Enhanced estimation of population mean using simple random sampling

Abstract

The present study suggests an enhanced class of estimators for the population mean estimation utilizing simple random sampling (SRS). The bias, mean square errors (MSE), and minimum MSE of the suggested estimators are computed to the approximation of order one. The efficiency conditions are obtained by comparing the MSE of the proposed and existing estimators. An empirical investigation is carried out using some real and artificially generated symmetric and asymmetric populations. The empirical findings are observed to be positive, clearly demonstrating the preponderance of the suggested estimators over the existing estimators.

KEYWORDS:

• Mean square error
• efficiency
• enhanced estimation
• numerical study
• simulation study

1 Introduction

The traditional ratio, regression, and exponential approaches have been widely utilized for population mean estimation in the recent years due to their simple structure and computational ease. With the help of various auxiliary variable-related parameters, including coefficients of variation, skewness and kurtosis, along with the population mean, the standard ratio and exponential estimators have been improved by various researchers to estimate the unknown population mean of the study variable. When applying the ratio, regression, and exponential estimation methods, the population characteristics of the auxiliary variable must also be available ahead of time. Watson (1937) considered auxiliary information and proposed the conventional regression estimator of population mean of the study variable which is best linear unbiased (BLU) estimator. Cochran (1940) proposed the conventional ratio estimator of population mean under SRS. Srivastava (1967) utilized auxiliary information and suggested a power ratio estimator of population mean under SRS. Walsh (1970) suggested a ratio type estimator for the population mean under SRS. Bahl and Tuteja (1991) proposed an exponential ratio estimator of population mean under SRS. Sisodia and Dwivedi (1981), Singh and Kakran (1993), Upadhyaya and Singh (1999), and Singh (2003a) utilized known parameters of auxiliary variable and suggested some modified ratio estimators of population mean under SRS. Singh et al. (2009) suggested a generalized exponential ratio type estimator of population mean which was later on enhanced by Yadav and Kadilar (2013) by utilizing Searls (1964) technique. Yan and Tian (2010) utilized the coefficient of skewness of auxiliary variable and presented a ratio method of estimation of population mean, whereas Subramani and Kumarapandiyan (2012) considered co-efficient of variation and median of an auxiliary variable and developed an estimation procedure of population mean. Jeelani et al. (2013) developed a modified ratio estimators of population mean utilizing a linear combination of co-efficient of skewness and quartile deviation, while Jerajuddin and Kishun (2016) suggested a modified ratio estimators for population mean utilizing size of the sample chosen from the population. Kadilar (2016) suggested an improved ratio cum exponential ratio estimator of population mean. Soponviwatkul and Lawson (2017) considered utilizing a coefficient of variation, correlation coefficient and a regression coefficient, and construct a new ratio estimator for estimating population mean under SRS. Ijaz and Ali (2018) suggested some improved ratio estimators for estimating population mean utilizing SRS. Yadav et al. (2019) adopted various auxiliary information and developed a class of population mean estimators under SRS. Yadav et al. (2019) examined the efficiency of their developed estimators using primary data of production of peppermint oil obtained from the crop from Banikodar Block of Barabanki District situated in Uttar Pradesh, India.

The studies discussed in previous paragraph are either equally or less efficient than the conventional regression (BLU) estimator. In this article, we propose an enhanced class of estimators for the population mean (PM) estimation of the study variable (SV) employing the data from auxiliary variable (AV) based on SRS. The aim of this paper is to:

1. Propose an enhanced class of estimators for the PM estimation under SRS that competes with the existing estimators, especially regression (BLU) estimator,

2. Compare theoretically the efficiency of the suggested estimators with the current estimators,

3. Exemplify the proposed estimators using some real-life populations,

4. Perform a simulation study using some artificially generated symmetric and asymmetric populations.

The current paper is further designed in the succeeding sections. In Section 2, methodology and terminologies used are explained. In Section 3, some prominent estimators along with their bias, MSE, and minimum MSE expressions are discussed. In Section 4, we suggest an enhanced class of estimators for PM estimation under SRS along with the bias, MSE, and minimum MSE expressions. The efficiency conditions are obtained in Section 5. In Section 6, an empirical illustration of the efficiency of the suggested estimators is provided by using some examples based on some real populations as well as some artificially generated populations. The paper is concluded in Section 7.

2 Methodology and notations

Let a population $\mathrm{P}=$ (${\mathrm{P}}_1,{\mathrm{P}}_2$,…, ${\mathrm{P}}_{\mathrm{N}})$ be the composition of the distinguishable items with a finite size N. Choose a sample of n size based on a finite population of N length by employing simple random sampling without replacement. Let y_i and x_i symbolize the SV and AV for the i^th (i $=$ 1, 2,…, N) unit from P. Let $\overline{y}={\sum }_{i=1}^n{y}_i/n$ and $\overline{Y}={\sum }_{i=1}^N{y}_i/N$ be the sample and population mean of SV y, respectively; $\overline{x}=$ ${\sum }_{i=1}^n{x}_i/n$ and $\overline{X}={\sum }_{i=1}^N{x}_i/N$ be the sample and population mean of AV x. Let M_x and M_y be the median of AV and SV, respectively. Let $s_{y=}\sqrt{{\sum }_{i=1}^n{(y_i-\overline{y})}^2/(n-1)}$ and $S_{y=}\sqrt{{\sum }_{i=1}^N{(y_i-\overline{Y})}^2/(N-1)}$ be the sample and population standard deviation of SV y. Let $s_x=\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2/(n-1)}$ and $S_x=\sqrt{{\sum }_{i=1}^N{(x_i-\overline{X})}^2/(N-1)}$ be the sample and population standard deviation of AV x. Let C ${}_{\mathrm{y}}=$S ${}_{\text{y/}}\overline{Y}$ and C_x = S_x/$\overline{X}$ be the population coefficient of variation of SV and AV, respectively. Let ρ_xy represent the population correlation coefficient between AV and SV.

The characteristics of the suggested estimators are further deduced using the notations below:

Let us consider the error terms as $e_0=(\overline{y}-\overline{Y})$/$\overline{Y}$ and $e_1=(\overline{x}-\overline{X})/\overline{X}$ such that the expected values of these error terms are zero and E(e ${}_0^2)=$ $f{C}_y^2,$ E(e ${}_1^2)=$ $f{C}_x^2$, and E(e₀e ${}_1)=$ $f{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y$, where $f=$1/n.

3 Existing estimators

The current section devotes to some prominent available estimators which are discussed below along with their MSE/minimum MSE expressions.

The usual unbiased estimator is given by: $$t_m=\overline{y},$$with the variance as $$V\left(t_m\right)=f{\overline{Y}}^2{C}_y^2.$$

The usual ratio estimator was proposed by Cochran (1940) which is given by: $$t_r=\overline{y}\frac{\overline{X}}{\overline{x}},$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_r\right)=f\overline{Y}(C_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_r\right)=f{\overline{Y}}^2(C_x^2+C_y^2-2{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\end{array}$$

The usual regression estimator was proposed by Watson (1937) which is given by $$t_{\mathit{\text{reg}}}=\overline{y}+\beta (\overline{X}-\overline{x}),$$with the bias and minimum MSE at the optimum value of scalar ${\beta }_{(\mathit{\text{opt}})}={\rho }_{\mathit{\text{xy}}}\frac{\overline{Y}}{\overline{X}}\frac{C_y}{C_x}$ as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{reg}}}\right)=0,\\ m\mathit{\text{inMSE}}\left(t_{\mathit{\text{reg}}}\right)=f{\overline{Y}}^2{C}_y^2(1-{\rho }_{\mathit{\text{xy}}}^2)\end{array}$$

Srivastava (1967) suggested the power ratio estimator as follows $$t_s=\overline{y}{\left(\frac{\overline{X}}{\overline{x}}\right)}^{\delta },$$with the bias and minimum MSE at the optimum value of scalar ${\delta }_{(\mathit{\text{opt}})}={\rho }_{\mathit{\text{xy}}}\frac{C_y}{C_x}$ as $$\begin{array}{c}\text{Bias}\left(t_s\right)=f\overline{Y}\delta \left(\frac{\left(\delta +1\right)}{2}{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y\right),\\ m\mathit{\text{inMSE}}\left(t_s\right)=f{\overline{Y}}^2{C}_y^2(1-{\rho }_{\mathit{\text{xy}}}^2).\end{array}$$

Walsh (1970) suggested the following ratio estimator as $$t_w=\overline{y}\left(\frac{\overline{X}}{\overline{X}+\theta (\overline{x}-\overline{X}}\right),$$with the bias and minimum MSE at the optimum value of scalar ${\theta }_{(\mathit{\text{opt}})}={\rho }_{\mathit{\text{xy}}}\frac{C_y}{C_x}$ as $$\begin{array}{c}\text{Bias}\left(t_w\right)=f\overline{Y}\theta (\theta C_x^2-{\rho }_{\mathrm{x}y}{C}_x{C}_y),\\ m\mathit{\text{inMSE}}\left(t_w\right)=f{\overline{Y}}^2{C}_y^2(1-{\rho }_{\mathit{\text{xy}}}^2).\end{array}$$

Sisodia and Dwivedi (1981) utilized the auxiliary information and suggested the following ratio estimator $$t_{\mathit{\text{sd}}}=\overline{y}\left[\frac{\overline{X}+C_x}{\overline{x}+C_x}\right],$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{sd}}}\right)=f\overline{Y}{\lambda }_1({\lambda }_1{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_{\mathit{\text{sd}}}\right)=f{\overline{Y}}^2(C_y^2+{\lambda }_1^2{C}_x^2-2{\lambda }_1{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\end{array}$$ where ${\lambda }_1=\frac{\overline{X}}{\overline{X}+C_x}$.

Bahl and Tuteja (1991) proposed the exponential ratio estimator as $$t_{\mathit{\text{re}}}=\overline{y}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left[\frac{\overline{X}-\overline{x}}{\overline{X}+\overline{x}}\right],$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{re}}}\right)=f\overline{Y}\left(\frac{3}{8}{C}_x^2-\frac{1}{2}{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y\right),\\ \mathit{\text{MSE}}\left(t_{\mathit{\text{re}}}\right)=f{\overline{Y}}^2\left(C_y^2+\frac{C_x^2}{4}-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y\right).\end{array}$$

Singh and Kakran (1993) used the coefficient of kurtosis and construct the following ratio estimator $$t_{\mathit{\text{sk}}}=\overline{y}\left[\frac{\overline{X}+{\beta }_2(\mathrm{x})}{\overline{X}+{\beta }_2(x)}\right],$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{sk}}}\right)=f\overline{Y}{\lambda }_2({\lambda }_2{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_{\mathit{\text{sk}}}\right)=f{\overline{Y}}^2(C_y^2+{\lambda }_2^2{C}_x^2-2{\lambda }_2{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\end{array}$$ where ${\lambda }_2=\frac{\overline{X}}{\overline{X}+{\beta }_2(x)}$.

Upadhyaya and Singh (1999) utilized the transformed auxiliary information and suggested the following estimators: $$\begin{array}{c}{t}_{u{p}_1}=\overline{y}\left[\frac{C_x\overline{X}+{\beta }_2(x)}{C_x\overline{X}+{\beta }_2(x)}\right],\\ t_{u{p}_2}=\overline{y}\left[\frac{{\beta }_2(x)\overline{X}+C_x}{{\beta }_2(x)\overline{X}+C_x}\right],\end{array}$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{u{p}_1}\right)=f\overline{Y}{\lambda }_3({\lambda }_3{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \text{Bias}\left(t_{u{p}_2}\right)=f\overline{Y}{\lambda }_4({\lambda }_4{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_{u{p}_1}\right)=f{\overline{Y}}^2(C_y^2+{\lambda }_3^2{C}_x^2-2{\lambda }_3{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_{u{p}_2}\right)=f{\overline{Y}}^2(C_y^2+{\lambda }_4^2{C}_x^2-2{\lambda }_4{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\end{array}$$ where ${\lambda }_3=\frac{C_x\overline{X}}{C_x\overline{X}+{\beta }_2(x)}$ and ${\lambda }_4=\frac{{\beta }_2(x)\overline{X}}{{\beta }_2(x)\overline{X}+C_x}$

Singh (2003a) examined the following ratio estimator: $$t_{s_1}=\overline{y}\left[\frac{\overline{X}+{\rho }_{\mathit{\text{xy}}}}{\overline{X}+{\rho }_{\mathit{\text{xy}}}}\right],$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{s_1}\right)=f\overline{Y}{\lambda }_5({\lambda }_5{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_{s_1}\right)=f{\overline{Y}}^2(C_y^2+{\lambda }_5^2{C}_x^2-2{\lambda }_5{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\end{array}$$ where ${\lambda }_5=\frac{\overline{X}}{\overline{X}+{\rho }_{\mathit{\text{xy}}}}$.

Singh et al. (2009) considered various auxiliary information and suggested following class of exponential estimator: $$t_{\mathit{\text{sn}}}=\overline{y}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left[\frac{\left(u\overline{X}+v\right)-(u\overline{x}+v)}{\left(u\overline{X}+v\right)+(u\overline{x}+v)}\right],$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{sn}}}\right)=f\overline{Y}(2{\tau }^2{C}_x^2-\tau {\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ \mathit{\text{MSE}}\left(t_{\mathit{\text{sn}}}\right)=f{\overline{Y}}^2(C_y^2+{\tau }^2{C}_x^2-2\tau {\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\end{array}$$ where $\tau =\frac{a\overline{X}}{2(a\overline{X}+b)}$.

Yadav and Kadilar (2013) suggested an improved class of exponential ratio estimator: $$t_{\mathit{\text{yk}}}=k\overline{y}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left[\frac{\left(u\overline{X}+v\right)-(u\overline{x}+v)}{\left(u\overline{X}+v\right)+(u\overline{x}+v)}\right]$$with the bias and minimum MSE at optimum value of $k_{(\mathit{\text{opt}})}=A/B$ as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{yk}}}\right)=\mathit{\text{kf}}\overline{Y}\left(2{\tau }^2{C}_x^2-\tau {\rho }_{\mathit{\text{xy}}}{C}_x{C}_y\right)+\overline{Y}(k-1),\\ \mathrm{\text{min}}.\mathit{\text{MSE}}\left(t_{\mathit{\text{yk}}}\right)={\overline{Y}}^2\left(1-\frac{A^2}{B}\right),\end{array}$$ where $\mathrm{A}=1+f(2{\tau }^2{C}_x^2-\tau {\rho }_{\mathit{\text{xy}}}{C}_x{C}_y)$ and $B=1+f(C_y^2+5{\tau }^2{C}_x^2-4\tau {\rho }_{\mathit{\text{xy}}}{C}_x{C}_y)$.

Here, u and v are either real amounts or some available population parametric values of AV x.

Kadilar (2016) developed an improved ratio cum exponential ratio class of estimator: $$t_{\mathit{\text{gk}}}=\overline{y}{\left(\frac{\overline{X}}{\overline{x}}\right)}^{\alpha }\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left[\frac{\overline{X}-\overline{x}}{\overline{X}+\overline{x}}\right],$$with the bias and minimum MSE at optimum value of scalar ${\alpha }_{\mathit{\text{opt}}}=\frac{(2{\rho }_{\mathit{\text{xy}}}{C}_y-C_x)}{2C_x}$ as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{gk}}}\right)=f\overline{Y}[\frac{\alpha \left(1+\alpha \right)}{2}{C}_x^2+\frac{\alpha }{2}{C}_x^2+\frac{3}{8}{C}_x^2\\ -\left(\alpha +\frac{1}{2}\right){\rho }_{\mathit{\text{xy}}}{C}_x{C}_y],\\ m\mathit{\text{inMSE}}\left(t_{\mathit{\text{gk}}}\right)=f{\overline{Y}}^2{C}_y^2(1-{\rho }_{\mathit{\text{xy}}}^2).\end{array}$$

Ijaz and Ali (2018) suggested the following class of estimators for the estimation of population mean: $$t_{\mathit{\text{ia}}}=w\overline{y}+(1-w)\overline{y}\frac{\overline{X}}{\overline{x}},$$with the bias and minimum MSE at optimum value of scalar $w_{(\mathit{\text{opt}})}=1-{\rho }_{\mathit{\text{xy}}}\frac{C_y}{C_x}$ as $$\begin{array}{c}\text{Bias}\left(t_{\mathit{\text{ia}}}\right)=(1-w)f\overline{Y}(C_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\\ m\mathit{\text{inMSE}}\left(t_{\mathit{\text{ia}}}\right)=f{\overline{Y}}^2{C}_y^2(1-{\rho }_{\mathit{\text{xy}}}^2).\end{array}$$

Yadav et al. (2019) suggested the following class of estimators for population mean under SRS: $$\begin{array}{c}{t}_{p_1}=\overline{y}\left[\frac{{\beta }_2\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}{{\beta }_2\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}\right],\\ t_{p_2}=\overline{y}\left[\frac{{\beta }_2\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x}{{\beta }_2\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x}\right]\prime \\ t_{p_3}=\overline{y}\left[\frac{{\beta }_1\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}{{\beta }_1\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}\right],\\ t_{p_4}=\overline{y}\left[\frac{{\beta }_1\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x}{{\beta }_1\left(x\right)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x}\right],\\ t_{p_5}=\overline{y}\left[\frac{n\overline{X}+{\rho }_{\mathit{\text{xy}}}}{n\overline{X}+{\rho }_{\mathit{\text{xy}}}}\right],\\ t_{p_6}=\overline{y}\left[\frac{n\overline{X}+C_x}{n\overline{X}+C_x}\right],\\ t_{p_7}=\overline{y}\left[\frac{n\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x}{n\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x}\right],\\ t_{p_8}=\overline{y}\left[\frac{n{\rho }_{\mathit{\text{xy}}}\overline{X}+C_x}{n{\rho }_{\mathit{\text{xy}}}\overline{X}+C_x}\right],\\ t_{p_9}=\overline{y}\left[\frac{n{C}_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}{n{C}_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}\right].\end{array}$$with the bias and MSE as $$\begin{array}{c}\text{Bias}\left(t_{p_i}\right)=f\overline{Y}{\phi }_i\left({\phi }_i{C}_x^2-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y\right),i=1,2,\dots ,9,\\ \mathit{\text{MSE}}\left(t_{p_i}\right)=f{\overline{Y}}^2(C_y^2+{\phi }_i^2{C}_x^2-2{\phi }_i{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),\end{array}$$

where ${\phi }_1=\frac{{\beta }_2(x)M_x\overline{X}}{{\beta }_2(x)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}},{\phi }_2=\frac{{\beta }_2(x)M_x\overline{X}}{{\beta }_2(x)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x},{\phi }_3=\frac{{\beta }_1(x)M_x\overline{X}}{{\beta }_1(x)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}},{\phi }_4=\frac{{\beta }_1(x)M_x\overline{X}}{{\beta }_1(x)M_x\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x},{\phi }_5=\frac{n\overline{X}}{n\overline{X}+{\rho }_{\mathit{\text{xy}}}},{\phi }_6=\frac{n\overline{X}}{n\overline{X}+C_x},{\phi }_7=\frac{n\overline{X}}{n\overline{X}+{\rho }_{\mathit{\text{xy}}}{C}_x},{\phi }_8=\frac{n{\rho }_{\mathit{\text{xy}}}\overline{X}}{n{\rho }_{\mathit{\text{xy}}}\overline{X}+C_x}$, and ${\phi }_9=\frac{n{C}_x\overline{X}}{n{C}_x\overline{X}+{\rho }_{\mathit{\text{xy}}}}$

4 Proposed estimators

Almost all estimators reviewed in the previous section are either less or equally efficient to the classical regression (BLU) estimator. This work aims to develop an enhanced class of estimators for the estimation of PM of SV using information on AV. Here is the proposed class of estimators: $$\mathrm{T}=k_1\overline{y}{\left(\frac{\overline{X}}{\overline{x}}\right)}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{\left(u\overline{X}+v\right)-(u\overline{x}+v)}{\left(u\overline{X}+v\right)+(u\overline{x}+v)}\right\},$$where k_j, j $=$ 1, 2 are constant to optimize the MSE, while u and v are either real amounts or some available population parametric values of AV x. Some sub-class of T are tabulated in Table 1.

Table 1 Several individuals from the proposed estimators.

Unknown members	u	v
$T_{(1)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left(\frac{\overline{X}-\overline{X}}{\overline{X}+\overline{X}}\right)$	1	0
$T_{(2)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(\overline{X}+C_x)-(\overline{X}+C_x)}{(\overline{X}+C_x)+(\overline{X}+C_x)}\right\}$	1	Cx
$T_{(3)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{({\beta }_2(x)\overline{X}+C_x)-({\beta }_2(x)\overline{X}+C_x)}{({\beta }_2(x)\overline{X}+C_x)+({\beta }_2(x)\overline{X}+C_x)}\right\}$	${\beta }_2(x)$	Cx
$T_{(4)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(C_x\overline{X}+{\beta }_2(x))-(C_x\overline{X}+{\beta }_2(x))}{(C_x\overline{X}+{\beta }_2(x))+(C_x\overline{X}+{\beta }_2(x))}\right\}$	Cx	${\beta }_2(x)$
$T_{(5)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(\overline{X}+{\rho }_{\mathit{\text{xy}}})-(\overline{X}+{\rho }_{\mathit{\text{xy}}})}{(\overline{X}+{\rho }_{\mathit{\text{xy}}})+(\overline{X}+{\rho }_{\mathit{\text{xy}}})}\right\}$	1	ρxy
$T_{(6)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(\overline{X}+{\beta }_2(x))-(\overline{X}+{\beta }_2(x))}{(\overline{X}+{\beta }_2(x))+(\overline{X}+{\beta }_2(x))}\right\}$	1	${\beta }_2(x)$
$T_{(7)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})-(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})}{(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})+(C_x\overline{X}+{\rho }_{\mathit{\text{xy}}})}\right\}$	Cx	ρxy
$T_{(8)}=k_1\overline{y}{(\frac{\overline{X}}{\overline{X}})}^{k_2}\hspace{0.17em}\mathrm{\text{exp}}\hspace{0.17em}\left\{\frac{({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})-({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})}{({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})+({\beta }_2(x)\overline{X}+{\rho }_{\mathit{\text{xy}}})}\right\}$	${\beta }_2(x)$	ρxy

By employing the notations provided in “Methodology and notations,” we can express the suggested estimator as $$\begin{array}{c}\mathrm{T}=k_1\overline{Y}\left(1+e_0\right)\left\{1-k_2{e}_1+\frac{k_2\left(k_2+1\right)}{2!}{e}_1^2-\dots \right\}\\ \left\{1-\lambda e_1\left(1-\lambda e_1+\frac{3}{2}{\lambda }^2{e}_1^2\right)\right\}\end{array}$$where $\lambda =2(u\overline{X}/(u\overline{X}+v))$.

Here, Taylor’s series expansion is used and the error terms having power greater than 2 are neglected. This provides the following expression: $$\begin{array}{c}T-\overline{Y}=\overline{Y}\left[k_1\right\{1+e_0-k_2{e}_1-\lambda e_1-k_2{e}_0{e}_1-\lambda e_0{e}_1\\ \hspace{1em}+\frac{k_2\left(k_2+1\right)}{2}{e}_1^2+k_2\lambda e_1^2+\frac{3}{2}{\lambda }^2{e}_1^2\}-1].\end{array}$$

Considering expectation both the sides to the above expression, we get $$\begin{array}{c}\mathit{\text{Bias}}\left(T\right)=f\overline{Y}\left[k_1\right\{1-\left(k_2+\lambda \right){\rho }_{\mathit{\text{xy}}}{C}_x{C}_y\\ \hspace{1em}+\frac{k_2\left(k_2+1\right)}{2}{C}_x^2+k_2\lambda C_x^2+\frac{3}{2}{\lambda }^2{C}_x^2\}-1].\end{array}$$

Squaring and considering expectation both sides to the above expression provides $$\begin{array}{c}\mathit{\text{MSE}}\left(T\right)={\overline{Y}}^2(1+k_1^2[1+f{C}_y^2+\{k_2\left(2k_2+1\right)\\ +4{\lambda }^2+4k_2\lambda \}\mathrm{f}{C}_x^2-4\left(k_2+\lambda \right)f{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y]\\ -2k_1[1+\{\frac{k_2\left(k_2+1\right)}{2}+k_2\lambda +\frac{3}{2}{\lambda }^2\}f{C}_x^2\\ -\left(k_2+\lambda \right)f{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y]).\end{array}$$

The above MSE equation can further be expressed as $$\text{MSE}(\mathrm{T})\text{=}{\overline{Y}}^2(1+k_1^2{F}_1-2k_1{F}_2),$$where $F_1=1+f{C}_y^2+\left\{k_2(2k_2+1)+4{\lambda }^2+4k_2\lambda \right\}\mathrm{f}{C}_x^2-4(k_2+\lambda )f{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y$ and $F_2=1+\left\{\frac{k_2(k_2+1)}{2}+k_2\lambda +\frac{3}{2}{\lambda }^2\right\}f{C}_x^2-(k_2+\lambda )f{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y$.

Minimizing the MSE(T) against k₁ provides the optimum value of k₁ as $$k_{1(\mathit{\text{opt}})}=\frac{F_2}{F_1}.$$

Using the amount of ${\mathrm{k}}_{1(\text{opt})}$ in the MSE(T), we obtain $$\text{MSE}{(T)}_{\mathrm{\text{min}}}={\overline{Y}}^2\left(1-\frac{F_2^2}{F_1}\right).$$

The optimization of k₁and k₂ simultaneously is very typical. Therefore, putting $k_1=$ 1 in the estimator T and minimize the MSE of T against k₂ provides the optimum value of k₂ as $$k_{2(\mathit{\text{opt}})}=\frac{{\rho }_{\mathit{\text{xy}}}{C}_y}{C_x}-\lambda .$$

5 Efficiency conditions

The MSE ${(T)}_{\mathrm{\text{min}}}$ is compared with the variance/MSE/minimum MSE of the estimators discussed in “Existing estimators” and obtained the efficiency conditions mentioned below. $$\begin{array}{c}\text{MSE}(\mathrm{T})<\mathrm{V}(t_m)\Rightarrow \frac{F_2^2}{F_1}>1-f{C}_y^2.\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_r)\Rightarrow \frac{F_2^2}{F_1}>1-f(C_y^2+C_x^2-2{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t^{\ast })\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f{C}_y^2\left(1-{\rho }_{\mathit{\text{xy}}}^2\right),\text{where }{t}^{\ast }=t_{\mathit{\text{reg}}},t_s,t_w,t_{\mathit{\text{gk}}},\text{and }{t}_{\mathit{\text{ia}}}.\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{\mathit{\text{sd}}})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\lambda }_1^2{C}_x^2-2{\lambda }_1{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{\mathit{\text{re}}})\Rightarrow \frac{F_2^2}{F_1}>1-f(C_y^2+\frac{C_x^2}{4}-{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{\mathit{\text{sk}}})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\lambda }_2^2{C}_x^2-2{\lambda }_2{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{u{p}_1})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\lambda }_3^2{C}_x^2-2{\lambda }_3{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{u{p}_2})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\lambda }_4^2{C}_x^2-2{\lambda }_4{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{s_1})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\lambda }_5^2{C}_x^2-2{\lambda }_5{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{\mathit{\text{sn}}})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\tau }^2{C}_x^2-2\tau {\rho }_{\mathit{\text{xy}}}{C}_x{C}_y).\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{\mathit{\text{yk}}})\Rightarrow \frac{F_2^2}{F_1}>\frac{A^2}{B}.\\ \text{MSE}(\mathrm{T})<\text{MSE}(t_{p_i})\Rightarrow \frac{F_2^2}{F_1}>1\\ \hspace{1em}-f(C_y^2+{\phi }_i^2{C}_x^2-2{\phi }_i{\rho }_{\mathit{\text{xy}}}{C}_x{C}_y),i=1,2,\dots ,9.\end{array}$$

Under these efficiency conditions, the proposed class of estimators represses the reviewed estimators. In practice, these efficiency conditions are verified through the empirical study performed in the next section.

6 Empirical study

This section presents an empirical study in three subsections. In Section 6.1, a numerical study is presented based on four different real populations, in Section 6.2, a simulation study is presented based on artificially generated symmetric and asymmetric populations, while in Section 6.3, discussion of empirical results is presented.

6.1 Numerical study

To improve the theoretical establishment, we have conducted an empirical investigation on four real populations. The descriptive information of these populations is tabulated in Table 2.

Table 2 Descriptive statistics of different populations.

Parameters	Population 1: Source: Sarndal et al. (2003), 652–659), $Y=$ Total number of seats (S82) in municipal council in 1982 and $X=$ Number of conservative seats (CS82) in municipal council in 1982.	Population 2: Source: Singh (2003b, 1115), $Y=$ Seasonal average price per pound during 1996 and $X=$ Seasonal average price per pound during 1995.	Population 3: Source: Alomair and Shahzad (2023), $Y=$ Humidity (%) in Karachi during 2022 and $X=$ Humidity (%) in Karachi during 2021.	Population 4: Source: Kadilar and Cingi (2003), $Y=\text{Quantity}$ of apples produced in the Black Sea area of Turkey and $X=$ Apple trees’ quantity in the Black Sea area of Turkey.
N	284	36	365	204
n	50	14	100	40
$\overline{Y}$	46.07	0.20	90.09	966.95
$\overline{X}$	9.09	0.18	90.82	26441.72
Mx	8.00	0.18	91	13650
Cy	0.27	0.39	0.08	2.47
Cx	0.54	0.40	0.08	1.71
${\rho }_{\mathit{\text{xy}}}$	0.68	0.87	0.75	0.71
${\beta }_1(x)$	1.24	0.77	–0.48	4.62
${\beta }_2(x)$	5.38	3.34	3.21	29.77

The optimum values of k₁ and k₂ are computed and the results are listed in Table 3 for all populations. Further, the range of k₁ (for fixed values of $k_2)$ and k₂ (for fixed values of $k_1)$ are computed and the results are listed in Tables 4 and 5, respectively, under which the proposed class of estimators T is more efficient than the existing estimators. It is noticed from Tables 4 and 5 that there is sufficient scope of choosing the scalars k₁ and k₂ to get better estimators than the existing estimators. From the common range of k₁ and k₂ and optimum values k₁ and k₂ for all populations, it is noticed that the scope of obtaining better estimators from the proposed class of estimators is wide even if the guessed values of the scalars k₁ and k₂ departs substantially from the exact optimum values k₁ and k₂.

Table 3 Optimum values of $(k_1,k_2)$ of the members of the proposed estimator for different populations.

Members of	Population 1	Population 2	Population 3	Population 4
estimator T
T(1)	(0.981,–1.653)	(0.964,–1.143)	(0.999,–1.220)	(0.805,–0.973)
T(2)	(0.983,–1.540)	(0.995, 0.228)	(0.999,–1.218)	(0.805,–0.973)
T(3)	(0.981,–1.631)	(0.986,–0.353)	(0.999,–1.219)	(0.805,–0.973)
T(4)	(0.996,–0.609)	(0.996, 0.812)	(0.999,–0.607)	(0.805,–0.972)
T(5)	(0.984,–1.513)	(0.997, 0.507)	(0.999,–1.203)	(0.805,–0.973)
T(6)	(0.993,–0.909)	(0.997, 0.751)	(0.999,–1.151)	(0.805,–0.971)
T(7)	(0.985,–1.409)	(0.997, 0.698)	(0.999,–1.031)	(0.805,–0.973)
T(8)	(0.982,–1.625)	(0.993, 0.027)	(0.999,–1.214)	(0.805,–0.973)

Table 4 Range of k₁ at the fixed value of k₂ for different populations.

Estimators	Population 1	Population 2	Population 3	Population 4
	($k_2=-1.65)$	($k_2=-1.14)$	($k_2=-1.22)$	($k_2=-0.97)$
tm	(0.95, 1.01)	(0.87, 1.06)	(0.994, 1.000)	(0.51, 1.100)
tr	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.000)	(0.63, 1.000)
treg	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
ts	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
tw	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
tsd	(0.94, 1.03)	(0.90, 1.03)	(0.998, 1.001)	(0.62, 0.999)
tre	(0.96, 1.00)	(0.92, 1.01)	(0.998, 1.002)	(0.58, 1.030)
tsk	(0.96, 1.00)	(0.88, 1.05)	(0.999, 1.001)	(0.62, 1.000)
$t_{u{p}_1}$	(0.94, 1.02)	(0.88, 1.05)	(0.998, 1.006)	(0.59, 1.020)
$t_{u{p}_2}$	(0.96, 1.00)	(0.89, 1.04)	(0.997, 1.003)	(0.52, 1.090)
$t_{s_1}$	(0.96, 1.01)	(0.94, 1.00)	(0.999, 1.000)	(0.57, 1.040)
tsn	(0.96, 1.00)	(0.92, 1.01)	(0.998, 1.002)	(0.58, 1.030)
tyk	(0.96, 1.00)	(0.92, 1.01)	(0.998, 1.002)	(0.60, 1.010)
tgk	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
tia	(0.97, 0.99)	(0.93, 0.99)	(0.999, 1.000)	(0.62, 0.999)
$t_{p_1}$	(0.93, 1.03)	(0.88, 1.04)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_2}$	(0.93, 1.03)	(0.90, 1.03)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_3}$	(0.93, 1.03)	(0.88, 1.05)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_4}$	(0.93, 1.03)	(0.88, 1.05)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_5}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_6}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_7}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_8}$	(0.93, 1.03)	(0.93, 1.00)	(0.998, 1.001)	(0.62, 1.000)
$t_{p_9}$	(0.93, 1.03)	(0.92, 1.01)	(0.999, 1.001)	(0.62, 1.000)

Table 5 Range of k₂ at the fixed value of k₁ for different populations.

Estimators	Population 1	Population 2	Population 3	Population 4
	($k_1=0.98)$	($k_1=0.97)$	($k_1=1.00)$	($k_1=0.80)$
tm	(–1.21,–2.08)	(–0.17,–2.09)	(–0.45,–1.99)	(–2.76, 1.15)
tr	(–0.93,–2.35)	(–0.75,–1.51)	(–1.00,–1.44)	(–2.09, 0.47)
treg	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
ts	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
tw	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
tsd	(–0.98,–2.30)	(–0.46,–1.79)	(–1.001,–1.43)	(–2.09, 0.47)
tre	(–1.35,–1.94)	(–0.62,–1.64)	(–0.941,–1.49)	(–2.29, 0.68)
tsk	(–1.26,–2.02)	(–0.22,–2.04)	(–1.04,–1.40)	(–2.09, 0.47)
$t_{u{p}_1}$	(–1.04,–2.24)	(–0.22,–2.03)	(–1.04,–1.39)	(–2.20, 0.62)
$t_{u{p}_2}$	(–1.35,–1.94)	(–0.34,–1.91)	(–1.05,–1.41)	(–2.70, 1.11)
$t_{s_1}$	(–1.25,–2.04)	(–0.75,–1.51)	(–1.08,–1.09)	(–2.40, 0.79)
tsn	(–1.35,–1.94)	(–0.62,–1.64)	(–0.95,–1.49)	(–2.29, 0.68)
tyk	(–1.35,–1.93)	(–0.62,–1.64)	(–0.95,–1.49)	(–2.22, 0.60)
tgk	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
tia	(–1.8901,–1.89)	(–0.78,–1.48)	(–1.22,–1.23)	(–2.09, 0.47)
$t_{p_1}$	(–0.93,–2.36)	(–0.28,–1.98)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_2}$	(–0.93,–2.36)	(–0.40,–1.86)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_3}$	(–0.94,–2.36)	(–0.20,–2.06)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_4}$	(–0.93,–2.35)	(–0.23,–2.02)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_5}$	(–0.93,–2.36)	(–0.76,–1.49)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_6}$	(–0.93,–2.36)	(–0.78,–1.48)	(–1.01,–1.43)	(–2.09, 0.47)
$t_{p_7}$	(–0.93,–2.36)	(–0.78,–1.48)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_8}$	(–0.93,–2.36)	(–0.78,–1.48)	(–1.01,–1.44)	(–2.09, 0.47)
$t_{p_9}$	(–0.93,–2.35)	(–0.65,–1.60)	(–1.01,–1.44)	(–2.09, 0.47)

We have also computed the bias, MSE, and PRE of the estimators and reported in Table 6. The following expression is used to calculate PRE: $$\mathit{\text{PRE}}=\frac{V(t_m)}{\mathit{\text{MSE}}(T^{\ast })}\times 100,$$where $$\begin{array}{c}{T}^{\ast }=t_m,t_r{t}_{\mathit{\text{reg}}},t_s,t_w,t_{\mathit{\text{sd}}},t_{\mathit{\text{re}}},t_{\mathit{\text{sk}}},t_{u{p}_1},t_{u{p}_2},t_{s_1},t_{\mathit{\text{sn}}},t_{\mathit{\text{yk}}},t_{\mathit{\text{gk}}},t_{\mathit{\text{ia}}},\\ t_{p_1},t_{p_2},t_{p_3},t_{p_4},t_{p_5},t_{p_6},t_{p_7},t_{p_8},t_{p_9}\text{ and }T\end{array}$$

Table 6 Bias, MSE and PRE for several estimators.

Estimators	Population 1			Population 2			Population 3			Population 4
	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE
t m	0.00	3.1	100.0	0.00	0.0004	100.0	0.0000	0.55	100.00	0.00	142775.0	100.00
t r	0.17	7.0	45.2	0.00	0.0001	397.6	0.0012	0.25	212.79	–1.87	70220.4	203.32
t reg	0.00	1.6	189.8	0.00	0.0001	434.9	0.0000	0.23	235.79	0.00	70172.4	203.46
ts	0.03	1.6	189.8	0.00	0.0001	434.9	0.0004	0.23	235.79	–0.96	70172.4	203.46
tw	–0.00	1.6	189.8	0.00	0.0001	434.9	0.0000	0.23	235.79	0.00	70172.4	203.46
tsd	0.15	6.1	51.7	–0.00	0.0002	185.6	0.0012	0.25	212.96	–1.88	70220.6	203.32
tre	0.05	1.9	161.3	–0.00	0.0001	275.1	–0.0000	0.27	200.68	–9.84	89267.9	159.93
tsk	0.04	2.6	119.1	–0.00	0.0004	110.0	0.0010	0.25	218.91	–1.95	70224.6	203.31
$t_{u{p}_1}$	0.12	5.2	60.4	–0.00	0.0004	110.4	0.3943	32.55	1.69	–18.43	83781.9	170.41
$t_{u{p}_2}$	–0.00	2.0	158.3	–0.00	0.0003	141.0	–0.0008	0.34	158.12	–2.37	138101.3	103.38
$t_{s_1}$	0.05	2.8	112.85	–0.00	0.0001	396.8	0.0007	0.24	225.10	–17.26	100042.3	142.71
tsn	0.08	1.9	161.3	0.00	0.0001	275.1	0.0006	0.27	200.68	–0.93	89267.9	159.93
tyk	–0.04	1.9	162.7	–0.00	0.0001	276.8	–0.0030	0.27	200.69	–84.42	81632.0	174.90
tgk	0.03	1.6	189.8	0.00	0.0001	434.9	0.0004	0.23	235.79	–0.96	70172.4	203.46
tia	0.06	1.6	189.8	0.00	0.0001	434.9	0.0009	0.23	235.79	–1.92	70172.4	203.46
$t_{p_1}$	0.17	6.9	45.3	–0.00	0.0003	123.4	0.0012	0.25	212.80	–1.87	70220.4	203.32
$t_{p_2}$	0.17	6.9	45.3	–0.00	0.0002	158.7	0.0012	0.25	212.79	–1.87	70220.4	203.32
$t_{p_3}$	0.17	6.8	46.0	–0.00	0.0004	105.3	0.0012	0.25	212.75	–1.87	70220.4	203.32
$t_{p_4}$	0.17	6.9	45.6	–0.00	0.0004	113.3	0.0012	0.25	212.79	–1.87	70220.4	203.32
$t_{p_5}$	0.17	6.9	45.3	–0.00	0.0001	412.5	0.0012	0.25	212.81	–1.87	70220.4	203.32
$t_{p_6}$	0.17	6.9	45.3	0.00	0.0001	434.7	0.0012	0.25	212.79	–1.88	70220.4	203.32
$t_{p_7}$	0.17	6.9	45.3	0.00	0.0001	433.8	0.0012	0.25	212.79	–1.87	70220.4	203.32
$t_{p_8}$	0.17	6.9	45.3	–0.00	0.0001	434.7	0.0012	0.25	212.79	–1.88	70220.4	203.32
$t_{p_9}$	0.17	6.9	45.5	–0.00	0.0001	301.5	0.0012	0.25	212.99	–1.87	70220.4	203.32
T(1)	0.81	0.9	343.9	0.00	0.0000	936.8	0.0176	0.23	236.21	166.68	16408.4	870.12
T(2)	0.71	1.0	292.2	0.00	0.0001	437.9	0.0176	0.23	236.21	166.66	16418.3	869.60
T(3)	0.79	0.9	331.6	0.00	0.0000	466.6	0.0176	0.23	236.21	166.68	16408.8	870.11
T(4)	0.14	1.6	193.7	0.00	0.0001	436.4	0.0075	0.23	235.89	166.45	16507.6	864.90
T(5)	0.69	1.1	282.8	–0.00	0.0001	436.1	0.0173	0.23	236.20	166.67	16412.5	869.91
T(6)	0.28	1.5	202.4	0.00	0.0001	436.2	0.0163	0.23	236.16	166.28	16578.5	861.20
T(7)	0.60	1.2	255.6	–0.00	0.0001	436.1	0.0140	0.23	236.07	166.67	16410.8	870.00
T(8)	0.78	0.9	328.6	0.00	0.0001	442.0	0.0175	0.23	236.21	166.68	16408.6	870.12

6.2 Simulation study

To extrapolate the theoretical findings as well as to strengthen the findings of numerical study, we conducted a simulation study based on hypothetically generated symmetric and asymmetric populations. In order to generated the data, following Singh and Horn (1998), we used the models listed below: $$\begin{array}{c}y=3.4+\sqrt{\left(1-{\rho }_{\mathit{\text{xy}}}\right)}{y}^{\ast }+{\rho }_{\mathit{\text{xy}}}\left(\frac{S_y}{S_x}\right)x^{\ast }\\ x=3.2+x^{\ast }\end{array}$$where $x^{\ast }$ and $y^{\ast }$ are possessing the corresponding distributions. In particular, we generated the following populations.

1. We generated a normal population of size N $=$1000 using $x^{\ast }\sim N(15,65)$ and $y^{\ast }\sim N(20,70)$.

2. We generated a gamma population of size N $=$1000 using $x^{\ast }\sim \mathit{\text{gamma}}(6.005,0.05)$ and $y^{\ast }\sim \mathit{\text{gamma}}(7.009,1.09)$.

We have taken a sample of size $n=$200 from the above populations. To observe the behavior of the proposed estimators, utilizing 15,000 replications, we have computed MSE and percent relative efficiency (PRE) for the varying values of correlation coefficient as 0.1, 0.5, and 0.9. The PRE is calculated with the help of following expression. $$\mathit{\text{PRE}}=\frac{\sum _{i=1}^{15,000}{\left(t_m-\overline{Y}\right)}^2}{\sum _{i=1}^{15,000}{\left(T^{\ast }-\overline{Y}\right)}^2}\times 100$$

The simulation findings are reported in Tables 7 and 8 for normal and gamma populations with bias, MSE, and PRE, respectively.

Table 7 Bias, MSE, and PRE of several estimators for simulated normal population.

${\rho }_{\mathit{\text{xy}}}$	0.1			0.5			0.9
Estimators	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE
t m	0.00000	0.0014	100.00	0.00000	0.0014	100.00	0.00000	0.0014	100.00
t r	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.17
t reg	0.00000	0.0014	100.00	0.00000	0.0011	123.14	0.00000	0.0002	489.86
ts	0.00005	0.0014	100.00	0.00001	0.0011	123.14	–0.00008	0.0002	489.86
tw	0.00000	0.0014	100.00	0.00000	0.0011	123.14	–0.00007	0.0002	489.86
tsd	0.00002	0.0018	75.31	0.00003	0.0012	122.20	–0.00001	0.0004	339.42
tre	0.00001	0.0015	91.55	–0.00000	0.0012	119.20	–0.00009	0.0007	186.90
tsk	0.00002	0.0018	76.35	0.00002	0.0012	122.54	–0.00001	0.0004	329.48
$t_{u{p}_1}$	0.00000	0.0015	92.93	–0.00003	0.0012	119.56	–0.00001	0.0008	172.05
$t_{u{p}_2}$	0.00000	0.0014	96.45	–0.00003	0.0012	113.59	–0.00001	0.0009	148.38
$t_{s_1}$	0.00000	0.0014	96.34	–0.00003	0.0012	114.69	–0.00001	0.0009	146.15
tsn	0.00001	0.0015	91.55	0.00002	0.0011	119.20	–0.00006	0.0007	186.90
tyk	–0.00007	0.0015	99.68	–0.00005	0.0011	125.53	–0.00003	0.0007	191.15
tgk	0.00000	0.0014	100.00	0.00001	0.0011	123.14	–0.00008	0.0002	489.86
tia	0.00000	0.0014	100.00	0.00003	0.0011	123.14	–0.00001	0.0002	489.86
$t_{p_1}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.03
$t_{p_2}$	0.00002	0.0019	72.89	0.00004	0.0012	121.47	–0.00001	0.0003	368.87
$t_{p_3}$	0.00002	0.0019	72.89	0.00004	0.0012	121.36	–0.00001	0.0003	374.87
$t_{p_4}$	0.00002	0.0019	72.89	0.00004	0.0012	121.27	–0.00001	0.0003	381.78
$t_{p_5}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.10
$t_{p_6}$	0.00002	0.0019	72.90	0.00004	0.0012	121.47	–0.00001	0.0003	369.01
$t_{p_7}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.03
$t_{p_8}$	0.00002	0.0018	77.78	0.00004	0.0012	121.47	–0.00001	0.0003	368.99
$t_{p_9}$	0.00002	0.0019	72.89	0.00004	0.0012	121.46	–0.00001	0.0003	369.14
T(1)	0.00007	0.0010	128.89	0.00006	0.0009	150.97	0.00007	0.0001	852.63
T(2)	0.00006	0.0011	124.33	0.00005	0.0009	147.53	0.00006	0.0001	721.11
T(3)	0.00007	0.0011	127.24	0.00005	0.0009	149.75	0.00007	0.0001	798.36
T(4)	0.00006	0.0011	125.56	0.00005	0.0009	147.86	0.00006	0.0001	759.96
T(5)	0.00007	0.0010	128.88	0.00005	0.0009	150.08	0.00007	0.0001	787.86
T(6)	0.00006	0.0011	122.65	0.00005	0.0009	145.69	0.00005	0.0002	689.73
T(7)	0.00007	0.0010	128.88	0.00005	0.0009	150.48	0.00007	0.0001	820.51
T(8)	0.00007	0.0010	128.88	0.00005	0.0009	150.67	0.00007	0.0001	828.79

Table 8 Bias, MSE, and PRE of several estimators for simulated gamma population.

${\rho }_{\mathit{\text{xy}}}$	0.1			0.5			0.9
Estimators	Bias	MSE	PRE	Bias	MSE	PRE	Bias	MSE	PRE
t m	0.00000	0.0008	100.00	0.00000	0.0008	100.00	0.00000	0.0008	100.00
t r	0.00001	0.0009	87.63	–0.00005	0.0005	142.93	–0.00001	0.0002	325.40
t reg	0.00000	0.0007	103.81	0.00000	0.0004	142.94	0.00000	0.0001	536.95
ts	0.00002	0.0007	103.81	–0.00002	0.0004	142.94	–0.00001	0.0001	536.95
tw	0.00000	0.0007	103.81	0.00002	0.0004	142.94	–0.00001	0.0001	536.95
tsd	0.00001	0.0009	88.52	–0.00003	0.0005	142.91	–0.00009	0.0002	317.93
tre	0.00005	0.0008	102.40	–0.00002	0.0006	129.00	–0.00001	0.0004	173.34
tsk	0.00008	0.0008	96.36	–0.00003	0.0006	140.12	–0.00008	0.0003	239.92
$t_{u{p}_1}$	0.00006	0.0014	56.95	0.00004	0.0008	98.34	–0.00001	0.0002	491.24
$t_{u{p}_2}$	–0.00001	0.0008	103.78	–0.00004	0.0006	117.86	–0.00009	0.0006	132.61
$t_{s_1}$	0.00009	0.0008	94.91	–0.00002	0.0005	140.92	–0.00001	0.0003	257.90
tsn	0.00008	0.0008	102.40	–0.00002	0.0006	129.00	–0.00006	0.0004	173.34
tyk	–0.00006	0.0008	102.50	–0.00005	0.0006	129.08	–0.00003	0.0004	173.39
tgk	0.00002	0.0007	103.81	–0.00002	0.0004	142.94	–0.00001	0.0001	536.95
tia	0.00005	0.0007	103.81	–0.00005	0.0004	142.94	–0.00002	0.0001	536.95
$t_{p_1}$	0.00001	0.0009	87.65	–0.00008	0.0005	142.93	–0.00001	0.0002	324.68
$t_{p_2}$	0.00001	0.0009	87.64	–0.00006	0.0005	142.93	–0.00001	0.0002	325.24
$t_{p_3}$	0.00001	0.0009	87.73	–0.00001	0.0005	142.93	–0.00001	0.0002	321.18
$t_{p_4}$	0.00001	0.0009	87.66	–0.00008	0.0005	142.93	–0.00001	0.0002	324.45
$t_{p_5}$	0.00001	0.0009	87.63	–0.00006	0.0005	142.93	–0.00001	0.0002	325.24
$t_{p_6}$	0.00001	0.0009	87.64	–0.00006	0.0005	142.93	–0.00001	0.0002	325.36
$t_{p_7}$	0.00001	0.0009	87.63	–0.00005	0.0005	142.93	–0.00001	0.0002	325.36
$t_{p_8}$	0.00001	0.0009	87.65	–0.00006	0.0005	142.93	–0.00001	0.0002	325.36
$t_{p_9}$	0.00001	0.0009	87.64	–0.00007	0.0005	142.93	–0.00001	0.0002	324.70
T(1)	0.00007	0.0007	104.25	0.00005	0.0004	143.50	0.00005	0.0001	539.96
T(2)	0.00007	0.0007	104.18	0.00005	0.0004	143.34	0.00004	0.0001	538.89
T(3)	0.00007	0.0007	104.20	0.00005	0.0004	143.35	0.00004	0.0001	539.01
T(4)	0.00001	0.0007	103.92	0.00006	0.0004	143.04	0.00004	0.0001	537.62
T(5)	0.00007	0.0007	104.19	0.00005	0.0004	143.31	0.00004	0.0001	538.46
T(6)	0.00004	0.0007	104.02	0.00003	0.0004	143.17	0.00002	0.0001	537.55
T(7)	0.00006	0.0007	104.14	0.00003	0.0004	143.20	0.00001	0.0001	537.49
T(8)	0.00007	0.0007	104.20	0.00005	0.0004	143.34	0.00004	0.0001	538.87

6.3 Discussion of empirical results

After carefully observing the findings of the numerical and simulation studies, we have drawn the following observations:

1. From Table 6, for each population, the outcomes show that the members ${\mathbf{T}}_{(\mathbf{j})},\mathbf{j}=\mathbf{1},\mathbf{2},\dots ,\mathbf{8}$ of the suggested estimator T obtain the lowest MSE and highest PRE as compare to the existing estimators such as the conventional mean estimator, usual ratio and regression estimators, Srivastava (1967) estimator, Walsh (1970) estimator, Sisodia and Dwivedi (1981) estimator, Bahl and Tuteja (1991) estimator, Singh and Kakran (1993) estimator, Upadhyaya and Singh (1999) estimators, Singh (2003a) estimator, Singh et al. (2009) estimators, Yadav and Kadilar (2013) estimators, Kadilar (2016) estimator, Ijaz and Ali (2018) estimator, and Yadav et al. (2019) estimators.

2. From Table 6, the member T₍₁₎ is found to be best among the proposed class of estimator T in each population.

3. The results reported in Table 7 are based on normal population which reveal that the members $T_{(j),}j=1,2,\dots ,8$ of the proposed estimator T repress the reviewed estimators with minimum MSE and maximum PRE for each value of ${\rho }_{\mathit{\text{xy}}}$. Moreover, as ${\rho }_{\mathit{\text{xy}}}$ increase, the MSE and PRE of the members of the proposed estimators also decreases and increases, respectively.

4. The results reported in Table 8 are based on gamma population which demonstrate that the members $T_{(j),}j=1,2,\dots ,8$ of the proposed estimator T repress the reviewed estimators with minimum MSE and maximum PRE for each value of ${\rho }_{\mathit{\text{xy}}}$. Moreover, as ${\rho }_{\mathit{\text{xy}}}$ increases, the MSE and PRE of the members of the proposed estimators also increase, respectively.

5. From Tables 7 and 8, the member T₍₁₎ is found to be best among the proposed class of estimator T in normal and gamma populations.

7 Conclusions

This paper suggested an enhanced class of estimators for the population mean estimation utilizing the simple random sampling scheme. The bias, mean square error, and minimum mean square error of the proposed class of estimators are obtained mathematically to the approximation of first order. By comparing the minimum MSE expression of the proposed estimators with the MSE/minimum MSE of the existing estimators, the efficiency conditions are determined. A numerical illustration involving four real populations and a simulation study involving artificially generated normal and gamma populations provide support for the efficiency conditions developed in “Efficiency conditions.” The empirical findings demonstrate that the proposed estimator represses the conventional estimators in each population which are further extended with the simulation findings. Therefore, it is advised to utilize the suggested estimators for the population mean estimation in practical issues.

Further, the suggested estimators can be examined for the population mean estimation using different sampling schemes like, stratified random sampling, stratified ranked set sampling, two-phase sampling, adaptive cluster sampling, probability proportion to size sampling, for more details refer the studies of Ahmad and Shabbir (2018), Ahmad et al. (2021), (2022), Iftikhar et al. (2022), Bhushan et al. (2022a), (2022b), (2023), Rana et al. (2022), Bhushan and Kumar (2023a), (2023b), Qureshi and Hanif (2019).

Acknowledgment

The authors are extremely grateful to the learned referees for their valuable comments and to Editor-in-Chief.

Disclosure statement

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