A simulation study: new optimal estimators for population mean by using dual auxiliary information in stratified random sampling

ABSTRACT

Recently, Haq et al. [A new estimator of finite population mean based on the dual use of the auxiliary information. Commun Stat Theory Methods. 2017;46(9):4425–4436] utilized the dual auxiliary information under simple random sampling only. Motivated by their idea, we initiated the dual use of auxiliary variable under a stratified random sampling scheme. Dual use of auxiliary variable consists: (1) the original auxiliary information and (2) the ranked auxiliary information. We proposed new optimal exponential-type estimators for the estimation of the finite population mean. Mathematical properties such as bias and mean squared error of the proposed estimators are derived. Monte Carlo simulation studies are included to successfully validate the theoretical results. Moreover, the applicability of the proposed estimators is highlighted through empirical interpretation with the help of a real-life data set. It is clearly identified from the numerical results that our proposed estimators are more efficient over the competitors.

KEYWORDS:

• Auxiliary variable
• bias
• mean squared error
• ranked auxiliary variable
• stratified random sampling

1. Introduction

One of the objectives of sample survey theory is to estimate the unknown population parameters of the study variable such as population total, mean, proportion, ratio and variance etc. A procedure is desirable that provides a precise estimator of the parameter of interest by surveying a suitably chosen sample of individuals. Supplementary/additional information provided by an auxiliary variable which is correlated with the study variable enhances the precision of the estimators. Survey statisticians take advantage of this information whenever it is available to explore the efficient estimators. Ratio, product, regression and their modified estimators are best examples in this regard.

An elaborate literature has grown for identifying more efficient estimators under different sampling designs, e.g. simple random sampling, stratified random sampling, cluster sampling, systematic sampling and etc. Simple random sampling does not produce administrative convenience and representative sample for a heterogeneous population. As it does not capture the diversity which is likely to be mined through stratified random sampling. Stratified random sampling is one of the possible ways to increase the precision of the estimates. It is a powerful and flexible method that is widely used in practice. Many researchers, such as Kadilar and Cingi [1,2], Koyuncu and Kadilar [3,4], Singh and Vishwakarma [5], Shabbir and Gupta [6], Haq and Shabbir [7], Singh and Solanki [8], Yadav et al. [9], Solanki and Singh [10,11], Aslam [12], Bhatti et al. [13], Javed et al. [14], Marin et al. [15–17], etc. have contributed to estimate the finite population mean under stratified random sampling scheme. All these contributions and alike published work under a stratified random sampling scheme are based on only the utilization of original auxiliary information. None of them tried the dual use of auxiliary information to enhance the estimation procedure.

Recently, Haq et al. [18] used an additional information of the auxiliary variable called ranked auxiliary variable to develop efficient estimators for the estimation of mean. These estimators are developed only to cope with the simple random sampling scheme.

Here, comes a new challenge/idea for exploring more optimal estimators using dual use of auxiliary information to deal with the stratified random sampling scheme. This challenge is successfully meet and new optimal estimators for finite population mean are developed under a stratified random sampling scheme in this article.

The remaining part of the paper is organized as follows: In section 2, procedures, notations and various estimators under stratified random sampling are introduced. In section 3, proposed estimators for estimating finite population mean using the original and ranked auxiliary information are defined. In section 4, an empirical study is carried out to evaluate the performance of the proposed estimators. Monte Carlo simulation studies are included to successfully validate the theoretical results in section 5. Finally, concluding remarks are enclosed in the last section.

2. Procedure, notations and review of literature

Consider $U=\left\{U_1,U_2,U_3,\dots ,U_N\right\}$ be a finite population of size $N$ and is divided into L homogenous strata with $h\text{th}$ stratum containing $N_h,\left(h=1,2,...,L\right)$ units with the condition that $\sum _{h=1}^L{N}_h=N.$ Under the condition $\sum _{h=1}^L{n}_h=n$, a sample of size $n_h$ is drawn under simple random sampling without replacement (SRSWOR) from $h\text{th}$ stratum. Let $\begin{array}{rl}\overline{Y}& ={\overline{Y}}_{st}=\sum _{h=1}^L{W}_h{\overline{Y}}_h\text{be the population mean of the}\\ & \text{study variable }\left(Y\right)\end{array}$ $\begin{array}{rl}\overline{X}& ={\overline{X}}_{st}=\sum _{h=1}^L{W}_h{\overline{X}}_h\text{be the population mean of the}\\ & \text{auxiliary variable }\left(X\right)\end{array}$ $\begin{array}{rl}\overline{Z}& ={\overline{Z}}_{st}=\sum _{h=1}^L{W}_h{\overline{Z}}_h\text{be the population mean of the}\\ & \text{ranked auxiliary variable }\left(Z\right)\end{array}$ $\begin{array}{rl}{\overline{y}}_{st}& =\sum _{h=1}^L{W}_h{\overline{y}}_h\text{ be the sample mean of the }Y\end{array}$ $\begin{array}{rl}{\overline{x}}_{st}& =\sum _{h=1}^L{W}_h{\overline{x}}_h\text{be the sample mean of the }X\end{array}$ $\begin{array}{rl}{\overline{z}}_{st}& =\sum _{h=1}^L{W}_h{\overline{z}}_h\text{ be the sample mean of the }Z\end{array}$ $\begin{array}{rl}{\overline{Y}}_h& =\sum _{i=1}^{N_h}\left(\frac{y_{hi}}{N_h}\right)\text{ be the population mean }\\ & \left(h\text{th stratum}\right)\text{ of the }Y\end{array}$ $\begin{array}{rl}{\overline{X}}_h& =\sum _{i=1}^{N_h}\left(\frac{x_{hi}}{N_h}\right)\text{ be the population mean }\\ & \left(h\text{th stratum}\right)\text{ of the }X\end{array}$ $\begin{array}{rl}{\overline{Z}}_h& =\sum _{i=1}^{N_h}\left(\frac{z_{hi}}{N_h}\right)\text{ be the population mean }\\ & \left(h\text{th stratum}\right)\text{ of the }Z\end{array}$ $\begin{array}{rl}{\overline{y}}_h& =\sum _{i=1}^{n_h}\left(\frac{y_{hi}}{n_h}\right)\text{ be the sample mean }\\ & \left(h\text{th stratum}\right)\text{ of the }Y\end{array}$ $\begin{array}{rl}{\overline{x}}_h& =\sum _{i=1}^{n_h}\left(\frac{x_{hi}}{n_h}\right)\text{ be the sample mean }\\ & \left(h\text{th stratum}\right)\text{ of the }X\end{array}$ $\begin{array}{rl}{\overline{z}}_h& =\sum _{i=1}^{n_h}\left(\frac{z_{hi}}{n_h}\right)\text{ be the sample mean }\\ & \left(h\text{th stratum}\right)\text{ of the }Z\end{array}$ $\begin{array}{rl}{W}_h& =\left(\frac{N_h}{N}\right)\text{be the known stratum weight}\end{array}$

We define the following relative error terms and their expectations to drive the expressions for bias, MSE and minimum MSE of the proposed estimators. ${\xi }_0=\frac{{\overline{y}}_{st}-\overline{Y}}{\overline{Y}},{\xi }_1=\frac{{\overline{z}}_{st}-\overline{Z}}{\overline{Z}}\text{ and}{\xi }_2=\frac{{\overline{x}}_{st}-\overline{X}}{\overline{X}},$such that $E\left({\xi }_0\right)=E\left({\xi }_1\right)=E\left({\xi }_2\right)=0,$

Let us define, (2.1) $V_{abc}=\sum _{h=1}^L{W}_h^{a+b+c}\frac{E\left[{\left({\overline{y}}_h-{\overline{Y}}_h\right)}^a{\left({\overline{z}}_h-{\overline{Z}}_h\right)}^b{\left({\overline{x}}_h-{\overline{X}}_h\right)}^c\right]}{{\overline{Y}}^a{\overline{Z}}^b{\overline{X}}^c}.$(2.1)

Using (2.1), we can write as: (2.2) $\left.\begin{array}{c}E\left({\xi }_0^2\right)=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yh}^2}{{\overline{Y}}^2}=V_{200}\\ E\left({\xi }_1^2\right)=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{zh}^2}{{\overline{Z}}^2}=V_{020}\\ E\left({\xi }_2^2\right)=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{xh}^2}{{\overline{X}}^2}=V_{002}\end{array}\right\},$(2.2) and (2.3) $\left.\begin{array}{c}E\left({\xi }_0{\xi }_1\right)=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yzh}}{\overline{Y}\overline{Z}}=V_{110}\\ E\left({\xi }_0{\xi }_2\right)=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yxh}}{\overline{Y}\overline{X}}=V_{101}\\ E\left({\xi }_1{\xi }_2\right)=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{zxh}}{\overline{Z}\overline{X}}=V_{011}\end{array}\right\},$(2.3) where $\begin{array}{rl}{\varphi }_h& =\left(\frac{1-f_h}{n_h}\right)\text{be the finite population}\\ & \text{correction factor}\end{array}$ $\begin{array}{rl}{f}_h& =\frac{n_h}{N_h}\text{be the sampling fracation for the stratum}\end{array}$ $\begin{array}{rl}{S}_{yh}^2& =\frac{{\sum }_{i=1}^{N_h}{\left(y_{hi}-{\overline{Y}}_h\right)}^2}{N_h-1}\text{ be the population variance}\\ & \left(h\text{th stratum}\right)\text{ of }Y\end{array}$ $\begin{array}{rl}{S}_{xh}^2& =\frac{{\sum }_{i=1}^{N_h}{\left(x_{hi}-{\overline{X}}_h\right)}^2}{N_h-1}\text{ the population variance }\\ & \left(h\text{th stratum}\right)\text{ of }X\end{array}$ $\begin{array}{rl}{S}_{zh}^2& =\frac{{\sum }_{i=1}^{N_h}{\left(z_{hi}-{\overline{Z}}_h\right)}^2}{N_h-1}\text{ be the population variance }\\ & \left(h\text{th stratum}\right)\text{ of }Z\end{array}$ $\begin{array}{rl}{S}_{yxh}& =\frac{{\sum }_{i=1}^{N_h}\left(y_{hi}-{\overline{Y}}_h\right)\left(x_{hi}-{\overline{X}}_h\right)}{N_h-1}\text{be the population}\\ & \text{covariance between }Y\text{ and }X\end{array}$ $\begin{array}{rl}{S}_{yzh}& =\frac{{\sum }_{i=1}^{N_h}\left(y_{hi}-{\overline{Y}}_h\right)\left(z_{hi}-{\overline{Z}}_h\right)}{N_h-1}\text{ be the population}\\ & \text{covariance between }Y\text{ and }Z\end{array}$ $\begin{array}{rl}{S}_{zxh}& =\frac{{\sum }_{i=1}^{N_h}\left(z_{hi}-{\overline{Z}}_h\right)\left(x_{hi}-{\overline{X}}_h\right)}{N_h-1}\text{ be the population}\\ & \text{covariance between }Z\text{ and }X\end{array}$ $\begin{array}{rl}{C}_{yst}& =\sum _{h=1}^L{W}_h{C}_{yh}\text{be the population coefficient}\\ & \text{ of variation }\left(h\text{th stratum}\right)\text{ of }Y\end{array}$ $\begin{array}{rl}{C}_{xst}& =\sum _{h=1}^L{W}_h{C}_{xh}\text{ be the population coefficient}\\ & \text{ of variation }\left(h\text{th stratum}\right)\text{ of }X\end{array}$ $\begin{array}{rl}{\text{β}}_{1xst}& =\sum _{h=1}^L{W}_h{\text{β}}_{1xh}\text{be the population coefficient}\\ & \text{ of skewness }\left(h\text{th stratum}\right)\text{ of }X\end{array}$ $\begin{array}{rl}{\text{β}}_{2xst}& =\sum _{h=1}^L{W}_h{\text{β}}_{2xh}\text{ be the population coefficient}\\ & \text{ of kurtosis }\left(h\text{th stratum}\right)\text{ of }X\end{array}$ $\begin{array}{rl}{\rho }_{yxst}& =\sum _{h=1}^L{W}_h{\rho }_{yxh}=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yxh}}{\sqrt{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yh}^2}\sqrt{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{xh}^2}}\\ & =\frac{V_{101}}{\sqrt{V_{200}}\sqrt{V_{002}}}\text{ be the population correlation}\\ & \text{ coefficient }\left(h\text{th stratum}\right)\text{ between }Y\text{ and }X\end{array}$ $\begin{array}{rl}{\rho }_{yzst}& =\sum _{h=1}^L{W}_h{\rho }_{yzh}=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yzh}}{\sqrt{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yh}^2}\sqrt{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{zh}^2}}\\ & =\frac{V_{110}}{\sqrt{V_{200}}\sqrt{V_{020}}}\text{ be the population correlation}\\ & \text{ coefficient }\left(h\text{th stratum}\right)\text{ between }Y\text{and }Z\end{array}$ $\begin{array}{rl}{\rho }_{zxst}& =\sum _{h=1}^L{W}_h{\rho }_{zxh}=\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{zxh}}{\sqrt{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{zh}^2}\sqrt{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{xh}^2}}\\ & =\frac{V_{011}}{\sqrt{V_{020}}\sqrt{V_{002}}}\text{ be the population correlation}\\ & \text{ coefficient }\left(h\text{th stratum}\right)\text{ between }Z\text{and }X\end{array}$Some well-known estimators for population mean under stratified random sampling scheme are detailed below. All these estimators are based on only original auxiliary information.

2.1. Usual unbiased, combined ratio and combined regression estimators are detailed below

(2.4) ${\hat{\overline{Y}}}_{st}=\sum _{h=1}^L{W}_h{\overline{y}}_h$(2.4) (2.5) ${\hat{\overline{Y}}}_{CR}={\overline{y}}_{st}\left(\frac{\overline{X}}{{\overline{x}}_{st}}\right)$(2.5) (2.6) $\begin{array}{rl}{\hat{\overline{Y}}}_{CReg}& =\left[{\overline{y}}_{st}+\hat{b}\left(\overline{X}-{\overline{x}}_{st}\right)\right],\text{ where }\hat{b}\\ & =\frac{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{yxh}}{{\sum }_{h=1}^L{W}_h^2{\varphi }_h{S}_{xh}^2}.\end{array}$(2.6)

2.2. Haq and Shabbir [7] proposed two exponential ratio-type families of estimators detailed below

(2.7) $\begin{array}{rl}{\hat{\overline{Y}}}_{HS1}& =\left[{\mu }_1{\overline{y}}_{st}+{\mu }_2\left(\overline{X}-{\overline{x}}_{st}\right)\right]\text{exp}\\ & \times \left(\frac{a_{st}\overline{X}+b_{st}}{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}-1\right).\end{array}$(2.7) (2.8) $\begin{array}{rl}{\hat{\overline{Y}}}_{HS2}& =\left[{\mu }_3{\overline{y}}_{st}+{\mu }_4\left(\overline{X}-{\overline{x}}_{st}\right)\right]\text{exp}\\ & \times \left(\frac{a_{st}\overline{X}+b_{st}}{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}-1\right)\\ & \times \left\{\left(\frac{1}{2}\right)\left(\frac{a_{st}\overline{X}+b_{st}}{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}\right.\right.\\ & +{\left.\left.\frac{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}{a_{st}\overline{X}+b_{st}}\right)\right\}}^2.\end{array}$(2.8) where η is the suitable constant, $a_{st}\left(\ne 0\right)$ and b _st are either real numbers or functions of known parameters of the auxiliary variable.

2.3. Singh and Solanki [8] proposed a family of estimators as given below

(2.9) $\begin{array}{rl}{\hat{\overline{Y}}}_{SS1}& =\left[{\mu }_5{\overline{y}}_{st}{\left\{\frac{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}{\left(a_{st}\overline{X}+b_{st}\right)}\right\}}^{{\omega }_1}\right.\\ & +\left.{\mu }_6{\overline{y}}_{st}{\left\{\frac{\left(a_{st}\overline{X}+b_{st}\right)}{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}\right\}}^{{\omega }_2}\right].\end{array}$(2.9)

where $\eta ,a_{st}\left(\ne 0\right)$ and $b_{st}$ are defined earlier.

Remark 2.1:

${\hat{\overline{Y}}}_{SS1}$ reduces to the ratio-type ${\hat{\overline{Y}}}_{SS1R}$, product-type ${\hat{\overline{Y}}}_{SS1P}$ and ratio-cum-product-type ${\hat{\overline{Y}}}_{SS1RP}$ estimators by placing the suitable values of the constants as: $({\omega }_1=0,{\omega }_2=1,\eta =1),({\omega }_1=0,{\omega }_2=-1,\eta =1)\text{ and }({\omega }_1=1,{\omega }_2=1,\eta =1)$, respectively.

2.4. Given below is the class of estimators suggested by Solanki and Singh [9]

(2.10) $\begin{array}{rl}{\hat{\overline{Y}}}_{SS2}& =\left[{\mu }_7{\overline{y}}_{st}{\left[\frac{a_{st}\overline{X}+b_{st}}{\eta \left(a_{st}{\overline{x}}_{st}+b_{st}\right)+\left(1-\eta \right)\left(a_{st}\overline{X}+b_{st}\right)}\right]}^{{\omega }_3}\right.\\ & +\left.{\mu }_8{\overline{y}}_{st}exp\left[\frac{{\omega }_4\left\{\left(a_{st}\overline{X}+b_{st}\right)-\left(a_{st}{\overline{x}}_{st}+b_{st}\right)\right\}}{\left(a_{st}\overline{X}+b_{st}\right)+\left(a_{st}{\overline{x}}_{st}+b_{st}\right)}\right]\right],\end{array}$(2.10)

where $\eta ,a_{st}\left(\ne 0\right)$ and $b_{st}$ are defined earlier.

Remark 2.2:

${\hat{\overline{Y}}}_{SS2}$ reduces the following different estimators by placing different values of $\eta ,{\omega }_3\text{and}{\omega }_4$ in (2.10) as:

1. ${\hat{\overline{Y}}}_{SS2R}$ for $\left(\eta =1,{\omega }_3=0,{\omega }_4=1\right)$.

2. ${\hat{\overline{Y}}}_{SS2P}$ for $\left(\eta =1,{\omega }_3=0,{\omega }_4=-1\right)$.

3. ${\hat{\overline{Y}}}_{SS2RR}$ for $\left(\eta =1,{\omega }_3=1,{\omega }_4=1\right)$.

4. ${\hat{\overline{Y}}}_{SS2PP}$. for $\left(\eta =1,{\omega }_3=-1,{\omega }_4=-1\right)$.

5. ${\hat{\overline{Y}}}_{SS2RP}$ for $\left(\eta =1,{\omega }_3=1,{\omega }_4=-1\right)$.

6. ${\hat{\overline{Y}}}_{SS2PR}$ for $\left(\eta =1,{\omega }_3=-1,{\omega }_4=1\right)$.

2.5. Recently, Solanki and Singh [10] defined an improved estimation given as

(2.11) $\begin{array}{rl}{\hat{\overline{Y}}}_{SS3}& =\left[{\mu }_9{\overline{y}}_{st}{\left(\frac{{\overline{X}}_{st}^{\ast }}{{\overline{x}}_{st}^{\ast }}\right)}^{{\omega }_5}exp\left\{\frac{{\omega }_6\left({\overline{X}}_{st}^{\ast }-{\overline{x}}_{st}^{\ast }\right)}{\left({\overline{X}}_{st}^{\ast }+{\overline{x}}_{st}^{\ast }\right)}\right\}\right.\\ & +\left.{\mu }_{10}{\overline{y}}_{st}{\left(\frac{{\overline{x}}_{st}^{\ast }}{{\overline{X}}_{st}^{\ast }}\right)}^{{\omega }_7}exp\left\{\frac{{\omega }_8\left({\overline{x}}_{st}^{\ast }-{\overline{X}}_{st}^{\ast }\right)}{\left({\overline{X}}_{st}^{\ast }+{\overline{x}}_{st}^{\ast }\right)}\right\}\right],\end{array}$(2.11) where ${\overline{X}}_{st}^{\ast }=\sum _{h=1}^L{W}_h\left(a_h{\overline{X}}_h+b_h\right),{\overline{x}}_{st}^{\ast }=\sum _{h=1}^L{W}_h\left(a_h{\overline{x}}_h+b_h\right)and{a}_h\left(\ne 0\right),b_h$ are either real number to parameters related to auxiliary variate X.

Remark 2.3:

For obtaining different class of estimators ${\hat{\overline{Y}}}_{SS3i}$, assume the different values of the constants ${\omega }_5,{\omega }_6,{\omega }_7\text{ and }{\omega }_8$. in Equation (2.11) as:

1. ${\hat{\overline{Y}}}_{SS31}$ for ${\omega }_5=-1,{\omega }_6=1,{\omega }_7=1\text{ and }{\omega }_8=1$.

2. ${\hat{\overline{Y}}}_{SS32}$ for ${\omega }_5=-1,{\omega }_6=1,{\omega }_7=1\text{ and }{\omega }_8=0$.

3. ${\hat{\overline{Y}}}_{SS33}$ for ${\omega }_5=+1,{\omega }_6=1,{\omega }_7=1\text{ and }{\omega }_8=1$.

Remark 2.4:

The optimal weights of ${\mu }_1,{\mu }_2,{\mu }_3,...,{\mu }_{10}$ are determined for minimizing the MSE’s of estimators mentioned in (2.7)–(2.11). $\begin{array}{rl}{\mu }_1& =\frac{V_{002}\left(2-{\delta }^2{V}_{002}\right)}{2\left\{-V_{101}^2+V_{002}\left(1+V_{200}\right)\right\}},\\ {\mu }_2& =\frac{\begin{array}{c}\overline{Y}\{2{\delta }^3{V}_{002}^2-2V_{101}(-1+\delta V_{101})\\ -\delta V_{002}\left(2+\delta V_{101}-2V_{200}\right)\}\end{array}}{2\overline{X}\left\{-V_{101}^2+V_{002}\left(1+V_{200}\right)\right\}},\\ {\mu }_3& =\frac{V_{002}\left(2+{\delta }^2{V}_{002}\right)}{2\left\{-V_{101}^2+V_{002}\left(1+2{\delta }^2{V}_{002}+V_{200}\right)\right\}},\\ {\mu }_4& =\frac{\begin{array}{c}\overline{Y}\{2{\delta }^3{V}_{002}^2-2V_{101}(-1+\delta V_{101})\\ +\delta V_{002}\left(-2+\delta V_{101}+2V_{200}\right)\}\end{array}}{2\overline{X}\left\{-V_{101}^2+V_{002}\left(1+2{\delta }^2{V}_{002}+V_{200}\right)\right\}},\\ {\mu }_5& =\frac{A_2{A}_4-A_5{A}_3}{A_2{A}_1-A_3^2},{\mu }_6=\frac{A_1{A}_5-A_4{A}_3}{A_2{A}_1-A_3^2},\\ {\mu }_7& =\frac{B_2{B}_4-B_5{B}_3}{B_2{B}_1-B_3^2},{\mu }_8=\frac{B_1{B}_5-B_4{B}_3}{B_2{B}_1-B_3^2},\\ {\mu }_9& =\frac{C_2{C}_4-C_5{C}_3}{C_2{C}_1-C_3^2},{\mu }_{10}=\frac{C_1{C}_5-C_4{C}_3}{C_2{C}_1-C_3^2},\end{array}$where $\begin{array}{rl}\tau & =\frac{a_{st}\overline{X}}{a_{st}\overline{X}+b_{st}},\delta =\eta \tau ,\end{array}$ $\begin{array}{rl}{A}_1& =\left[1+V_{200}+4\eta {\omega }_1\tau V_{101}+{\omega }_1\left(2{\omega }_1-1\right){\eta }^2{\tau }^2{V}_{002}\right],\end{array}$ $\begin{array}{rl}{A}_2& =\left[1+V_{200}-4\eta {\omega }_2\tau V_{101}+{\omega }_2\left(2{\omega }_2+1\right){\eta }^2{\tau }^2{V}_{002}\right],\end{array}$ $\begin{array}{rl}{A}_3& =\left[1+V_{200}+2\eta \left({\omega }_1-{\omega }_2\right)\tau V_{101}\right.\\ & +\left.\left(\frac{{\eta }^2{\tau }^2}{2}\right)\left({\omega }_1-{\omega }_2\right)\left({\omega }_1-{\omega }_2-1\right)V_{002}\right],\end{array}$ $\begin{array}{rl}{A}_4& =\left[1+\eta {\omega }_1\tau V_{101}+\frac{{\omega }_1\left({\omega }_1-1\right)}{2}{\eta }^2{\tau }^2{V}_{002}\right],\end{array}$ $\begin{array}{rl}{A}_5& =\left[1-\eta {\omega }_2\tau V_{101}+\frac{{\omega }_2\left({\omega }_2+1\right)}{2}{\eta }^2{\tau }^2{V}_{002}\right],\end{array}$ $\begin{array}{rl}{B}_1& =\left[1+V_{200}+{\eta }^2{\tau }^2\left(2{\omega }_3^2+{\omega }_3\right)V_{002}-4\eta {\omega }_3\tau V_{101}\right],\end{array}$ $\begin{array}{rl}{B}_2& =\left[1+V_{200}+\frac{{\tau }^2\left({\omega }_4^2+{\omega }_4\right)}{2}{V}_{002}-2{\omega }_4\tau V_{101}\right],\end{array}$ $\begin{array}{rl}{B}_3& =\left[1+V_{200}+\left(\frac{\begin{array}{c}{\tau }^2[{\left(2\eta {\omega }_3+{\omega }_4\right)}^2\\ +2\left(2{\eta }^2{\omega }_3+{\omega }_4\right)]\end{array}}{8}\right)V_{002}\right.\\ & -\left.\tau \left(2\eta {\omega }_3+{\omega }_4\right)V_{101}\phantom{\frac{\begin{array}{c}{\tau }^2[{\left(2\eta {\omega }_3+{\omega }_4\right)}^2\\ +2\left(2{\eta }^2{\omega }_3+{\omega }_4\right)]\end{array}}{8}}\right],\end{array}$ $\begin{array}{rl}{B}_4& =\left[1+\frac{{\eta }^2{\tau }^2\left({\omega }_3^2+{\omega }_3\right)}{2}{V}_{002}-\eta {\omega }_3\tau V_{101}\right],\end{array}$ $\begin{array}{rl}{B}_5& =\left[1+\frac{{\tau }^2\left({\omega }_4^2+2{\omega }_4\right)}{8}{V}_{002}-\frac{{\omega }_4\tau }{2}{V}_{101}\right],\end{array}$ $\begin{array}{rl}{C}_1& =1+\frac{1}{{\overline{Y}}^2}\sum _{h=1}^L{W}_h^2{\varphi }_h\left\{S_{yh}^2-2k_1{a}_h{R}^{\ast }{S}_{yxh}\phantom{\frac{k_1\left(k_1+1\right)}{2}}\right.\\ & +\left.\frac{k_1\left(k_1+1\right)}{2}{a}_h^2{R}^{\ast 2}{S}_{xh}^2\right\},\end{array}$ $\begin{array}{rl}{C}_2& =1+\frac{1}{{\overline{Y}}^2}\sum _{h=1}^L{W}_h^2{\varphi }_h\left\{S_{yh}^2+2k_2{a}_h{R}^{\ast }{S}_{yxh}\phantom{\frac{k_1\left(k_1+1\right)}{2}}\right.\\ & +\left.\frac{k_2\left(k_2-1\right)}{2}{a}_h^2{R}^{\ast 2}{S}_{xh}^2\right\},\end{array}$ $\begin{array}{rl}{C}_3& =1+\frac{1}{{\overline{Y}}^2}\sum _{h=1}^L{W}_h^2{\varphi }_h\left\{S_{yh}^2+\left(k_2-k_1\right)a_h{R}^{\ast }{S}_{yxh}\phantom{\frac{k_1\left(k_1+1\right)}{2}}\right.\\ & +\left.\frac{\left(k_2-k_1\right)\left(k_2-k_1-2\right)}{8}{a}_h^2{R}^{\ast 2}{S}_{xh}^2\right\},\end{array}$ $\begin{array}{rl}{C}_4& =1-\frac{k_1}{2\overline{Y}}\sum _{h=1}^L{W}_h^2{\varphi }_h\left\{R^{\ast }{a}_h{S}_{yxh}-\frac{\left(k_1+2\right)}{4}{a}_h^2{R}^{\ast 2}{S}_{xh}^2\right\},\end{array}$ $\begin{array}{rl}{C}_5& =1+\frac{k_2}{2\overline{Y}}\sum _{h=1}^L{W}_h^2{\varphi }_h\left\{R^{\ast }{a}_h{S}_{yxh}+\frac{\left(k_2-2\right)}{4}{a}_h^2{R}^{\ast 2}{S}_{xh}^2\right\},\end{array}$ $\begin{array}{rl}{k}_1& =\left(2{\omega }_5+{\omega }_6\right),k_2=\left(2{\omega }_7+{\omega }_8\right),\\ k_3& =\frac{S_{yxh}}{S_{xh}^2},R^{\ast }=\left(\frac{\overline{Y}}{{\overline{X}}_{st}^{\ast }}\right).\end{array}$

Remark 2.5:

By placing the suitable weights in corresponding estimators, we have the following minimum MSE’s of above-said estimators. (2.12) $MSE\left({\hat{\overline{Y}}}_{st}\right)=Var\left({\hat{\overline{Y}}}_{st}\right)={\overline{Y}}^2{V}_{200}.$(2.12) (2.13) $MSE\left({\hat{\overline{Y}}}_{CR}\right)={\overline{Y}}^2\left(V_{200}+V_{002}-2V_{101}\right).$(2.13) (2.14) $MSE\left({\hat{\overline{Y}}}_{CReg}\right)={\overline{Y}}^2{V}_{200}\left(1-{\rho }_{yxst}^2\right).$(2.14) (2.15) $MS{E}_{min}\left({\hat{\overline{Y}}}_{HS1}\right)\cong \frac{\begin{array}{c}{\overline{Y}}^2[4V_{101}^2+V_{002}\{{\delta }^4{V}_{002}^2-4{\delta }^2{V}_{101}^2\\ +4\left(-1+{\delta }^2{V}_{002}\right)V_{200}\left\}\right]\end{array}}{4\left\{V_{101}^2-V_{002}\left(1+V_{200}\right)\right\}}.$(2.15) (2.16) $MS{E}_{min}\left({\hat{\overline{Y}}}_{HS2}\right)\cong \frac{\begin{array}{c}{\overline{Y}}^2[4V_{101}^2+V_{002}\{9{\delta }^4{V}_{002}^2-4{\delta }^2{V}_{101}^2\\ +4\left(-1+{\delta }^2{V}_{002}\right)V_{200}\left\}\right]\end{array}}{4\left\{V_{101}^2-V_{002}\left(1+2{\delta }^2{V}_{002}+V_{200}\right)\right\}}.$(2.16) (2.17) $MS{E}_{min}\left({\hat{\overline{Y}}}_{SS1}\right)\cong {\overline{Y}}^2\left[1-\frac{A_2{A}_4^2-2A_4{A}_3{A}_5+A_5^2{A}_1}{A_2{A}_1-A_3^2}\right].$(2.17) (2.18) $MS{E}_{min}\left({\hat{\overline{Y}}}_{SS2}\right)\cong {\overline{Y}}^2\left[1-\frac{B_2{B}_4^2-2B_4{B}_3{B}_5+B_5^2{B}_1}{B_2{B}_1-B_3^2}\right].$(2.18) (2.19) $MS{E}_{min}\left({\hat{\overline{Y}}}_{SS3}\right)\cong {\overline{Y}}^2\left[1-\frac{C_1{C}_5^2-2C_4{C}_3{C}_5+C_4^2{C}_2}{C_2{C}_1-C_3^2}\right].$(2.19)

3. Proposed estimators

In this section, two new exponential-type estimators are proposed for the estimation of population mean using dual auxiliary information in stratified random sampling. Dual auxiliary information refers to the double use of auxiliary variable (i) the original/actual measurements of the auxiliary variable and (ii) the use of ranks of the auxiliary variable. Mathematical properties such as bias and mean square error (MSE) of the proposed estimators are derived up to first order of approximation. The bias of an estimator is the difference between the estimator's expected value and the true value of the parameter being estimated i.e. $Bias\left(\hat{\overline{Y}}\right)=E\left(\hat{\overline{Y}}-\overline{Y}\right)$ and MSE can be defined as the divergence of the estimator values from the true parameter value i.e. $MSE\left(\hat{\overline{Y}}\right)=E(\hat{\overline{Y}}-\overline{Y}{)}^2.$

3.1. First proposed estimator

(3.1) $\begin{array}{rl}{\hat{\overline{Y}}}_{P1}& =\frac{{\mu }_{11}}{2}\left(\frac{\overline{X}}{{\overline{x}}_{st}}+\frac{{\overline{x}}_{st}}{\overline{X}}\right){\hat{\overline{Y}}}_{B{T}_{st},Avg}+{\mu }_{12}\left(\overline{Z}-{\overline{z}}_{st}\right)\\ & +{\mu }_{13}\left(\overline{X}-{\overline{x}}_{st}\right)\text{exp}\left(\frac{\overline{X}-{\overline{x}}_{st}}{\overline{X}+{\overline{x}}_{st}}\right),\end{array}$(3.1) where ${\mu }_{11},{\mu }_{12}\text{and}{\mu }_{13}$ are the suitably chosen weights.

The bias and MSE of ${\hat{\overline{Y}}}_{P1}$ are given below (3.2) $Bias\left({\hat{\overline{Y}}}_{P1}\right)\cong \left({\mu }_{11}-1\right)\overline{Y}+\frac{V_{002}}{2}\left({\mu }_{13}\overline{X}+\frac{5{\mu }_{11}\overline{Y}}{4}\right),$(3.2) and (3.3) $\begin{array}{rl}MSE\left({\hat{\overline{Y}}}_{P1}\right)& \cong {\overline{Y}}^2\left[1+{\mu }_{11}^2\left(1+V_{200}+\frac{5}{4}{V}_{002}\right)\right.\\ & +{\mu }_{12}^2{\tilde{R}}^2{V}_{020}+{\mu }_{13}^2{\acute{R}}^2{V}_{002}\\ & -{\mu }_{11}\left(2+\frac{5}{4}{V}_{002}\right)\\ & +\phantom{2+\frac{5}{4}{V}_{002}}2{\mu }_{12}\tilde{R}\left({\mu }_{13}\acute{R}{V}_{011}-{\mu }_{11}{V}_{110}\right)\\ & -\left.{\mu }_{13}\acute{R}{V}_{002}\left(2\acute{\kappa }{\mu }_{11}-{\mu }_{11}+1\right)\right],\end{array}$(3.3) where $\acute{\kappa }={\rho }_{yxst}\frac{C_{yst}}{C_{xst}},\acute{R}=\frac{\overline{X}}{\overline{Y}}\text{and}\tilde{R}=\frac{\overline{Z}}{\overline{Y}}.$

The optimal weights ${\mu }_{11},{\mu }_{12}\text{and}{\mu }_{13}$ are obtained by minimizing Equation (3.3), so ${\mu }_{11\left(\text{opt}\right)}=\frac{E_1{E}_2-2V_{002}{V}_{020}{E}_3}{E_2{E}_4-2E_3^2},$ ${\mu }_{12\left(\text{opt}\right)}=\frac{\begin{array}{c}2V_{110}\left(E_1{E}_2-2V_{002}{V}_{020}{E}_3\right)\\ +V_{011}\left(E_1{E}_3-V_{002}{V}_{020}{E}_4\right)\end{array}}{2\tilde{R}{V}_{020}\left(E_2{E}_4-2E_3^2\right)},$ ${\mu }_{13\left(\text{opt}\right)}=\frac{V_{002}{V}_{020}{E}_4-E_1{E}_3}{2\acute{R}\left(E_2{E}_4-2E_3^2\right)}.$

Inserting optimal weights of ${\mu }_{11},{\mu }_{12}\text{and}{\mu }_{13}$ in Equation (3.3), the minimum MSE of the proposed estimator is (3.4) $\begin{array}{rl}& MS{E}_{min}\left({\hat{\overline{Y}}}_{P1\left(st\right)}\right)\\ & \cong \frac{{\overline{\mathrm{Y}}}^2}{4V_{020}{F}_1^2}\left[\begin{array}{l}4V_{020}{F}_1^2+(4V_{020}+4V_{020}{V}_{200}\\ -4V_{110}^2+5V_{002}{V}_{020})F_2^2\\ +(V_{002}{V}_{020}-V_{011}^2)F_3^2\\ -V_{020}\left(8+5V_{002}\right)F_1{F}_2\\ +2\left(2V_{110}{V}_{011}-V_{020}{V}_{002}\right.\\ \left.\left(2\stackrel{\mathrm{\prime }}{\kappa }-1\right)\right)F_2{F}_3-2V_{002}{V}_{020}{F}_1{F}_3\end{array}\right],\end{array}$(3.4) where $\begin{array}{rl}& E_1=8V_{020}+5V_{002}{V}_{020},\\ & E_2=V_{002}{V}_{020}-V_{011}^2,\\ & E_3=2V_{110}{V}_{011}-V_{020}{V}_{002}\left(2\acute{\kappa }-1\right),\\ & E_4=8V_{020}\left(1+V_{200}\right)+10V_{002}{V}_{020}-8V_{110}^2,\\ & F_1=E_2{E}_4-2E_3^2,\\ & F_2=E_1{E}_2-2V_{002}{V}_{020}{E}_3,\\ & F_3=V_{002}{V}_{020}{E}_4-E_1{E}_3\end{array}$

3.2. Second proposed estimator

(3.5) $\begin{array}{rl}{\hat{\overline{Y}}}_{P2}& ={\mu }_{14}{\overline{y}}_{st}+{\mu }_{15}\left(\overline{Z}-{\overline{z}}_{st}\right)\\ & +{\mu }_{16}\left(\overline{X}-{\overline{x}}_{st}\right)\text{exp}\left(\frac{2\left(\overline{X}-{\overline{x}}_{st}\right)}{\overline{X}+{\overline{x}}_{st}}\right).\end{array}$(3.5) where ${\mu }_{14},{\mu }_{15}\text{ and }{\mu }_{16}$ are the suitably chosen weights.

The bias and MSE of ${\hat{\overline{Y}}}_{P2}$ are given below (3.6) $Bias\left({\hat{\overline{Y}}}_{P2}\right)\cong \overline{Y}\left[\left({\mu }_{14}-1\right)+{\mu }_{16}{V}_{002}\acute{R}\right],$(3.6) and (3.7) $\begin{array}{rl}MSE\left({\hat{\overline{Y}}}_{P2}\right)& \cong {\overline{Y}}^2\left[1+{\mu }_{14}^2\left(1+V_{200}\right)+{\mu }_{15}^2{\tilde{R}}^2{V}_{020}\right.\\ & +{\mu }_{16}^2{\acute{R}}^2{V}_{002}-2{\mu }_{14}-2{\mu }_{16}\acute{R}{V}_{002}\\ & -2{\mu }_{14}{\mu }_{16}\acute{R}{V}_{002}\left(\acute{\kappa }-1\right)\\ & +\left.2{\mu }_{15}{\mu }_{16}\acute{R}\tilde{R}{V}_{011}-2{\mu }_{14}{\mu }_{15}\tilde{R}{V}_{110}\right],\end{array}$(3.7)

By minimizing Equation (3.7), the optimal weights ${\mu }_{14},{\mu }_{15}\text{ and }{\mu }_{16}$ are as under: $\begin{array}{rl}{\mu }_{14\left(\text{opt}\right)}& =\frac{\begin{array}{c}\left(1+V_{200}\right)V_{020}{E}_5-E_6(V_{011}+V_{011}{V}_{200}\\ +V_{110}{E}_7)+V_{002}\left(\acute{\kappa }-1\right)E_7{E}_8\end{array}}{\left(1+V_{200}\right)\left(E_5{E}_8-E_6^2\right)},\\ {\mu }_{15\left(\text{opt}\right)}& =\frac{V_{110}{E}_5-E_6{E}_7}{\tilde{R}\left(E_5{E}_8-E_6^2\right)}\\ {\mu }_{16\left(\text{opt}\right)}& =\frac{E_7{E}_8-V_{110}{E}_6}{\acute{R}\left(E_5{E}_8-E_6^2\right)},\end{array}$

Inserting optimal weights of ${\mu }_{14},{\mu }_{15}\text{ and }{\mu }_{16}$ in Equation (3.7), the minimum MSE of the proposed estimator is (3.8) $\begin{array}{rl}& MS{E}_{min}\left({\hat{\overline{Y}}}_{P2\left(st\right)}\right)\cong \frac{{\overline{\mathrm{Y}}}^2}{\left(1+V_{200}\right)F_4^2}\\ & \times \left[\begin{array}{l}\left(1+V_{200}\right)F_4^2+V_{002}\left(1+V_{200}\right)F_5^2\\ +V_{020}\left(1+V_{200}\right)F_6^2+F_7^2\\ -2\left(V_{002}{F}_4-V_{011}{F}_6\right)\left(1+V_{200}\right)F_5\\ -2\left(V_{110}{F}_6+V_{002}\left(\acute{\kappa }-1\right)F_5+F_4\right)F_7\end{array}\right],\end{array}$(3.8) where $\begin{array}{rl}& E_5=V_{002}\left(1+V_{200}\right)-V_{002}^2{\left(\acute{\kappa }-1\right)}^2,\\ & E_6=V_{011}\left(1+V_{200}\right)-V_{110}{V}_{002}\left(\acute{\kappa }-1\right),\\ & E_7=V_{002}\left(1+V_{200}\right)+V_{002}\left(\acute{\kappa }-1\right),\\ & E_8=V_{020}\left(1+V_{200}\right)-V_{110}^2,\\ & F_4=E_5{E}_8-E_6^2,\\ & F_5=E_7{E}_8-V_{110}{E}_6,\\ & F_6=V_{110}{E}_5-E_6{E}_7,\\ & F_7=V_{020}\left(1+V_{200}\right)E_5-V_{011}\left(1+V_{200}\right)E_6\\ & -V_{110}{E}_6{E}_7+V_{002}\left(\acute{\kappa }-1\right)E_7{E}_8.\end{array}$

4. Application on a real data

In this section, we compare the performance of newly proposed estimators with the traditional unbiased, combined ratio and combined regression estimators and existing estimators, i.e. Haq and Shabbir [7], Singh and Solanki [8] and Solanki and Singh [10,11]. We considered a real-life data set of Turkey (2007) used by Koyuncu and Kadilar [3]. For the remaining characteristics of the data set, interested readers may refer to Koyuncu and Kadilar [3]. Necessary data statistics are given in Table 1.

Table 1. Data statistics.

	$h\text{th}$ Stratum
Values	1	2	3	4	5	6
$N_h$	127	117	103	170	205	201
$n_h$	31	21	29	38	22	39
${\overline{Y}}_h$	703.7402	413.00	573.17	424.66	267.03	393.84
${\overline{X}}_h$	20,804.5906	9211.79	14,309.3	9478.85	5569.95	12,997.59
${\overline{Z}}_h$	64.0000	59	52	85.5	103	101
$S_{yh}$	883.8348	644.922	1033.467	810.585	403.654	711.723
$S_{xh}$	30,486.7514	15,180.769	27,549.697	18,218.931	8497.776	23,094.141
$S_{zh}$	36.8057	33.9190	29.8775	49.2188	59.3225	58.1678
$S_{yxh}$	25,237,153.52	9,747,942.85	28,294,397.04	14,523,885.53	3,393,591.75	15,864,573.97
$S_{yzh}$	26,804.3730	14,403.5689	19,568.9314	25,375.9556	15,794.0515	24,275.7125
$S_{zxh}$	879,077.4127	335,602.2414	513,448.8922	577,739.0621	335,511.1225	827,795.6600
$C_{xh}$	1.4653	1.6479	1.9253	1.9221	1.5256	1.7768
$C_{yh}$	1.2559	1.5616	1.8031	1.9088	1.5116	1.8071
$C_{zh}$	0.5751	0.5749	0.5746	0.5757	0.5759	0.5759
${\rho }_{yxh}$	0.9366	0.9956	0.9938	0.9835	0.9893	0.9652
${\rho }_{yzh}$	0.8239	0.6584	0.6338	0.6361	0.6596	0.5864
${\rho }_{zxh}$	0.7834	0.6518	0.6238	0.6443	0.6656	0.6162
${\text{β}}_{2xh}$	4.5929	18.5425	15.4459	10.1620	21.9467	23.1140
${\text{β}}_{1xh}$	2.1638	3.8671	3.7479	3.1212	4.0844	4.4111

We calculated the MSEs of the proposed and competing exponential-type estimators and are presented in Table A1. Table A1 reveals that the proposed estimators have smaller MSE values i.e. (57.0590 and 67.9338) among all the reviewed exponential-type estimators i.e. ${\hat{\overline{Y}}}_{HS1},{\hat{\overline{Y}}}_{HS2},{\hat{\overline{Y}}}_{SS1R},{\hat{\overline{Y}}}_{SS1P},{\hat{\overline{Y}}}_{SS1RP},{\hat{\overline{Y}}}_{SS2R},{\hat{\overline{Y}}}_{SS2P},{\hat{\overline{Y}}}_{SS2RR},{\hat{\overline{Y}}}_{SS2PP},{\hat{\overline{Y}}}_{SS2RP},{\hat{\overline{Y}}}_{SS2PR},{\hat{\overline{Y}}}_{SS31},{\hat{\overline{Y}}}_{SS32}\text{and}{\hat{\overline{Y}}}_{SS33}$.

5. Simulation study based on real data

In the previous section, it is clearly observed that proposed estimators are efficient over the competing estimators. In addition, this superiority is assessed through a Monte Carlo simulation study using R software. Again, the real population presented in Table 1 is used. We considered different sample sizes $\left(n=180,230\text{ and }280\right)$ through the proportional allocation method. The steps of a simulation study to find the average MSE of an estimator are as follows:

Step 1: Select a bivariate stratified sample of size $n$ using SRSWOR from the bivariate stratified population.

Step 2: Use sample data from step 1 to find the MSE of all the estimators under study.

Step 3: The whole procedure is repeated 30,000 times and obtain 30,000 values i.e. $\hat{y}$ for MSEs.

Step 4: Average MSE of each estimator is calculated as: $MSE=\frac{{\sum }_{i=1}^{30000}{\left(\hat{y}-\overline{Y}\right)}^2}{30000}.$

Tables A2–A4 present the minimum mean square errors provided by the simulation study. It is quite obvious, as in the previous section, that the proposed estimators ${\hat{\overline{Y}}}_{P1}\text{ and }{\hat{\overline{Y}}}_{P2}$ have the least MSEs over all the competing estimators under study in different sample sizes i.e. $n=180,230\text{ and }280.$

The sequel of the above findings, the performance of the proposed estimators ${\hat{\overline{Y}}}_{P1}\text{ and }{\hat{\overline{Y}}}_{P2}$ is the best among all the reviewed estimators under study.

6. Concluding remarks

Several estimators for the estimation of finite population mean based only on original auxiliary information under stratified random sampling are available in the literature. Haq et al. [18] built up a family of estimators for evaluating the population mean under simple random sampling scheme by using additional information of the auxiliary variable called ranked auxiliary variable. First time in this manuscript, new optimal estimators are suggested for the estimation of population mean by using the original and the ranked auxiliary information under a stratified random sampling scheme. Mathematical properties such as bias, mean square error (MSE) and minimum MSE of the proposed estimators are derived up to the first degree of approximation. Both real-life applications and simulation studies are performed to access the potentiality of the proposed estimators over the competitors. Numerical findings confirmed that the proposed estimators have the minimum mean square errors than all the other existing estimators such as usual unbiased, combined ratio, combined regression, Haq and Shabbir [7], Singh and Solanki [8] and Solanki and Singh [10,11]. Therefore, new proposed estimators under stratified random sampling are very attractive to the survey statisticians.

The possible extension of this current work to estimate the: (1) finite population mean under other sampling designs like stratified double sampling and different rank set sampling schemes, etc.; (2) other unknown finite population parameters including median, variance, interquartile range and proportions, etc.; (3) population mean of a sensitive variable in the presence of sensitive and non-sensitive auxiliary information.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix

Table A1. Minimum MSEs of different estimators based on a real population.

$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{HS1}$	${\hat{\overline{Y}}}_{HS2}$	${\hat{\overline{Y}}}_{SS1R}$	${\hat{\overline{Y}}}_{SS1P}$	${\hat{\overline{Y}}}_{SS1RP}$	${\hat{\overline{Y}}}_{SS2R}$	${\hat{\overline{Y}}}_{SS2P}$
1	0	183.482	115.870	193.418	175.925	192.847	194.156	185.521
1	$\sum _{h=1}^L{W}_h{\rho }_h$	183.485	115.895	193.419	175.925	192.848	194.156	185.520
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	183.488	115.914	193.420	175.925	192.849	194.156	185.519
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	183.536	116.292	193.432	175.925	192.875	194.152	185.509
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	183.494	115.964	193.421	175.925	192.853	194.155	185.518
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	183.514	116.118	193.426	175.925	192.863	194.154	185.514
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	183.484	115.882	193.418	175.925	192.848	194.156	185.520
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	183.483	115.877	193.419	175.925	192.847	194.156	185.521
$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{SS2RR}$	${\hat{\overline{Y}}}_{SS2PP}$	${\hat{\overline{Y}}}_{SS2RP}$	${\hat{\overline{Y}}}_{SS2PR}$	${\hat{\overline{Y}}}_{SS31}$	${\hat{\overline{Y}}}_{SS32}$	${\hat{\overline{Y}}}_{SS33}$	${\hat{\overline{Y}}}_{P1}$	${\hat{\overline{Y}}}_{P2}$
1	0	193.648	143.775	193.648	191.975	108.859	143.775	172.099	57.059	67.934
1	$\sum _{h=1}^L{W}_h{\rho }_h$	193.649	143.781	193.649	191.974	108.870	143.780	172.110	57.059	67.934
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	193.649	143.784	193.649	191.974	108.879	143.784	172.119	57.059	67.934
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	193.657	143.864	193.657	191.957	109.054	143.864	172.302	57.059	67.934
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	193.650	143.795	193.156	191.971	108.902	143.795	172.144	57.059	67.934
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	193.653	143.827	193.653	191.965	107.0285	132.792	161.589	57.059	67.934
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	193.649	143.778	193.152	191.975	178.566	214.558	231.179	57.059	67.934
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	193.648	143.777	193.152	191.975	178.563	214.557	231.175	57.059	67.934

Table A2. Minimum MSEs of different estimators based on a simulation study ($n=180$).

$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{HS1}$	${\hat{\overline{Y}}}_{HS2}$	${\hat{\overline{Y}}}_{SS1R}$	${\hat{\overline{Y}}}_{SS1P}$	${\hat{\overline{Y}}}_{SS1RP}$	${\hat{\overline{Y}}}_{SS2R}$	${\hat{\overline{Y}}}_{SS2P}$
1	0	172.008	104.345	181.844	163.382	181.228	182.417	173.304
1	$\sum _{h=1}^L{W}_h{\rho }_h$	171.478	103.978	181.294	162.897	180.683	181.871	172.790
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	172.610	104.599	182.500	163.926	181.885	183.076	173.907
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	171.054	104.128	180.793	162.449	180.194	181.350	172.295
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	171.047	103.276	180.894	162.377	180.282	181.465	172.325
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	172.405	104.791	182.243	163.748	181.635	182.811	173.681
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	172.502	104.963	182.322	163.991	181.701	182.909	173.861
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	172.031	104.454	181.851	163.484	181.230	182.433	173.367
$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{SS2RR}$	${\hat{\overline{Y}}}_{SS2PP}$	${\hat{\overline{Y}}}_{SS2RP}$	${\hat{\overline{Y}}}_{SS2PR}$	${\hat{\overline{Y}}}_{SS31}$	${\hat{\overline{Y}}}_{SS32}$	${\hat{\overline{Y}}}_{SS33}$	${\hat{\overline{Y}}}_{P1}$	${\hat{\overline{Y}}}_{P2}$
1	0	182.084	130.465	181.559	179.980	94.757	130.465	160.744	52.425	63.241
1	$\sum _{h=1}^L{W}_h{\rho }_h$	181.534	130.064	181.011	179.447	94.430	130.059	160.228	52.425	63.241
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	182.741	130.814	182.216	180.624	94.870	130.805	161.282	52.425	63.241
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	181.028	129.823	180.516	178.921	94.240	129.733	159.780	52.425	63.241
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	181.133	129.374	180.611	179.022	93.526	129.354	159.726	52.425	63.241
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	182.481	130.814	181.961	180.366	94.973	130.761	161.075	52.425	63.241
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	182.565	131.214	182.034	180.498	95.642	131.212	161.205	52.425	63.241
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	182.094	130.0	181.563	180.015	95.064	130.668	160.734	52.425	63.241

Table A3. Minimum MSEs of different estimators based on a simulation study ($n=230$).

$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{HS1}$	${\hat{\overline{Y}}}_{HS2}$	${\hat{\overline{Y}}}_{SS1R}$	${\hat{\overline{Y}}}_{SS1P}$	${\hat{\overline{Y}}}_{SS1RP}$	${\hat{\overline{Y}}}_{SS2R}$	${\hat{\overline{Y}}}_{SS2P}$
1	0	128.404	93.426	133.472	124.181	133.169	133.784	129.180
1	$\sum _{h=1}^L{W}_h{\rho }_h$	128.481	93.299	133.575	124.241	133.268	133.889	129.264
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	127.542	92.471	132.617	123.297	132.311	132.927	128.310
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	127.622	92.729	132.682	123.301	132.387	132.976	128.328
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	127.589	92.586	132.658	123.338	132.355	132.966	128.348
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	128.157	93.117	133.235	123.888	132.934	133.541	128.910
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	128.108	92.972	133.199	123.843	132.895	133.508	128.873
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	128.172	93.044	133.259	123.915	132.955	133.569	128.940
$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{SS2RR}$	${\hat{\overline{Y}}}_{SS2PP}$	${\hat{\overline{Y}}}_{SS2RP}$	${\hat{\overline{Y}}}_{SS2PR}$	${\hat{\overline{Y}}}_{SS31}$	${\hat{\overline{Y}}}_{SS32}$	${\hat{\overline{Y}}}_{SS33}$	${\hat{\overline{Y}}}_{P1}$	${\hat{\overline{Y}}}_{P2}$
1	0	133.592	107.684	133.332	132.562	89.970	107.684	122.608	50.152	55.855
1	$\sum _{h=1}^L{W}_h{\rho }_h$	133.696	107.656	133.432	132.663	89.842	107.653	122.630	50.152	55.855
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	132.737	106.751	132.475	131.702	88.977	106.747	121.706	50.152	55.855
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	132.798	152	132.545	131.731	88.894	106.707	121.801	50.152	55.855
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	132.778	106.810	132.517	131.738	89.043	106.800	121.769	50.152	55.855
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	133.354	107.328	133.095	132.308	89.493	107.302	122.309	50.152	55.855
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	133.318	107.248	133.058	132.276	89.429	107.247	122.292	50.152	55.855
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	133.379	107.335	133.118	132.340	89.532	107.334	122.350	50.152	55.855

Table A4. Minimum MSEs of different estimators based on a simulation study ($n=280$).

$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{HS1}$	${\hat{\overline{Y}}}_{HS2}$	${\hat{\overline{Y}}}_{SS1R}$	${\hat{\overline{Y}}}_{SS1P}$	${\hat{\overline{Y}}}_{SS1RP}$	${\hat{\overline{Y}}}_{SS2R}$	${\hat{\overline{Y}}}_{SS2P}$
1	0	99.582	79.330	102.510	97.205	102.340	102.693	100.059
1	$\sum _{h=1}^L{W}_h{\rho }_h$	99.737	79.475	102.668	97.347	102.499	102.850	100.208
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	99.387	79.074	102.323	96.990	102.154	102.505	99.857
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	99.757	79.451	102.695	97.340	102.528	102.874	100.215
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	99.885	79.536	102.829	97.469	102.661	103.010	100.348
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	99.637	79.441	102.558	97.262	102.389	102.740	100.110
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	100.067	79.711	103.011	97.679	102.840	103.195	100.548
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	99.407	79.147	102.337	97.014	102.168	102.518	99.875
$a_{st}$	$b_{st}$	${\hat{\overline{Y}}}_{SS2RR}$	${\hat{\overline{Y}}}_{SS2PP}$	${\hat{\overline{Y}}}_{SS2RP}$	${\hat{\overline{Y}}}_{SS2PR}$	${\hat{\overline{Y}}}_{SS31}$	${\hat{\overline{Y}}}_{SS32}$	${\hat{\overline{Y}}}_{SS33}$	${\hat{\overline{Y}}}_{P1}$	${\hat{\overline{Y}}}_{P2}$
1	0	102.578	87.820	102.431	101.996	77.804	87.820	96.240	46.135	49.474
1	$\sum _{h=1}^L{W}_h{\rho }_h$	102.735	87.946	102.589	102.149	77.912	87.945	96.399	46.135	49.474
1	$\sum _{h=1}^L{W}_h{C}_{xh}$	102.390	87.566	102.244	101.804	77.503	87.563	96.034	46.135	49.474
1	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	102.762	87.899	102.618	102.167	77.771	87.873	96.370	46.135	49.474
1	$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	102.896	88.013	102.751	102.302	77.911	88.007	96.535	46.135	49.474
$\sum _{h=1}^L{W}_h{C}_{xh}$	$\sum _{h=1}^L{W}_h{\text{β}}_{2xh}$	102.625	87.898	102.479	102.044	77.874	87.883	96.274	46.135	49.474
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{C}_{xh}$	103.079	88.244	102.931	102.495	78.173	88.243	96.708	46.135	49.474
$\sum _{h=1}^L{W}_h{\text{β}}_{1xh}$	$\sum _{h=1}^L{W}_h{\rho }_h$	102.404	87.611	102.258	101.817	77.578	87.611	96.072	46.135	49.474