Exponential ratio estimator of the median: An alternative to the regression estimator of the median under stratified sampling

1 Introduction

Auxiliary information is commonly used in survey sampling in order to increase precision while estimating the population parameters. So, whenever this information is available and every researcher wants to utilize it in order to get more precise results. However various authors have put their sincere efforts to do the same for details see Kadilar and Cingi (2003) who amended the estimators in Upadhyaya and Singh (1999) to the sampling design stratified random sampling. Also, Singh and Vishwakarma (2008), Sharma and Singh (2015), Verma et al. (2015) suggested a new family of estimators in Stratified random sampling and Subzar et al (2018) have anticipated different estimators using Non-Conventional measures of dispersion for estimating finite population mean in stratified random sampling. Recently Lone et al (2021) have also proposed a general class of ratio estimator for estimating the finite population variance. So this type of sampling procedure is used when the population under study is heterogeneous, it is then usually used to make substrata, which are homogeneous within and heterogeneous between, with a view to producing smaller bound on the error of estimation for a fixed cost of the survey. Having been motivated by the above research works, the present study focuses on how we will get precise results from a heterogonous population and even that is skewed. However, we will get by using the linear regression estimator, but one thing is clear while using OLS (ordinary least square) our results are not precise because it is sensitive to extreme values. But our present study focuses on obtaining reliable results from the data having skewed distribution.

Consider a finite population with auxiliary variate $\left(U_t\right)$ and study variable $\left(V_t\right)$ which is divided into $\left(M\right)$ strata containing $\left(D_t\right)$ units in each $t^{\mathit{th}}$ stratum such that ${\sum }_{t=1}^M{D}_t=D$ . Let $\left(u_{\mathit{ti}}\right)$ and $\left(v_{\mathit{ti}}\right)$ represent the sample medians corresponding to the population medians $\left(Q_{\mathit{ut}}\right)$ and with $Q_{\mathit{vt}}$ correlation coefficient between $\left({\widehat{Q}}_{\mathit{ut}}\right)$ and ${\widehat{Q}}_{\mathit{vt}}$ as ${\rho }_{\mathit{ct}}$ . Let $f_{\mathit{ut}}\left(u_t\right)$ and $f_{\mathit{vt}}\left(v_t\right)$ be the marginal densities; $f_{\mathit{ut}}\left(Q_{\mathit{ut}}\right)$ and $f_{\mathit{vt}}\left(Q_{\mathit{vt}}\right)$ the probability density functions of the variables. Let ${\widehat{Q}}_{\mathit{ust}}={\sum }_{t=1}^M{W}_t{\widehat{Q}}_{\mathit{ut}}$ and ${\widehat{Q}}_{\mathit{vst}}={\sum }_{t=1}^M{W}_t{\widehat{Q}}_{\mathit{vt}}$ be the respective weighted sample medians corresponding to the population medians $Q_u=Q_{\mathit{ust}}={\sum }_{t=1}^M{W}_t{Q}_{\mathit{ut}}$ and $Q_v=Q_{\mathit{vst}}={\sum }_{t=1}^M{W}_t{Q}_{\mathit{vt}}$ where $W_t=D_t/D$ , $f_t=d_t/d$ where $d_t$ is the sample size from stratum $t=1,2,\dots ,M$ and $d$ is the total sample size. Also, let

Such that

2 Review of some existing estimators for population median in stratified sampling scheme

For the above described population different authors have proposed different estimators for population median in stratified sampling in different years whose literature is mentioned in this section, given as under.

1. The usual unbiased sample median estimator (Gross; 1980) given by ${\widehat{Q}}_{\mathit{vst}}$ with variance

• The classical ratio median estimator (Kuk and Mak; 1989) given by.

• The product median estimator [(Robson;1957), (Murthy;1964)] is defined as.

• The exponential ratio type median estimator (Bahl and Tuteja; 1991) defined as

With bias $B\left({\widehat{Q}}_{\exp PS}\right)={\sum }_{t=1}^M\left[\left(W_t{\zeta }_t{Q}_{\mathit{vt}}{C}_{\mathit{Qut}}^2\left(3-4K_t\right)\right)/8\right]$ and Mean square error

• The exponential product type median estimator (Bahl and Tuteja; 1991) defined as

With bias $B\left({\widehat{Q}}_{\exp PS}\right)={\sum }_{t=1}^M\left[\left(W_t{\zeta }_t{Q}_{\mathit{vt}}{C}_{\mathit{Qut}}^2\left(4K_t-1\right)\right)/8\right]$ and Mean square error

• The chain ratio type median estimator (Kadilar and Cingi; 2003) defined as.

3 The proposed median estimator in stratified random sampling scheme

While taking the motivation from the existing estimators and suggesting the new class of Exponential Ratio estimator of the median using ancillary information and the combination of scalars. The obtainable estimators are the members of the suggested class of estimator and is given as

Where $\left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)$ are the scalars chosen suitably, such that ${ħ}_{\iota t}$ and ${\theta }_{\iota t}$ fulfil the condition

In order to derive the estimated expressions of bias and mean square error for the suggested class of estimator we have expressed (3.1) in terms of (1.1), then we obtain $$\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)=\sum _{i=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{{ħ}_{\iota t}-{\theta }_{\iota t}{\left(\frac{Q_{\mathit{ut}}\left(1+{\varphi }_{1t}\right)}{Q_{\mathit{ut}}}\right)}^{{\vartheta }_{\iota t}}\exp \left[\frac{{\tau }_{\iota t}\left[Q_{\mathit{ut}}\left(1+{\varphi }_{1t}\right)\right]-Q_{\mathit{ut}}}{2Q_{\mathit{ut}}\left(1+{\varphi }_{1t}/2\right)}\right]\right\}\right]$$ $$=\sum _{i=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{{ħ}_{\iota t}-{\theta }_{\iota t}{\left(1+{\varphi }_{1t}\right)}^{{\vartheta }_t}\exp \left[\frac{{\tau }_{\iota t}{Q}_{\mathit{ut}}{\varphi }_{1t}}{2Q_{\mathit{ut}}\left(1+{\varphi }_{1t}/2\right)}\right]\right\}\right]$$ $$=\sum _{i=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{{ħ}_{\iota t}-{\theta }_{\iota t}{\left(1+{\varphi }_{1t}\right)}^{{\vartheta }_{\iota t}}\exp \left[\frac{{\tau }_{\iota t}{\varphi }_{1t}}{2}{\left(1+{\varphi }_{1t}/2\right)}^{-1}\right]\right\}\right]$$

Assuming that $\left|{\varphi }_{1t}\right|<1$ and expanding ${\left(1+{\varphi }_{1t}\right)}^{{\vartheta }_{\iota t}}$ , $\left[\frac{{\tau }_{\iota t}{\varphi }_{1t}}{2}{\left(1+{\varphi }_{1t}/2\right)}^{-1}\right]$ and ${\left(1+{\varphi }_{1t}/2\right)}^{-1}$ , we have $$\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\partial }_{\iota t}\right)=\sum _{t=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{\begin{array}{c}{ħ}_{\iota t}-{\theta }_{\iota t}\left[1+{\vartheta }_{\iota t}{\varphi }_{1t}+\frac{{\vartheta }_{\iota t}\left({\vartheta }_{\iota t}-1\right)}{2}{\varphi }_{1t}^2+\dots \right]\times \\ \left[1+\frac{{\tau }_{\iota t}{\varphi }_{1t}}{2}{\left(1+{\varphi }_{1t}/2\right)}^{-1}+\frac{{\tau }_{\iota t}^2{\varphi }_{1t}^2}{8}{\left(1+{\varphi }_{1t}/2\right)}^{-2}\right]\end{array}\right\}\right]$$ $$\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)=\sum _{t=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{\begin{array}{c}{ħ}_{\iota t}-{\theta }_{\iota t}\left[1+{\vartheta }_{\iota t}{\varphi }_{1t}+\frac{{\vartheta }_{\iota t}\left({\vartheta }_{\iota t}-1\right)}{2}{\varphi }_{1t}^2+\dots \right]\times \\ \left[1+\left[\frac{{\tau }_{\iota t}{\varphi }_{1t}}{2}\left(1-\frac{{\varphi }_{1t}}{2}+\frac{{\varphi }_{1t}^2}{4}-\dots \right)\right]+\frac{{\tau }_{\iota t}^2{\varphi }_{1t}^2}{8}\left(1+{\varphi }_{1t}\right)\right]\end{array}\right\}\right]$$ $$\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)=\sum _{t=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{\begin{array}{c}{ħ}_{\iota t}-{\theta }_t\left[1+{\vartheta }_{\iota t}{\varphi }_{1t}+\frac{{\vartheta }_{\iota t}\left({\vartheta }_{\iota t}-1\right)}{2}{\varphi }_{1t}^2+\dots \right]\times \\ \left[1+\frac{{\tau }_{\iota t}{\varphi }_{1t}}{2}-\frac{{\tau }_{\iota t}{\varphi }_{1t}^2}{4}+\frac{{\tau }_{\iota t}^2{\varphi }_{1t}^2}{8}\right]\end{array}\right\}\right]$$ $$\begin{gathered} \cong \sum _{t=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)-\frac{{\lambda }_{\iota t}{\varphi }_{1t}}{2}\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)-\frac{{\lambda }_{\iota t}{\varphi }_{1t}^2}{8}\left(4{\vartheta }_{\iota t}^2-4{\vartheta }_{\iota t}-2{\tau }_{\iota t}+{\tau }_{\iota t}^2+4{\vartheta }_{\iota t}{\tau }_{\iota t}\right)\right\}\right] \\ \cong \sum _{t=1}^M{W}_t\left[Q_{\mathit{vt}}\left(1+{\varphi }_{0t}\right)\left\{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)-\frac{{\theta }_{\iota t}\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right){\varphi }_{1t}}{2}-\frac{{\theta }_{\iota t}\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}-2\right){\varphi }_{1t}^2}{8}\right\}\right] \end{gathered}$$

Neglecting terms of ${\varphi }_{\mathit{it}}\left(i=0or1\right)$ having power greater than two, we have

Therefore,

Then, taking expectation of (3.4) and using (1.1), we obtain the bias for the proposed class of ratio median estimators $\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)$ to the first degree of approximation as $$B\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]=E\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)-Q_{\mathit{vt}}\right]$$ $$\cong \sum _{t=1}^M{W}_t\left[\left({ħ}_{\iota t}-{\theta }_{\iota t}-1\right)Q_{\mathit{vt}}+\frac{\left(1-f_t\right)}{d_t}{Q}_{\mathit{vt}}\left\{-\frac{{\theta }_{\iota t}\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)}{2}\left[\frac{\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}-2\right)}{4}{C}_{\mathit{Qut}}^2+{\rho }_{\mathit{ct}}{C}_{\mathit{Qv}}{C}_{\mathit{Qu}}\right]\right\}\right]$$

While squaring equation (3.4) on both sides, ignoring terms of ${\varphi }_{\mathit{it}}\left(i=0or1\right)$ having power greater than two, taking expectation, and using (1.1), we obtain the mean square error for the suggested class of ratio median estimator to the first degree of approximation as $$\begin{gathered} MSE\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]=E{\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)-Q_{\mathit{vt}}\right]}^2 \\ \cong E{\left\{\sum _{t=1}^M{W}_t\left[Q_{\mathit{vt}}\left\{\left({ħ}_{\iota t}-{\theta }_{\iota t}-1\right)\right\}-\frac{{\theta }_{\iota t}\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)}{2}+\left({ħ}_{\iota t}-{\theta }_{\iota t}\right){\varphi }_{1t}\right]\right\}}^2 \end{gathered}$$

Now to examine the optimality condition for the suggested class of ratio median estimator, let $$\frac{\partial MSE\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]}{\partial {\theta }_{\iota t}}=0$$

So that $${\theta }_{\iota t}^2{C}_{\mathit{Qut}}^2\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)=2K_{\iota t}{\theta }_{\iota t}\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)C_{\mathit{Qut}}^2$$ $$\Rightarrow \left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)=\frac{2K_t{\theta }_{\iota t}\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}{{\theta }_{\iota t}}$$

Putting (3.7) in (3.5), we have.

$\frac{2K_t}{{\theta }_{\iota t}}=\left(2{\vartheta }_{\iota t}+{\tau }_{\iota t}\right)$ and ${ħ}_{\iota t}=\left(1+{\theta }_{\iota t}\right)$ .

Then $$B\left(\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right)=E\left[\left(\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right)-Q_{\mathit{vt}}\right]$$ $$\cong \sum _{t=1}^M{W}_t\left[\left(1+{\theta }_{\iota t}-{\theta }_{\iota t}-1\right)Q_{\mathit{vt}}+\frac{1-f_t}{d_t}{Q}_{\mathit{vt}}{C}_{\mathit{Qut}}^2\left\{\frac{-{\theta }_{\iota t}\frac{2K_t}{{\theta }_{\iota t}}}{2}\left[\frac{\frac{2K_t}{{\theta }_{\iota t}}-2}{4}+K_t\right]\right\}\right]$$ $$\cong \sum _{t=1}^M{W}_t\frac{1-f_t}{d_t}{Q}_{\mathit{vt}}{C}_{\mathit{Qut}}^2\left\{\frac{-2K_t}{2}\left[\frac{2K_t-2{\theta }_{\iota t}}{4{\theta }_{\iota t}}+K_t\right]\right\}$$

Hence substituting (3.7) in (3.6), we obtain $$MSE\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]\cong \sum _{t=1}^M{W}_t^2\left[\begin{array}{c}{Q}_{\mathit{vt}}^2{\left({ħ}_{\iota t}-{\theta }_{\iota t}-1\right)}^2+\\ \frac{\left(1-f_t\right)}{d_t}{Q}_{\mathit{vt}}^2\left[\begin{array}{c}{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}^2{C}_{\mathit{Mvt}}^2+\frac{2K_t\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}{4{\lambda }_{\iota t}}{C}_{\mathit{Mut}}^2\times \\ \left\{\frac{2K_t{\theta }_{\iota t}^2\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}{4{\theta }_{\iota t}}-4K_t{\theta }_{\iota t}\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)\right\}\end{array}\right]\end{array}\right]$$ $$\cong \sum _{t=1}^M{W}_t^2\left[Q_{\mathit{vt}}^2{\left({ħ}_{\iota t}-{\theta }_{\iota t}-1\right)}^2+\frac{\left(1-f_t\right)}{d_t}{Q}_{\mathit{vt}}^2\left[{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}^2{C}_{\mathit{Mvt}}^2+\frac{K_t\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}{2{\theta }_{\iota t}}{C}_{\mathit{Mut}}^2\left\{-2K_t{\theta }_{\iota t}\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)\right\}\right]\right]$$ $$\cong \sum _{t=1}^M{W}_t^2\left[Q_{\mathit{vt}}^2{\left({ħ}_{\iota t}-{\theta }_{\iota t}-1\right)}^2+\frac{\left(1-f_t\right)}{d_t}{Q}_{\mathit{vt}}^2\left[{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}^2{C}_{\mathit{Mvt}}^2-{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}^2{K}_t^2{C}_{\mathit{Mut}}^2\right]\right]$$ $$\cong \sum _{t=1}^M{W}_t^2\left[Q_{\mathit{vt}}^2{\left({ħ}_{\iota t}-{\theta }_{\iota t}-1\right)}^2+\frac{\left(1-f_t\right)}{d_t}{Q}_{\mathit{vt}}^2{\left({ħ}_{\iota t}-{\theta }_{\iota t}\right)}^2\left[C_{\mathit{Mvt}}^2-K_t^2{C}_{\mathit{Mut}}^2\right]\right]$$

Thus substituting (3.2) in (3.9) we obtain the asymptotic optimum MSE for the Proposed class of ratio median estimator $\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)$ as $$\mathit{MS}{E}_{\mathit{opt}}\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]\cong \sum _{t=1}^M{W}_t^2\left[\frac{\left(1-f_t\right)}{d_t}{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2\left(1-{\rho }_{\mathit{ct}}^2\right)\right]$$

Thus the mean square error of the suggested class of exponential ratio estimator of the median in stratified random sampling scheme at optimal condition has the same proficiency as the unbiased linear regression estimator as the mean square error expression at the optimal condition of the suggested estimator is the same as the expression of unbiased linear regression estimator.

4 Some existing members of the proposed class of estimators

How the existing estimators which are mentioned in this study fit into the suggested class of exponential ratio estimator of the median are presented in this section.

Note. As $\left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)$ take on definite distinctive values, the suggested class of exponential ratio estimator of the median giving the same expressions of bias and mean square error which the estimators give mentioned in section 2. Thus, the study provides unified treatment towards the properties of the existing members of the suggested class of estimators point out in this study (see for details Table 1).

5 Efficiency comparisons among members of the proposed class of estimators

Here, the mean square error of the suggested exponential ratio estimator of the median at optimum condition is equated with the mean square error of some existing median estimators.

1. Comparison of $\mathit{MS}{E}_{\mathit{opt}}\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ with sample median estimator ${\widehat{Q}}_{\mathit{vst}}$ .

Comparing (2.1) and (3.9), $\left(\mathit{optimum}\right)\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ will be more efficient than ${\widehat{Q}}_{\mathit{vst}}$ $$\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2-\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2\left(1-{\rho }_{\mathit{ct}}^2\right)>0$$

${\sum }_{t=1}^M{W}_t^2{\theta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2{\rho }_{\mathit{ct}}^2>0$ , Which is always true.

• b)

  Comparison of $\mathit{MS}{E}_{\mathit{opt}}\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ with classical ratio median estimator ${\widehat{Q}}_{\mathit{RS}}$ .

Comparing (2.2) and (3.9), $\left(\mathit{optimum}\right)\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ will be more efficient than ${\widehat{Q}}_{\mathit{RS}}$ $$\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2\left[C_{\mathit{Qvt}}^2+C_{\mathit{Qut}}^2\left(1-2K_t\right)\right]-\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2\left(1-{\rho }_{\mathit{ct}}^2\right)>0$$

$\Rightarrow {\left(1-K_t\right)}^2>0$ , Which is always true.

• c)

  Comparison of $\mathit{MS}{E}_{\mathit{opt}}\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ with classical product median estimator ${\widehat{Q}}_{\mathit{PS}}$ .

Comparing (2.3) and (3.9), $\left(\mathit{optimum}\right)\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ will be more efficient than ${\widehat{Q}}_{\mathit{PS}}$ $$\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2\left[C_{\mathit{Qvt}}^2+C_{\mathit{Qut}}^2\left(1+2K_t\right)\right]-\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2\left(1-{\rho }_{\mathit{ct}}^2\right)>0$$

$\Rightarrow {\left(1+K_t\right)}^2>0$ , Which is always true.

• d)

  Comparison of $\mathit{MS}{E}_{\mathit{opt}}\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ with classical exponential ratio median estimator ${\widehat{Q}}_{\exp RS}$ .

Comparing (2.4) and (3.9), $\left(\mathit{optimum}\right)\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ will be more efficient than ${\widehat{Q}}_{\exp RS}$ $$\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2\left[C_{\mathit{Qvt}}^2+\frac{C_{\mathit{Qut}}^2}{4}\left(1-4K_t\right)\right]-\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2\left(1-{\rho }_{\mathit{ct}}^2\right)>0$$

$\Rightarrow {\left(1-2K_t\right)}^2>0$ , Which is always true.

• e)

  Comparison of $\mathit{MS}{E}_{\mathit{opt}}\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ with classical exponential product median estimator ${\widehat{Q}}_{\exp PS}$ .

Comparing (2.5) and (3.9), $\left(\mathit{optimum}\right)\left[\psi \left({ħ}_{\iota t},{\theta }_{\iota t},{\vartheta }_{\iota t},{\tau }_{\iota t}\right)\right]$ will be more efficient than ${\widehat{Q}}_{\exp PS}$ $$\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2\left[C_{\mathit{Qvt}}^2+\frac{C_{\mathit{Qut}}^2}{4}\left(1+4K_t\right)\right]-\sum _{t=1}^M{W}_t^2{\zeta }_t{Q}_{\mathit{vt}}^2{C}_{\mathit{Mvt}}^2\left(1-{\rho }_{\mathit{ct}}^2\right)>0$$