An efficient class of estimators based on ranked set sampling

Abstract

This article proposes an efficient class of estimators for population mean in ranked set sampling framework which includes the usual mean estimator, Upadhyaya et al. (1985) type estimator, Samawi and Muttlak (1996) estimator, Yu and Lam (1997) estimator, Khoshnevisan et al. (2007) type estimators, Bouza (2008) estimator, Koyuncu and Kadilar (2009) type estimators, Kadilar et al. (2009) estimator, Al-Omari et al. (2009) estimator, Singh et al. (2014) estimator, Mehta and Mandowara (2016) estimators, Mandowara and Mehta (2016) estimators and Bhushan and Kumar (2020a; b) estimators. The bias and mean square error of the proposed estimators are reported up to first order of approximation. It has been proven, both theoretically and empirically, that the proposed estimators provide more accurate results than the existing estimators.

Appendix A

The MSE of the existing estimators is given below.

$$\begin{aligned} MSE(t_{m})&={\bar{Y}}^2\Delta _{2,0} \end{aligned}$$

(A.1)

$$\begin{aligned} MSE(t_{r})&={\bar{Y}}^2\bigg [\Delta _{2,0}+\Delta _{0,2}-2\Delta _{1,1}\bigg ] \end{aligned}$$

(A.2)

$$\begin{aligned} MSE(t_{lr})&={\bar{Y}}^2\left[ \Delta _{2,0}+{\beta }^2\Delta _{0,2}-2\beta \Delta _{1,1}\right] \end{aligned}$$

(A.3)

$$\begin{aligned} {min}MSE(t_{lr})&={\bar{Y}}^2\bigg [\Delta _{2,0}-\frac{\Delta _{1,1}^2}{\Delta _{0,2}}\bigg ] \end{aligned}$$

(A.4)

$$\begin{aligned} MSE(t_{kc})&= {\bar{Y}}^2 \bigg [(k-1)^2+\Delta _{0,2}+k^2\Delta _{2,0}-2k\Delta _{1,1}\bigg ] \end{aligned}$$

(A.5)

$$\begin{aligned} {min}MSE(t_{kc})&={\bar{Y}}^2\bigg [(k^*-1)^2+\Delta _{0,2}+k^{*2}\Delta _{2,0}-2k^{*}\Delta _{1,1}\bigg ] \end{aligned}$$

(A.6)

$$\begin{aligned} MSE(t_{o_i})&={\bar{Y}}^2\Delta _{2,0}+K_i^2{\bar{X}}^2\Delta _{0,2}-2K_i{\bar{X}}{\bar{Y}}\Delta _{1,1},~i=1,3 \end{aligned}$$

(A.7)

$$\begin{aligned} MSE(t_s)&={\bar{Y}}^2 \left[ \begin{array}{l} (1+\lambda ^2_1+\lambda ^2_2)+\lambda ^2_1\Delta _{2,0}+\lambda ^2_2\{ \Delta _{2,0}+g(2g+1)\theta ^2\alpha ^2\Delta _{0,2} \}\\ +2\lambda _1\lambda _2\{\Delta _{2,0}-2\theta g\alpha \Delta _{1,1}+\frac{g(g+1)}{2}\theta ^2\alpha ^2\Delta _{0,2}\}\\ -2\lambda _2\{\frac{g(g+1)}{2}\theta ^2\alpha ^2\Delta _{0,2}-g\theta \alpha \Delta _{1,1}\}-2(\lambda _1+\lambda _2)+2\lambda _1\lambda _2 \end{array} \right] \end{aligned}$$

(A.8)

$$\begin{aligned} {min}MSE(t_s)_I&={\bar{Y}}^2\bigg [\Delta _{2,0}-\frac{\Delta _{1,1}^2}{\Delta _{0,2}}\bigg ],~\text {when}~\lambda _1+\lambda _2=1 \end{aligned}$$

(A.9)

$$\begin{aligned} {min}MSE(t_s)_{II}&={\bar{Y}}^2\left[ 1-\frac{\big (\begin{array}{l}B_s-2C_sD_s+A_sD_s^2 \end{array} \big )}{\big (A_sB_s-C_s^2\big )}\right] ~\text {when}~\lambda _1+\lambda _2 \ne 1 \end{aligned}$$

(A.10)

$$\begin{aligned} MSE(t_{mm_i})&={\bar{Y}}^2\big [\Delta _{2,0}+\delta _i^2\Delta _{0,2}-2\delta _i\Delta _{1,1}\big ],~i=1,2,3 \end{aligned}$$

(A.11)

$$\begin{aligned} MSE(t_{mm_4})&={\bar{Y}}^2\big [\Delta _{2,0}+\delta _4^2\Delta _{0,2}+2\delta _4\Delta _{1,1}\big ] \end{aligned}$$

(A.12)

$$\begin{aligned} MSE(t_{mm_5})&={\bar{Y}}^2\big [\Delta _{2,0}+(1-2\phi )^2t_3^2\Delta _{0,2}+2(1-2\phi )t_3\Delta _{1,1}\big ] \end{aligned}$$

(A.13)

$$\begin{aligned} minMSE(t_{mm_5})&={\bar{Y}}^2\big [\Delta _{2,0}+(1-2\phi _0)^2t_3^2\Delta _{0,2}+2(1-2\phi _0)t_3\Delta _{1,1}\big ] \end{aligned}$$

(A.14)

$$\begin{aligned} MSE(t_{sk_i})&={\bar{Y}}^2\left[ \Delta _{2,0}+I_i^2\Delta _{0,2}-2I_i\Delta _{1,1}\right] ,~i=1,3 \end{aligned}$$

(A.15)

$$\begin{aligned} MSE(t_{1})&={\bar{Y}}^2(\alpha _{1}-1)^2+ {\bar{Y}}^2\alpha _1^2\Delta _{0,2}+{\beta _1}^2{\bar{X}}^2\nu ^2\Delta _{2,0}+2\alpha _1\beta _1{\bar{X}}{\bar{Y}}\nu \Delta _{1,1} \end{aligned}$$

(A.16)

$$\begin{aligned} MSE(t_{2})&={\bar{Y}}^2\left[ \begin{array}{l} 1+\alpha ^2_2\left( 1+\Delta _{0,2}+ \beta _2(2\beta _2+1)\nu ^2\Delta _{2,0}-4\beta _2\nu \Delta _{1,1} \right) \\ -2\alpha _2 \left( 1-\beta _2\nu \Delta _{1,1}+\frac{\beta _2(\beta _2+1)}{2}\nu ^2\Delta _{2,0} \right) \end{array} \right] \end{aligned}$$

(A.17)

$$\begin{aligned} MSE(t_{3})&={\bar{Y}}^2\left[ \begin{array}{l} 1+\alpha ^2_{3}\left( 1+ \Delta _{0,2}+3\beta ^2_3\nu ^2\Delta _{2,0}-4\beta _3\nu \Delta _{1,1} \right) \\ -2\alpha _3 \left( 1+\beta ^2_3\nu ^2\Delta _{2,0}-\beta _3\nu \Delta _{1,1}\right) \end{array}\right] \end{aligned}$$

(A.18)

$$\begin{aligned} minMSE(t_i)&={\bar{Y}}^2\left[ 1-\alpha _{i(opt)}\right] ={\bar{Y}}^2\left[ 1-\frac{Q_i^2}{P_i}\right] ,~i=1,3 \end{aligned}$$

(A.19)

$$\begin{aligned} minMSE(t_2)&={\bar{Y}}^2\left[ 1-\frac{Q_2^2}{P_2}\right] \end{aligned}$$

(A.20)

$$\begin{aligned} MSE(t_{g_1})&={\bar{Y}}^2\left[ \Delta _{2,0}+\eta ^2\Delta _{0,2}+2\eta \Delta _{1,1}\right] \end{aligned}$$

(A.21)

$$\begin{aligned} MSE(t_{g_2})&={\bar{Y}}^2\left[ \Delta _{2,0}+\nu ^2\zeta ^2\Delta _{0,2}+2\zeta \nu \Delta _{1,1}\right] \end{aligned}$$

(A.22)

$$\begin{aligned} minMSE(t_{g_i})&={\bar{Y}}^2\left[ \Delta _{2,0}-\frac{\Delta _{1,1}^2}{\Delta _{0,2}}\right] ,~i=1,2 \end{aligned}$$

(A.23)

The optimum values of constants involved in the estimators can be obtained by minimizing with respect to the constants as

$$\begin{aligned} \beta _{(opt)}&=\frac{{R}{\Delta }_{1,1}}{{\Delta }_{0,2}} \end{aligned}$$

(A.24)

$$\begin{aligned} k_{(opt)}&=\frac{\big (1+\Delta _{1,1}\big )}{\big (1+\Delta _{2,0}\big )}=k^* (\text {say}) \end{aligned}$$

(A.25)

$$\begin{aligned} \lambda _{1_{(opt)}}&=1-\frac{\Delta _{1,1}}{g\theta \alpha \Delta _{0,2}},~\text {when}~ \lambda _1+\lambda _2=1 \end{aligned}$$

(A.26)

$$\begin{aligned} \lambda _{1_{(opt)}}&=\frac{[B_s-C_sD_s]}{[A_sB_s-C_s^2]},~\text {when}~ \lambda _1+\lambda _2\ne 1 \end{aligned}$$

(A.27)

$$\begin{aligned} \lambda _{2_{(opt)}}&=\frac{[A_sD_s-C_s]}{[A_sB_s-C_s^2]} \end{aligned}$$

(A.28)

$$\begin{aligned} \phi _{(opt)}&=\frac{(t_3+k)}{2t_3}=\phi _0(say) \end{aligned}$$

(A.29)

$$\begin{aligned} \alpha _{i(opt)}&=\frac{Q_i}{P_i},~i=1,2,3 \end{aligned}$$

(A.30)

$$\begin{aligned} \beta _{1(opt)}&=-\frac{{\bar{Y}}}{{\bar{X}}}\frac{\Delta _{1,1}}{\Delta _{2,0}}\alpha _{1(opt)} \end{aligned}$$

(A.31)

$$\begin{aligned} \beta _{i(opt)}&=\frac{\Delta _{1,1}}{\Delta _{2,0}},~i=2,3 \end{aligned}$$

(A.32)

$$\begin{aligned} \eta _{(opt)}&=-\frac{\Delta _{1,1}}{\Delta _{0,2}}=\zeta _{(opt)} \end{aligned}$$

(A.33)

where $K_i={{\bar{Y}}}/{({\bar{X}}+q_i)},~i=1,3$,  $A_s=1+\Delta _{2,0}$,  $B_s=1+\Delta _{2,0}+g(2g+1)\theta ^2\alpha ^2\Delta _{0,2}-4g\theta \alpha \Delta _{1,1}$,  $C_s=1+\Delta _{2,0}-2g\theta \alpha \Delta _{1,1}+\{{g(g+1)}/{2}\}\theta ^2\alpha ^2\Delta _{0,2}$,  $D_s=1+\{{g(g+1)}/{2}\}\theta ^2\alpha ^2\Delta _{0,2}-g\theta \alpha \Delta _{1,1}$,  $I_i={({\bar{X}}+q_i)}/{(2{\bar{X}}+q_i)^2},~i=1,3$,  $\delta _1={\bar{X }}/{({\bar{X}}+C_x)},~ \delta _2={{\bar{X}}}/{(\bar{X }+\beta _{2}(x))}$,  $\delta _3=\delta _4=t_3={{\bar{X}}C_x}/{(\bar{X }C_x+\beta _{2}(x))}$,  $P_i=1+\Delta _{0,2}-{\Delta _{1,1}^2}/{\Delta _{2,0}},~i=1,3$,  $Q_i=1,~i=1,3$,  $P_2=1+\Delta _{0,2}+\nu \Delta _{1,1}-{2\Delta _{1,1}^2}/{\Delta _{2,0}}$,  $Q_2=1+{\nu \Delta _{1,1}}/{2}-{\Delta _{1,1}^2}/{2\Delta _{2,0}}$.