{x}^{2}+2K_{4}^{2}C_{z}^{2}+K_{3}\left( C_{x}^{2}-4\rho _{yx}C_{y}C_{x}\right) +K_{4}\left( C_{z}^{2}-4\rho _{yz}C_{y}C_{z}\right) + \\ 4K_{3}K_{4}\rho _{zx}C_{z}C_{x} \end{array} \right\} \\ B_{3}= & {} 1+f_{r}\left\{ \dfrac{K_{3}^{2}}{2}C_{x}^{2}+\dfrac{K_{4}^{2}}{2} C_{z}^{2}+\dfrac{K_{3}}{2}\left( C_{x}^{2}-2\rho _{yx}C_{y}C_{x}\right) \right. \\&\left. + \dfrac{K_{4}}{2}\left( C_{z}^{2}-2\rho _{yz}C_{y}C_{z}\right) +K_{3}K_{4}\rho _{zx}C_{z}C_{x}\right\} \\ \end{aligned}$$
$$\begin{aligned} A_{4}= & {} 1+f_{r}\left\{ C_{y}^{2}+3\delta _{3}^{2}C_{x}^{2}+3\delta _{4}^{2}C_{z}^{2}-4\delta _{3}\rho _{yx}C_{y}C_{x}-4\delta _{4}\rho _{yz}C_{y}C_{z}+4\delta _{3}\delta _{4}\rho _{zx}C_{z}C_{x}\right\} \\ B_{4}= & {} 1+f_{r}\left\{ \delta _{3}^{2}C_{x}^{2}+\delta _{4}^{2}C_{z}^{2}-\delta _{3}\rho _{yx}C_{y}C_{x}-\delta _{4}\rho _{yz}C_{y}C_{z}+\delta _{3}\delta _{4}\rho _{zx}C_{z}C_{x}\right\} \\ A_{3mult}= & {} 1+f_{r}\left\{ C_{y}^{2}+2\sum \limits _{j=1}^{p}K_{j}^{2}C_{j}^{2}+\sum \limits _{j=1}^{p}K_{j}\left( C_{x_{j}}^{2}-4\rho _{yx_{j}}C_{y}C_{x_{j}}\right) \right. \\&\left. +4\sum \sum \limits _{i>j}^{p}K_{i}K_{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ B_{3mult}= & {} 1+f_{r}\left\{ \dfrac{1}{2}\sum \limits _{j=1}^{p}K_{j}^{2}C_{x_{j}}^{2}+\dfrac{1}{2}\sum \limits _{j=1}^{p}K_{j}\left( C_{x_{j}}^{2}-2\rho _{yx_{j}}C_{y}C_{x_{j}}\right) \right. \\&\left. +\sum \sum \limits _{i>j}^{p}K_{i}K_{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ A_{4mult}= & {} 1+f_{r}\left\{ C_{y}^{2}+3\sum \limits _{j=1}^{p}\delta _{j}^{2}C_{x_{j}}^{2}-4\sum \limits _{j=1}^{p}\delta _{j}\rho _{yx_{j}}C_{y_{j}}C_{x_{j}}+4\sum \sum \limits _{i>j}^{p}\delta _{i}\delta _{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ B_{4mult}= & {} 1+f_{r}\left\{ \sum \limits _{j=1}^{p}\delta _{j}^{2}C_{x_{j}}^{2}-\sum \limits _{j=1}^{p}\delta _{j}\rho _{yx_{j}}C_{y_{j}}C_{x_{j}}+\sum \sum \limits _{i>j}^{p}\delta _{i}\delta _{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \end{aligned}$$
$$\begin{aligned} A_{5}= & {} 1+f_{r}C_{y}^{2}+f_{rn}\left\{ \begin{array}{c} 2K_{5}^{2}C_{x}^{2}+2K_{6}^{2}C_{z}^{2}+K_{5}\left( C_{x}^{2}-4\rho _{yx}C_{y}C_{x}\right) +K_{6}\left( C_{z}^{2}-4\rho _{yz}C_{y}C_{z}\right) + \\ 4K_{5}K_{6}\rho _{zx}C_{z}C_{x} \end{array} \right\} \end{aligned}$$
$$\begin{aligned} B_{5}= & {} 1+f_{rn}\left\{ \begin{array}{c} \dfrac{K_{5}^{2}}{2}C_{x}^{2}+\dfrac{K_{6}^{2}}{2}C_{z}^{2}+\dfrac{K_{5}}{2} \left( C_{x}^{2}-2\rho _{yx}C_{y}C_{x}\right) +\dfrac{K_{6}}{2}\left( C_{z}^{2}-2\rho _{yz}C_{y}C_{z}\right) + \\ K_{5}K_{6}\rho _{zx}C_{z}C_{x} \end{array} \right\} \end{aligned}$$
$$\begin{aligned} A_{6}= & {} 1+f_{r}C_{y}^{2}+f_{rn}\left\{ 3\delta _{5}^{2}C_{x}^{2}+3\delta _{6}^{2}C_{z}^{2}-4\delta _{5}\rho _{yx}C_{y}C_{x}-4\delta _{6}\rho _{yz}C_{y}C_{z}+4\delta _{5}\delta _{6}\rho _{zx}C_{z}C_{x}\right\} \\ B_{6}= & {} 1+f_{rn}\left\{ \delta _{5}^{2}C_{x}^{2}+\delta _{6}^{2}C_{z}^{2}-\delta _{5}\rho _{yx}C_{y}C_{x}-\delta _{6}\rho _{yz}C_{y}C_{z}+\delta _{5}\delta _{6}\rho _{zx}C_{z}C_{x}\right\} \\ A_{5mult}= & {} 1+f_{r}C_{y}^{2}+f_{rn}\left\{ 2\sum \limits _{j=1}^{p}K_{j}^{2}C_{j}^{2}+\sum \limits _{j=1}^{p}K_{j}\left( C_{x_{j}}^{2}-4\rho _{yx_{j}}C_{y}C_{x_{j}}\right) \right. \\&\left. +4\sum \sum \limits _{i>j}^{p}K_{i}K_{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ B_{5mult}= & {} 1+f_{rn}\left\{ \dfrac{1}{2}\sum \limits _{j=1}^{p}K_{j}^{2}C_{x_{j}}^{2}+\dfrac{1}{2}\sum \limits _{j=1}^{p}K_{j}\left( C_{x_{j}}^{2}-2\rho _{yx_{j}}C_{y}C_{x_{j}}\right) \right. \\&\left. +\sum \sum \limits _{i>j}^{p}K_{i}K_{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \\ A_{6mult}= & {} 1+f_{r}C_{y}^{2}+f_{rn}\left\{ 3\sum \limits _{j=1}^{p}\delta _{j}^{2}C_{x_{j}}^{2}-4\sum \limits _{j=1}^{p}\delta _{j}\rho _{yx_{j}}C_{y_{j}}C_{x_{j}}\right. \\&\left. +4\sum \sum \limits _{i>j}^{p}\delta _{i}\delta _{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \end{aligned}$$
$$\begin{aligned} B_{6mult}= & {} 1+f_{rn}\left\{ \sum \limits _{j=1}^{p}\delta _{j}^{2}C_{x_{j}}^{2}-\sum \limits _{j=1}^{p}\delta _{j}\rho _{yx_{j}}C_{y_{j}}C_{x_{j}}+\sum \sum \limits _{i>j}^{p}\delta _{i}\delta _{j}\rho _{x_{i}x_{j}}C_{x_{i}}C_{x_{j}}\right\} \end{aligned}$$
The optimum value of the constants involve in the estimators are
$$\begin{aligned} K_{1}= & {} K_{3}=K_{5}=\frac{C_{y}\left( \rho _{yx}-\rho _{yz}\rho _{zx}\right) }{C_{x}\left( 1-\rho _{zx}^{2}\right) }, K_{2}=K_{4}=K_{6}=\frac{C_{y}\left( \rho _{yz}-\rho _{yx}\rho _{zx}\right) }{ C_{z}\left( 1-\rho _{zx}^{2}\right) }\\ \delta _{1}= & {} \delta _{3}=\delta _{5}=\frac{C_{y}\left( \rho _{yx}-\rho _{yz}\rho _{zx}\right) }{C_{x}\left( 1-\rho _{zx}^{2}\right) }, \delta _{2}=\delta _{4}=\delta _{6}=\frac{C_{y}\left( \rho _{yz}-\rho _{yx}\rho _{zx}\right) }{ C_{z}\left( 1-\rho _{zx}^{2}\right) }. \end{aligned}$$
Appendix 2
library(MASS)
library(e1071)
set.seed(989898);
\(N=2000; n = 256; r = 204;\)
\(sd.vec = c(10,10,10);royx =0.8;royz=0.7;rozx=0.49\)
\(cor.mat = matrix(c(1,royx,royz,royx,1,rozx,royz,rozx,1), ncol = 3)\)
\(cov.mat = diag(sd.vec) \%*\% cor.mat \%*\% diag(sd.vec)\)
\(dat1<-mvrnorm(n=2000, mu = c(5,5,5), Sigma=cov.mat, empirical =FALSE);\)
\(dat<-as.data.frame(dat1)\)
\(Y=dat\)V1; X=dat\(V2; Z=dat\)V3
\(\hbox {Mx}=\hbox {mean(X)};\hbox {My}=\hbox {mean(Y)};\hbox {Mz}=\hbox {mean(Z)}\)
\(\hbox {Vx}=\hbox {var(X)};\hbox {Vy}=\hbox {var(Y)};\hbox {Vz}=\hbox {var(Z)}; \hbox {Sx}<-\hbox {sd(X)};\hbox {Sy}<-\hbox {sd(Y)};\hbox {Sz}<-\hbox {sd(Z)};\)
\(\hbox {Cx}<-\hbox {Sx/Mx};\hbox {Cy}<-\hbox {Sy/My;Cz}<-\hbox {Sz/Mz;Sxy}=\hbox {cov(Y,X)}; \hbox {Syz}=\hbox {cov(Y,Z)};\hbox {Szx}=\hbox {cov(Z,X)}\)
\(\hbox {ryx}=\hbox {cor(Y,X)};\hbox {ryz}=\hbox {cor(Y,Z)};\hbox {rzx}=\hbox {cor(Z,X)}\)
$$\begin{aligned}&Ryzx<-(ryx^2)+(ryz^2)-(2*ryx*ryz*rzx)/(1-(rzx^2))\\&\hbox {R}<-(\hbox {My/Mx);fn}<-(1/\hbox {n}-1/\hbox {N});\hbox {fr}<-(1/\hbox {r}-1/\hbox {N});\hbox {frn}<-(1/\hbox {r}-1/\hbox {n});\\&b1=b3=b5=((Sy*(ryx-(ryz*rzx)))/(Sx*(1-(rzx^2))));\\&b2=b4=b6=((Sy*(ryz-(ryx*rzx)))/(Sz*(1-(rzx^2))));\\&m1<-(fr*Sy^2);m2<-(Sy^2*(frn+(fn*(1-Ryzx^2))));\\&m3<-(Sy^2*fr*(1-Ryzx^2));m4<-(Sy^2*(fn+(frn*(1-Ryzx^2))));\\&M1<-(My^2*m2)/(My^2+m2);M2<-(My^2*m3)/(My^2+m3);M3<\\&-(My^2*m4)/(My^2+m4);\\&g1<-((My^2)/((My^2)+(Sy^2*(frn+(fn*(1-(Ryzx^2)))))));\\&d1<-((g1*Sy*(ryx-(ryz*rzx)))/(Sx*(1-(rzx^2))));\\&d2<-((g1*Sy*(ryz-(ryx*rzx)))/(Sz*(1-(rzx^2))));\\&g2<-((My^2)/((My^2)+(Sy^2*(fr*(1-(Ryzx^2))))));\\&d3<-((g2*Sy*(ryx-(ryz*rzx)))/(Sx*(1-(rzx^2))));\\&d4<-((g2*Sy*(ryz-(ryx*rzx)))/(Sz*(1-(rzx^2))));\\&g3<-((My^2)/((My^2)+(Sy^2*(fn+(frn*(1-(Ryzx^2)))))));\\&d5<-((g3*Sy*(ryx-(ryz*rzx)))/(Sx*(1-(rzx^2)))); \\&d6<-((g3*Sy*(ryz-(ryx*rzx)))/(Sz*(1-(rzx^2))));\\&K1=K3=K5=((Cy*(ryx-(ryz*rzx)))/(Cx*(1-(rzx^2))));\\&K2=K4=K6=((Cy*(ryz-(ryx*rzx)))/(Cz*(1-(rzx^2))));\\&a1=a3=a5=((Cy*(ryx-(ryz*rzx)))/(Cx*(1-(rzx^2))));\\&a2=a4=a6=((Cy*(ryz-(ryx*rzx)))/(Cz*(1-(rzx^2))));\\&A1<-(1+(fr*Cy^2)+fn*((2*(K1^2)*(Cx^2))+(2*(K2^2)*(Cz^2))\\&\quad +K1* ((Cx^2)-(4*ryx*Cy*Cx))\\&\quad + K2*((Cz^2)-(4*ryz*Cy*Cz))+(4*K1*K2*rzx*Cz*Cx)));\\&B1<-(1+(fn*(((K1^2/2)*(Cx^2))+((K2^2/2)*(Cz^2))+(K1/2)*((Cx^2)\\&\quad -(2*ryx*Cy*Cx))\\&\quad + (K2/2)*((Cz^2)-(2*ryz*Cy*Cz))+(K1*K2*rzx*Cz*Cx)))); \end{aligned}$$
$$\begin{aligned}&A2<-(1+fr*((Cy^2)+(2*(K3^2)*(Cx^2))+(2*(K4^2)*(Cz^2))\\&\quad +K3*((Cx^2)-(4*ryx*Cy*Cx))\\&\quad + K4*((Cz^2)-(4*ryz*Cy*Cz))+(4*K3*K4*rzx*Cz*Cx)));\\&B2<-(1+fr*(((K3^2/2)*(Cx^2))+((K4^2/2)*(Cz^2))+((K3/2)*((Cx^2)\\&\quad -(2*ryx*Cy*Cx)))\\&\quad + ((K4/2)*((Cz^2)-(2*ryz*Cy*Cz)))+(K3*K4*rzx*Cz*Cx)));\\&A3<-(1+(fr*Cy^2)+frn*((2*(K5^2)*(Cx^2))+(2*(K6^2)*(Cz^2))\\&+(K5*((Cx^2)-(4*ryx*Cy*Cx)))\\&\quad + (K6*((Cz^2)-(4*ryz*Cy*Cz)))+(4*K5*K6*rzx*Cz*Cx)));\\&B3<-(1+(frn*(((K5^2/2)*(Cx^2))+((K6^2/2)*(Cz^2))\\&\quad +((K5/2)*((Cx^2)\\&\quad - (2*ryx*Cy*Cx)))+((K6/2)*((Cz^2)-(2*ryz*Cy*Cz)))\\&\quad +(K5*K6*rzx*Cz*Cx))))\\&A4<-(1+(fr*Cy^2)+(fn*((3*(a1^2)*(Cx^2))+(3*(a2^2)*(Cz^2))\\&\quad -(4*a1*ryx*Cy*Cx)\\&\quad - (4*a2*ryz*Cy*Cz)+(4*a1*a2*rzx*Cz*Cx))));\\&B4<-(1+(fn*(((a1^2)*(Cx^2))+((a2^2)*(Cz^2))-(a1*ryx*Cy*Cx)\\&\quad - (a2*ryz*Cy*Cz)+(a1*a2*rzx*Cz*Cx))));\\&A5<-(1+(fr*(Cy^2+(3*(a3^2)*(Cx^2))+(3*(a4^2)*(Cz^2))\\&\quad -(4*a3*ryx*Cy*Cx)\\&\quad - (4*a4*ryz*Cy*Cz)+(4*a3*a4*rzx*Cz*Cx))));\\&B5<-(1+(fr*(((a3^2)*(Cx^2))+((a4^2)*(Cz^2))-(a3*ryx*Cy*Cx)\\&\quad - (a4*ryz*Cy*Cz)+(a3*a4*rzx*Cz*Cx))));\\&A6<-(1+(fr*Cy^2)+(frn*((3*(a5^2)*(Cx^2))+(3*(a6^2)*(Cz^2))\\&\quad -(4*a5*ryx*Cy*Cx)\\&\quad - (4*a6*ryz*Cy*Cz)+(4*a5*a6*rzx*Cz*Cx))));\\ \end{aligned}$$
$$\begin{aligned}&B6<-(1+(frn*(((a5^2)*(Cx^2))+((a6^2)*(Cz^2))-(a5*ryx*Cy*Cx)\\&\quad - (a6*ryz*Cy*Cz)+(a5*a6*rzx*Cz*Cx))));\\&g4<-(B1/A1);g5<-(B2/A2);g6<-(B3/A3);\\&g7<-(B4/A4);g8<-(B5/A5);g9<-(B6/A6);\\&M4<-My^2*(1-(B1^2/A1));M5<-My^2*(1-(B2^2/A2));\\&M6<-My^2*(1-(B3^2/A3));\\&M7<-My^2*(1-(B4^2/A4));M8<-My^2*(1-(B5^2/A5));\\&M9<-My^2*(1-(B6^2/A6));\\&p1<-(m1/m2)*100;p1,P1<-(m1/M1)*100;P1,P4<-(m1/M4)*100;P4\\&P7<-(m1/M7)*100;P7,p2<-(m1/m3)*100;p2,P2<-(m1/M2)*100;P2\\&P5<-(m1/M5)*100;P5,P8<-(m1/M8)*100;P8,p3<-(m1/m4)*100;p3\\&P3<-(m1/M3)*100;P3,P6<-(m1/M6)*100;P6,P9<-(m1/M9)*100;P9 \end{aligned}$$
R Code for Simulation Study
$$\begin{aligned} M10= & {} NA;M11=NA;M12=NA;M13=NA;M14=NA;M15=NA;\\ M16= & {} NA;M21=NA;M22=NA; M23=NA;\\ M24= & {} NA;M25=NA; M26=NA;M31=NA;M32=NA;M33=NA;\\ M34= & {} NA;M35=NA;M36=NA; \end{aligned}$$
$$\begin{aligned}&for(i in 1:50000)\{\\&smp1=c(sample(1:1000, 256,replace=F))\\&\hbox {mar1}=\hbox {dat}[\hbox {smp1},];\\&\hbox {smp2}=\hbox {c}(\hbox {sample}(1:256,204,\hbox {replace}=\hbox {F}));\\&\hbox {mar}2=\hbox {mar1}[\hbox {smp2},];\\&\hbox {Yn}=\hbox {mar1}[,1];\hbox {Xn}=\hbox {mar1}[,2];\hbox {Zn}=\hbox {mar1}[,3];\\&\hbox {YY}=\hbox {mar2}[,1];\hbox {XX}=\hbox {mar2}[,2];\hbox {ZZ}=\hbox {mar}2[,3];\\&\hbox {ybn}=\hbox {mean(Yn)};\hbox {xbn}=\hbox {mean(Xn)};\hbox {zbn}=\hbox {mean(Zn)};\\&\hbox {ybr}=\hbox {mean(YY)};\hbox {xbr}=\hbox {mean(XX)};\hbox {zbr}=\hbox {mean(ZZ)};\\&yr<-mean(YY);\\&M10[i]<-((yr-mean(Y))^2);\\&t1<-mean(YY)+b1*(mean(X)-mean(Xn))+b2*(mean(Z)-mean(Zn));\\&M11[i]<-((t1-mean(Y))^2);\\&t2<-mean(YY)+b3*(mean(X)-mean(XX))+b4*(mean(Z)-mean(ZZ)); \end{aligned}$$
$$\begin{aligned}&M12[i]<-((t2-mean(Y))^2);\\&t3<-mean(YY)+b5*(mean(Xn)-mean(XX))+b6*(mean(Zn)-mean(ZZ));\\&M13[i]<-((t3-mean(Y))^2); \\&\hbox {T1}<-(\hbox {g1}*\hbox {mean(YY)})+\hbox {d1}*(\hbox {mean(X)}\\&\quad -\hbox {mean(Xn)})+\hbox {d2}* (\hbox {mean(Z)}-\hbox {mean(Zn)});\\&M14[i]<-((T1-mean(Y))^2); \\&T2<-(g2*mean(YY))+d3*(mean(X)-mean(XX))+d4*(mean(Z)-mean(ZZ));\\&M15[i]<-((T2-mean(Y))^2);\\&T3<-(g3*mean(YY))+d5*(mean(Xn)-mean(XX))\\&\quad +d6*(mean(Zn)-mean(ZZ));\\&M16[i]<-((T3-mean(Y))^2);\\&t4<-mean(YY)*((mean(X)/mean(Xn))^K1)*((mean(Z)/mean(Zn))^K2); \\&M21[i]<-((t4-mean(Y))^2);\\&t5<-mean(YY)*((mean(X)/mean(XX))^K3)*((mean(Z)/mean(ZZ))^K4); \end{aligned}$$
$$\begin{aligned}&M22[i]<-((t5-mean(Y))^2);\\&t6<-mean(YY)*((mean(Xn)/mean(XX))^K5)*((mean(Zn)/mean(ZZ))^K6);\\&M23[i]<-((t6-mean(Y))^2);\\&T4<-(g4*mean(YY))*((mean(X)/mean(Xn))^K1)*((mean(Z)/mean(Zn))^K2);\\&M24[i]<-((T4-mean(Y))^2);\\&T5<-(g5*mean(YY))*((mean(X)/mean(XX))^K3)*((mean(Z)/mean(ZZ))^K4);\\&M25[i]<-((T5-mean(Y))^2);\\&T6<-(g6*mean(YY))*((mean(Xn)/mean(XX))^K5)*((mean(Zn)/mean(ZZ))^K6); \\&M26[i]<-((T6-mean(Y))^2);\\&t7<-mean(YY)*(mean(X)/((a1*mean(Xn))+((1-a1)*mean(X))))\\&\quad *(mean(Z)/((a2*mean(Zn))+((1-a2)*mean(Z))));\\&M31[i]<-((t7-mean(Y))^2);\\&t8<-mean(YY)*(mean(X)/((a3*mean(XX))+((1-a3)*mean(X))))\\&\quad *(mean(Z)/((a4*mean(ZZ))+((1-a4)*mean(Z))));\\ \end{aligned}$$
$$\begin{aligned}&M32[i]<-((t8-mean(Y))^2);\\&t9<-mean(YY)*(mean(X)/((a5*mean(Xn))+((1-a5)*mean(XX))))\\&\quad *(mean(Z)/((a6*mean(Zn))+((1-a6)*mean(ZZ))));\\&M33[i]<-((t9-mean(Y))^2);\\&T7<-(g7*mean(YY))*(mean(X)/((a1*mean(Xn))+((1-a1)*mean(X))))\\&\quad *(mean(Z)/((a2*mean(Zn))+((1-a2)*mean(Z))));\\&M34[i]<-((T7-mean(Y))^2)\\&T8<-(g8*mean(YY))*(mean(X)/((a3*mean(XX))+((1-a3)*mean(X))))\\&\quad *(mean(Z)/((a4*mean(ZZ))+((1-a4)*mean(Z)))); \\&M35[i]<-((T8-mean(Y))^2); \\&T9<-(g9*mean(YY))*(mean(X)/((a5*mean(Xn))+((1-a5)*mean(XX))))\\&\quad *(mean(Z)/((a6*mean(Zn))+((1-a6)*mean(ZZ))));\\&M36[i]<-((T9-mean(Y))^2);\}\ \end{aligned}$$
$$\begin{aligned} m10= & {} sum(M10)/50000;m11=sum(M11)/50000;m12=sum(M12)/50000;\\ m13= & {} sum(M13)/50000;m14=sum(M14)/50000;m15=sum(M15)/50000;\\ m16= & {} sum(M16)/50000;m21=sum(M21)/50000;m22=sum(M22)/50000;\\ m23= & {} sum(M23)/50000;m24=sum(M24)/50000;m25=sum(M25)/50000;\\ m26= & {} sum(M26)/50000;m31=sum(M31)/50000;m32=sum(M32)/50000;\\ m33= & {} sum(M33)/50000;m34=sum(M34)/50000;m35=sum(M35)/50000;\\ m36= & {} sum(M36)/50000; \end{aligned}$$
$$\begin{aligned}&m10;m11;m12;m13;m14;m15;m16;m21;m22;m23;m24;m25;\\&m26;m31;m32;m33;m34;m35;m36; \end{aligned}$$
PRE10=100
#Strategy1
$$\begin{aligned} PRE11= & {} m10/m11*100;PRE11\\ PRE14= & {} m10/m14*100;PRE14\\ PRE21= & {} m10/m21*100;PRE21\\ PRE24= & {} m10/m24*100;PRE24\\ PRE31= & {} m10/m31*100;PRE31\\ PRE34= & {} m10/m34*100;PRE34\\ \end{aligned}$$
#Strategy2
$$\begin{aligned} PRE12= & {} m10/m12*100;PRE12\\ PRE15= & {} m10/m15*100;PRE15\\ PRE22= & {} m10/m22*100;PRE22\\ PRE25= & {} m10/m25*100;PRE25\\ PRE32= & {} m10/m32*100;PRE32\\ PRE35= & {} m10/m35*100;PRE35\\ \end{aligned}$$
#Strategy3
$$\begin{aligned} PRE13= & {} m10/m13*100;PRE13\\ PRE16= & {} m10/m16*100;PRE16\\ PRE23= & {} m10/m23*100;PRE23\\ PRE26= & {} m10/m26*100;PRE26\\ PRE33= & {} m10/m33*100;PRE33\\ PRE36= & {} m10/m36*100;PRE36 \end{aligned}$$
