New Techniques for Estimating Finite Population Variance Using Ranks of Auxiliary Variable in Two-Stage Sampling

Abstract

This article presents a new set of estimators designed to estimate the finite population variance of a study variable in two-phase sampling. These estimators utilize the information about extreme values and ranks of an auxiliary variable. Through a first-order approximation, we investigate the properties of these estimators, including biases and mean squared errors (MSEs). Furthermore, a comprehensive simulation study is conducted to assess their performance and validate our theoretical insights. Results demonstrate that our proposed class of estimators performs better in terms of percent relative efficiency (PRE) across various simulation scenarios compared to existing estimators. In addition, in the application section, we utilize three data sets to further validate the performance of our proposed estimators against conventional unbiased variance estimators, ratio and regression estimators, as well as other existing methods.

1. Introduction

In sampling theory, it is standard practice to incorporate auxiliary variables alongside the study variable to enhance design and improve the estimator efficiency by using their relationship. However, in many practical scenarios, information about auxiliary variables is not available before conducting a survey. In such cases, a two-phase sampling technique is preferred. Two-phase sampling, also known as double sampling, involves two distinct phases to select a sample from a population. Two-phase sampling is a cost-effective sampling scheme, so it plays an important role in sample surveys and is widely used when the auxiliary information is not available in advance. A brief review of two-phase sampling was first introduced by Neyman [1]. Following that, the topic was not investigated until the research conducted by Sukhatme [2]. In recent years, two-phase sampling has received much interest due to its abilities of screening variables at a low cost. Some studies on two-phase sampling include [3,4,5,6,7,8,9,10].

Variation is an inherent phenomenon of nature, and the estimation problem of finite population variance is a significant concern. Das [11] first discussed the use of auxiliary information to estimate population variance, which was extended by Isaki [12]. The accuracy of the estimators can be improved by carefully using auxiliary information. To estimate population variance, Bahl and Tuteja [13] suggested a few ratio- and product-type exponential estimators. Many scholars have provided a number of estimators for finite population variance, including [14,15,16,17,18,19,20,21,22].

There may be abnormal observation results in the sampling survey data when extreme values are part of the sample, which may lead to biased results. In this sense, some researchers have worked on extreme values and provided different kinds of transformations to estimate population characteristics. Through a linear transformation, Mohanty [23] provided two estimators given the known minimum and largest observations of the auxiliary variable. After that, these works were not examined any more until the work of Khan [24]. They used several finite population mean estimators and the idea of employing extreme values in them. Daraz et al. [25] enhanced the estimate of the finite population mean under extreme values by employing a stratified random sampling technique. Through transformations of extreme values, Daraz and Khan [26] proposed many efficient classes of estimators to estimate population variance. A recent study by Daraz et al. [27] looked into the characteristics of finite population variance and presented various types of estimators with minimal mean squared errors. By using the extreme values of the auxiliary variable, Daraz et al. [28] suggested double exponential ratio-type estimators to discuss the efficiency of the estimators for estimating population variance. In order to address the accuracy of the estimators through the linear transformation of extreme values and ranks of auxiliary variables, Daraz et al. [29] obtained a class of efficient estimators by utilizing the dual use of auxiliary variables under simple random sampling. For more details, see [30,31,32] and the references therein.

The significance of classical estimators tends to decrease in terms of the mean squared error $\left(MSE\right)$ when handling extreme values in a data set. The temptation to exclude such data from the sample may happen. Including these data in the process of determining population characteristics is important for properly addressing this difficulty. If the auxiliary variable and the study variable are related, the rankings of the auxiliary variable are associated with the study variable. Thus, these rankings can be employed as an effective way to increase the accuracy of the estimator. In this article, the extreme values of the auxiliary variable are kept in the data and are used as supplementary information to increase the accuracy of the proposed class of estimators. Inspired by [26,27,28,29], we use the transformations technique to provide a new class of estimators by utilizing the known information on the extreme values and the ranks of the auxiliary variable to estimate the finite population variance in two-phase sampling.

The following sections comprise this article: The concepts and notations are presented in Section 2. A number of existing estimators are included in Section 3. In Section 4, we describe our suggested class of estimators. The mathematical comparison is given in Section 5. To evaluate the theoretical results presented in Section 5, we simulate six distinct artificial populations using different probability distributions in Section 6. We also provide numerical examples in this section to validate our theoretical findings. In conclusion, the results are discussed along with suggestions for further research, as offered in Section 7.

2. Concepts and Notations

Let us consider a finite population $\varphi =\left({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_N\right)$ of size N units. Let the $i^{th}$ unit values of the dependent (study) variable Y, the independent (auxiliary) variable X, and the corresponding ranks of the independent variable R be represented by $y_i$, $x_i$, and $r_i$, respectively. The population variances for these variables are defined as

$$S_y^2={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N{\left(Y_i-\overline{Y}\right)}^2,$$

$$S_x^2={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N{\left(X_i-\overline{X}\right)}^2$$

and

$$S_r^2={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N{\left(R_i-\overline{R}\right)}^2,$$

where

$$\overline{Y}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{Y}_i,\overline{X}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{X}_i, \mathrm{and}\overline{R}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{R}_i,$$

are the corresponding population means of $Y,X,$ and R. For these variables $(Y,X,R)$, the population coefficients of variation are as follows:

$$C_y={\displaystyle \frac{S_y}{\overline{Y}}},C_x={\displaystyle \frac{S_x}{\overline{X}}}, \mathrm{and}{C}_r={\displaystyle \frac{S_r}{\overline{R}}},$$

respectively. Additionally, we have knowledge about the population correlation coefficients that exist between Y and $X,$ Y and $R,$ and X and R, as follows:

$${\rho }_{yx}={\displaystyle \frac{S_{yx}}{S_y{S}_x}},{\rho }_{yr}={\displaystyle \frac{S_{yr}}{S_y{S}_r}}, \mathrm{and}{\rho }_{xr}={\displaystyle \frac{S_{xr}}{S_x{S}_r}},$$

where

$$S_{yx}={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N\left(Y_i-\overline{Y}\right)\left(X_i-\overline{X}\right),$$

$$S_{yr}={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N\left(Y_i-\overline{Y}\right)\left(R_i-\overline{R}\right)$$

and

$$S_{xr}={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N\left(x_i-\overline{X}\right)\left(R_i-\overline{R}\right),$$

are the population covariances, respectively.

In this paper, we provide a set of estimators to estimate the finite population variance $S_y^2$ of Y in the presence of the auxiliary variable X. The definition of the two-phase sampling scheme is

• A sample of size (ń < N) from the first phase is chosen in order to estimate the population variance $S_x^2$.
• For the second phase, a sample size of (n < ń) is chosen in order to observe both y and x, respectively.

The biases and mean squared errors for various estimators can be derived by defining the following terms:

$${\xi }_0=\left({\displaystyle \frac{s_y^2-S_y^2}{S_y^2}}\right), {\xi }_1=\left({\displaystyle \frac{s_x^2-S_x^2}{S_x^2}}\right), {\xi }_2=\left({\displaystyle \frac{{\acute{s}}_x^2-S_x^2}{S_x^2}}\right), {\xi }_3=\left({\displaystyle \frac{s_r^2-S_r^2}{S_r^2}}\right), {\xi }_4=\left({\displaystyle \frac{{\acute{s}}_r^2-S_r^2}{S_r^2}}\right)$$

such that $E\left({\xi }_i\right)=0$ for $i=0,1,2,34$.

$$E\left({\xi }_0^2\right)=\eta {\Delta }_{400}^{*},E\left({\xi }_1^2\right)=\eta {\Delta }_{040}^{*},E\left({\xi }_2^2\right)={\eta }^{\prime }{\Delta }_{040}^{*},$$

$$E\left({\xi }_3^2\right)=\eta {\Delta }_{004}^{*},E\left({\xi }_4^2\right)={\eta }^{\prime }{\Delta }_{004}^{*},E\left({\xi }_0{\xi }_1\right)=\eta {\Delta }_{220}^{*},$$

$$E\left({\xi }_0{\xi }_2\right)={\eta }^{\prime }{\Delta }_{220}^{*},E\left({\xi }_0{\xi }_3\right)=\eta {\lambda }_{202}^{*},E\left({\xi }_0{\xi }_4\right)={\eta }^{\prime }{\Delta }_{202}^{*},$$

$$E\left({\xi }_1{\xi }_2\right)={\eta }^{\prime }{\Delta }_{040}^{*},E\left({\xi }_1{\xi }_3\right)=\eta {\Delta }_{022}^{*},E\left({\xi }_1{\xi }_4\right)={\eta }^{\prime }{\Delta }_{022}^{*},$$

$$E\left({\xi }_2{\xi }_3\right)={\eta }^{\prime }{\Delta }_{022}^{*},E\left({\xi }_2{\xi }_4\right)={\eta }^{\prime }{\Delta }_{022}^{*},E\left({\xi }_3{\xi }_4\right)={\eta }^{\prime }{\Delta }_{004}^{*},$$

where

$$\begin{gathered} {\Delta }_{400}^{*}=({\Delta }_{400}-1),{\Delta }_{040}^{*}=({\Delta }_{040}-1),{\Delta }_{004}^{*}=({\Delta }_{004}-1),{\Delta }_{220}^{*}=({\Delta }_{220}-1),{\Delta }_{202}^{*}=({\Delta }_{202}-1), \\ {\Delta }_{022}^{*}=({\Delta }_{022}-1),\eta =\left({\displaystyle \frac{1}{n}}-{\displaystyle \frac{1}{N}}\right),{\eta }^{\prime }=\left({\displaystyle \frac{1}{\acute{n}}}-{\displaystyle \frac{1}{N}}\right), \mathrm{and}{\eta }^{″}=\left({\displaystyle \frac{1}{n}}-{\displaystyle \frac{1}{\acute{n}}}\right). \end{gathered}$$

Also,

$${\Delta }_{lqs}={\displaystyle \frac{{\phi }_{lqs}}{{\phi }_{200}^{l/2}{\phi }_{020}^{q/2}{\phi }_{002}^{s/2}}},$$

where

$${\phi }_{lqs}={\displaystyle \frac{{\sum }_{i=1}^N{\left(Y_i-\overline{Y}\right)}^l{\left(X_i-\overline{X}\right)}^q{\left(R_i-\overline{R}\right)}^s}{N-1}}.$$

Here, ${\Delta }_{400}={\beta }_{2\left(y\right)},{\Delta }_{040}={\beta }_{2\left(x\right)}$, and ${\Delta }_{004}={\beta }_{2\left(r\right)}$ are the population coefficients of kurtosis.

3. Literature Review

In this section, our next step is to compare and contrast existing estimators for finite population variances with the proposed class of estimators.

The usual variance estimator of ${\widehat{S}}_y^2=s_y^2$ for population variance is given by

$$Var\left({\widehat{S}}_y^2\right)=\eta S_y^4{\Delta }_{400}^{*}.$$

(1)

Isaki [12] suggested a ratio estimator for population variance ${\widehat{S}}_R^2,$ which is given by

$${\widehat{S}}_R^2=s_y^2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right).$$

(2)

The bias and $MSE$ of ${\widehat{S}}_R^2$ are expressed as follows:

$$Bias\left({\widehat{S}}_R^2\right)\cong {\eta }^{″}{S}_y^4\left({\Delta }_{040}^{*}-{\Delta }_{220}^{*}\right)$$

(3)

and

$$MSE\left({\widehat{S}}_R^2\right)\cong S_y^4\left(\eta {\Delta }_{400}^{*}+{\eta }^{″}{\Delta }_{040}^{*}-2{\eta }^{″}{\Delta }_{220}^{*}\right).$$

(4)

Watson [33] proposed the linear regression estimator ${\widehat{S}}_{lr}^2$, which is defined as

$${\widehat{S}}_{lr}^2=s_y^2+b_{(s_y^2,s_x^2)}\left({\acute{s}}_x^2-s_x^2\right),$$

(5)

where $b_{(s_y^2,s_x^2)}={\displaystyle \frac{s_y^2{\widehat{\Delta }}_{220}^{*}}{s_x^2{\widehat{\Delta }}_{040}^{*}}}$ is the sample regression coefficient.

The $MSE$ of the estimator ${\widehat{S}}_{lr}^2$ is expressed as follows:

$$MSE\left({\widehat{S}}_{lr}^2\right)\cong S_y^4{\Delta }_{400}^{*}\left(\eta -{\eta }^{″}{\rho }_{yx}^{*2}\right),$$

(6)

where ${\rho }_{yx}^{*}={\displaystyle \frac{{\Delta }_{220}^{*}}{\sqrt{{\Delta }_{400}^{*}}\sqrt{{\Delta }_{040}^{*}}}}$.

Bahal and Tuteja [13] introduced an exponential ratio type estimator ${\widehat{S}}_{bt}^2,$ which is defined as follows:

$${\widehat{S}}_{bt}^2=s_y^{2}exp\left({\displaystyle \frac{{\acute{s}}_x^2-s_x^2}{{\acute{s}}_x^2+s_x^2}}\right).$$

(7)

The bias and $MSE$ of ${\widehat{S}}_{bt}^2$ are expressed as follows:

$$Bias\left({\widehat{S}}_{bt}^2\right)\cong {\displaystyle \frac{1}{2}}{\eta }^{″}{S}_y^2\left({\displaystyle \frac{3{\Delta }_{040}^{*}}{4}}-{\Delta }_{220}^{*}\right)$$

(8)

and

$$MSE\left({\widehat{S}}_{bt}^2\right)\cong S_y^4\left[\eta {\Delta }_{400}^{*}+{\eta }^{″}\left({\displaystyle \frac{{\Delta }_{040}^{*}}{4}}-{\Delta }_{220}^{*}\right)\right].$$

(9)

A ratio-type estimator called ${\widehat{S}}_{us}^2$ developed by Upadhyaya and Singh [14], which employs the kurtosis of an auxiliary variable, is expressed as follows:

$${\widehat{S}}_{us}^2=s_y^2\left({\displaystyle \frac{S_x^2+{\Delta }_{040}}{s_x^2+{\Delta }_{040}}}\right).$$

(10)

The bias and $MSE$ of ${\widehat{S}}_{us}^2$ are expressed as follows:

$$Bias\left({\widehat{S}}_{us}^2\right)\cong {\eta }^{″}g{S}_y^2\left(g{\Delta }_{040}^{*}-{\Delta }_{220}^{*}\right)$$

(11)

and

$$MSE\left({\widehat{S}}_{us}^2\right)\cong S_y^4\left[\eta {\Delta }_{400}^{*}+{\eta }^{″}\left(g^2{\Delta }_{040}^{*}-2g{\Delta }_{220}^{*}\right)\right],$$

(12)

where $g={\displaystyle \frac{S_x^2}{S_x^2+{\Delta }_{040}}}$.

Kadilar and Cingi [16] suggested that some ratio estimators are defined as

$${\widehat{S}}_{c{k}_1}^2=s_y^2\left({\displaystyle \frac{{\acute{s}}_x^2+C_x}{s_x^2+C_x}}\right),$$

(13)

$${\widehat{S}}_{c{k}_2}^2=s_y^2\left({\displaystyle \frac{{\Delta }_{040}{\acute{s}}_x^2+C_x}{{\Delta }_{040}{s}_x^2+C_x}}\right)$$

(14)

and

$${\widehat{S}}_{c{k}_3}^2=s_y^2\left({\displaystyle \frac{C_x{\acute{s}}_x^2+{\Delta }_{040}}{C_x{s}_x^2+{\Delta }_{040}}}\right).$$

(15)

The bias and $MSE$ of ${\widehat{S}}_{c{k}_i}^2(i=1,2,3)$ are expressed as follows:

$$Bias\left({\widehat{S}}_{c{k}_i}^2\right)\cong {\eta }^{″}{t}_i{S}_y^2\left(t_i{\Delta }_{040}^{*}-{\Delta }_{220}^{*}\right),$$

(16)

and

$$MSE\left({\widehat{S}}_{c{k}_i}^2\right)\cong S_y^4\left[\eta {\Delta }_{400}^{*}+{\eta }^{″}\left(t_i^2{\lambda }_{040}^{*}-2t_i{\Delta }_{220}^{*}\right)\right],$$

(17)

where $t_1={\displaystyle \frac{S_x^2}{S_x^2+C_x}},t_2={\displaystyle \frac{{\Delta }_{040}{S}_x^2}{{\Delta }_{040}{S}_x^2+C_x}},$ and $t_3={\displaystyle \frac{C_x{S}_x^2}{C_x{S}_x^2+{\Delta }_{040}}}$.

4. Proposed Class of Estimators

This section presents an improved class of estimators inspired by prior works [26,27,28,29]. These estimators employ minimum and maximum values of auxiliary variables, along with their ranks, in two-phase sampling to estimate the variance of the finite population. The suggested estimator is defined as

$${\widehat{S}}_T^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{{\gamma }_1\left({\acute{s}}_x^2-s_x^2\right)}{{\gamma }_1\left(\acute{s}+s_x^2\right)+2{\gamma }_2}}\right\}\right]exp\left[{\theta }_2\left\{{\displaystyle \frac{{\gamma }_3\left({\acute{s}}_r^2-s_r^2\right)}{{\gamma }_3\left({\acute{s}}_r^2+s_r^2\right)+2{\gamma }_4}}\right\}\right],$$

(18)

where $\left({\theta }_i,i=1,2\right)$ are known constants values either (1 or 2), and $\left({\gamma }_i,i=1,2,3,4\right)$ are the parameters of the auxiliary variables. The minimum and maximum observations of the independent (auxiliary) variable are represented by $(x_m,x_M)$, while the minimum and maximum observations of the ranks of the independent variable are represent by $(R_m,R_M)$. The known values of ${\gamma }_1,{\gamma }_2$ are given in Table 1, ${\gamma }_3=1$, and ${\gamma }_4=R_M-R_m$. We can introduce the different classes of the recommended estimator from (18), which are listed in Table 1.

Table 1. Some classes of the proposed estimator.

Subsets of the Proposed Estimator ${\widehat{\mathit{S}}}_{\mathit{T}}^2$	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$
${\widehat{S}}_{T_1}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{-{\beta }_{2\left(x\right)}\left({\acute{s}}_x^2-s_x^2\right)}{-{\beta }_{2\left(x\right)}\left({\acute{s}}_x^2+s_x^2\right)+2(x_M-x_m)}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$-{\beta }_{2\left(x\right)}$	$x_M-x_m$
${\widehat{S}}_{T_2}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{c_x\left({\acute{s}}_x^2-s_x^2\right)}{c_x\left({\acute{s}}_x^2+s_x^2\right)+2(x_M-x_m)}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$c_x$	$x_M-x_m$
${\widehat{S}}_{T_3}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{(x_M-x_m)\left({\acute{s}}_x^2-s_x^2\right)}{(x_M-x_m)\left({\acute{s}}_x^2+s_x^2\right)+2c_x}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$x_M-x_m$	$c_x$
${\widehat{S}}_{T_4}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{(x_M-x_m)\left({\acute{s}}_x^2-s_x^2\right)}{(x_M-x_m)\left({\acute{s}}_x^2+s_x^2\right)-2c_x}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$x_M-x_m$	$-c_x$
${\widehat{S}}_{T_5}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{(x_M-x_m)\left({\acute{s}}_x^2-s_x^2\right)}{(x_M-x_m)\left({\acute{s}}_x^2+s_x^2\right)+2{\beta }_{2\left(x\right)}}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$x_M-x_m$	${\beta }_{2\left(x\right)}$
${\widehat{S}}_{T_6}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{{\beta }_{2\left(x\right)}\left({\acute{s}}_x^2-s_x^2\right)}{{\beta }_{2\left(x\right)}\left({\acute{s}}_x^2+s_x^2\right)+2(x_M-x_m)}}\right\}\right]exp\left[{\theta }_2\delta \right]$	${\beta }_{2\left(x\right)}$	$x_M-x_m$
${\widehat{S}}_{T_6}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{(x_M-x_m)\left({\acute{s}}_x^2-s_x^2\right)}{(x_M-x_m)\left({\acute{s}}_x^2+s_x^2\right)-2{\beta }_{2\left(x\right)}}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$x_M-x_m$	$-{\beta }_{2\left(x\right)}$
${\widehat{S}}_{T_8}^2=s_y^{2}exp\left[{\theta }_1\left\{{\displaystyle \frac{-c_x\left({\acute{s}}_x^2-s_x^2\right)}{-c_x\left({\acute{s}}_x^2+s_x^2\right)+2(x_M-x_m)}}\right\}\right]exp\left[{\theta }_2\delta \right]$	$-c_x$	$x_M-x_m$

where $\delta =\left({\displaystyle \frac{{\acute{s}}_r^2-s_r^2}{{\acute{s}}_r^2+s_r^2+2(R_M-R_m)}}\right).$

Now, we discuss the properties of the new proposed class of estimators; we rewrite (18) in terms of errors to obtain the bias and the $MSE$ of ${\widehat{S}}_T^2$, i.e.,

$${\widehat{S}}_T^2=S_y^2\left(1+{\xi }_0\right)exp\left[{\displaystyle \frac{b_1\left({\xi }_2-{\xi }_1\right)}{2}}{\left(1+{\displaystyle \frac{b_1}{2}}\left({\xi }_1+{\xi }_2\right)\right)}^{-1}\right]exp\left[{\displaystyle \frac{b_2\left({\xi }_4-{\xi }_3\right)}{2}}{\left(1+{\displaystyle \frac{b_2}{2}}\left({\xi }_3+{\xi }_4\right)\right)}^{-1}\right]$$

(19)

where

${\theta }_1={\theta }_2=1,b_1={\displaystyle \frac{{\gamma }_1{S}_x^2}{{\gamma }_1{S}_x^2+{\gamma }_2}},$ and $b_2={\displaystyle \frac{S_r^2}{S_r^2+{\gamma }_4}}$.

Applying the Taylor series to the first approximation order, we obtain

$$\begin{array}{c}{\widehat{S}}_T^2-S_y^2\cong S_y^2\left[{\xi }_0-{\displaystyle \frac{b_1}{2}}\left({\xi }_1-{\xi }_2\right)-{\displaystyle \frac{b_2}{2}}\left({\xi }_3-{\xi }_4\right)+{\displaystyle \frac{3b_1^2}{8}}{\xi }_1^2-{\displaystyle \frac{b_1^2}{8}}{\xi }_2^2\right.\hfill \\ +{\displaystyle \frac{b_2^2}{8}}{\xi }_3^2-{\displaystyle \frac{b_2^2}{8}}{\xi }_4^2-{\displaystyle \frac{{\xi }_1}{2}}{\xi }_0{\xi }_1+{\displaystyle \frac{b_1}{2}}{\xi }_0{\xi }_2-{\displaystyle \frac{b_2}{2}}{\xi }_0{\xi }_3\hfill \\ +{\displaystyle \frac{b_2}{2}}{\xi }_0{\xi }_4-{\displaystyle \frac{b_1^2}{2}}{\xi }_1{\xi }_2+{\displaystyle \frac{b_1{b}_2}{4}}{\xi }_1{\xi }_3-{\displaystyle \frac{b_1{b}_2}{4}}{\xi }_1{\xi }_4\hfill \\ \left.-{\displaystyle \frac{b_1{b}_2}{4}}{\xi }_2{\xi }_3+{\displaystyle \frac{b_1{b}_2}{4}}{\xi }_2{\xi }_4-{\displaystyle \frac{b_2^2}{2}}{\xi }_3{\xi }_4\right].\hfill \end{array}$$

(20)

Using Equation (20), the bias of ${\widehat{S}}_T^2$ is given by

$$\begin{array}{c}Bias\left({\widehat{S}}_T^2\right)\cong \eta S_y^2\left[{\displaystyle \frac{3b_1^2}{8}}{\Delta }_{040}^{*}+{\displaystyle \frac{3b_2^2}{8}}{\Delta }_{004}^{*}-{\displaystyle \frac{b_1}{2}}{\Delta }_{220}^{*}-{\displaystyle \frac{b_2}{2}}{\Delta }_{202}^{*}\right.\left.+{\displaystyle \frac{b_1{b}_2}{2}}{\Delta }_{022}^{*}\right]\hfill \\ -{\eta }^{\prime }{S}_y^2\left[{\displaystyle \frac{3b_1^2}{8}}{\Delta }_{040}^{*}+{\displaystyle \frac{3b_2^2}{8}}{\Delta }_{004}^{*}-{\displaystyle \frac{b_1}{2}}{\Delta }_{220}^{*}-{\displaystyle \frac{b_2}{2}}{\Delta }_{202}^{*}\right.\left.+{\displaystyle \frac{b_1{b}_2}{2}}{\lambda }_{022}^{*}\right].\hfill \end{array}$$

After the simple simplifications, we get

$$Bias\left({\widehat{S}}_T^2\right)\cong {\eta }^{″}{S}_y^2\left[{\displaystyle \frac{3}{8}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}\right)-{\displaystyle \frac{1}{2}}\left(b_1{\Delta }_{220}^{*}+b_2{\Delta }_{202}^{*}-b_1{b}_2{\Delta }_{022}^{*}\right)\right],$$

(21)

where ${\eta }^{″}=\eta -{\eta }^{\prime }$.

In order to obtain a first-order approximation of the $MSE$, we squared both sides of Equation (20) and then applied the expected value, which is given by the following equation:

$$\begin{array}{c}MSE\left({\widehat{S}}_T^2\right)\cong \eta S_y^4\left[{\Delta }_{400}^{*}+{\displaystyle \frac{b_1^2}{4}}{\Delta }_{040}^{*}+{\displaystyle \frac{b_2^2}{4}}{\Delta }_{004}^{*}-b_1{\Delta }_{220}^{*}-b_2{\Delta }_{202}^{*}\right.\left.+{\displaystyle \frac{b_1{b}_2}{2}}{\Delta }_{022}^{*}\right]\hfill \\ -{\eta }^{\prime }{S}_y^4\left[{\displaystyle \frac{b_1^2}{4}}{\Delta }_{040}^{*}+{\displaystyle \frac{b_2^2}{4}}{\Delta }_{004}^{*}-b_1{\Delta }_{220}^{*}-b_2{\Delta }_{202}^{*}\right.\left.+{\displaystyle \frac{t_4{t}_5}{2}}{\lambda }_{022}^{*}\right].\hfill \end{array}$$

After the simplification, we get

$$MSE\left({\widehat{S}}_T^2\right)\cong S_y^4\left[\eta {\Delta }_{400}^{*}+{\displaystyle \frac{{\eta }^{″}}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right].$$

(22)

5. Mathematical Comparison

This section covers the comparisons between the suggested estimator ${\widehat{S}}_T^2$ and several existing estimators, such as ${\widehat{S}}_y^2,{\widehat{S}}_R^2,{\widehat{S}}_{lr}^2,{\widehat{S}}_{bt}^2,{\widehat{S}}_{us}^2$, and ${\widehat{S}}_{c{k}_i}^2$.

(i)

Comparison of the estimators given in Equations (1) and (22):

$$\begin{gathered} Var\left({\widehat{S}}_y^2\right)>MSE\left({\widehat{S}}_T^2\right)\mathrm{if} \\ \left(\eta -{\eta }^{\prime }\right)\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)<0 \end{gathered}$$

For $\eta -{\eta }^{\prime }>0,$ that is, $\eta >{\eta }^{\prime }$

$$\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)<0$$

(23)

Similarly, $\eta -{\eta }^{\prime }<0,$ that is, $\eta <{\eta }^{\prime }$

$$\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)>0$$

(24)

If Conditions (23) or (24) hold true, the suggested estimator ${\widehat{S}}_T^2$ demonstrates a higher efficiency in comparison to $MSE\left({\widehat{S}}_y^2\right)$.

(ii)

Comparison of the estimators given in Equations (4) and (22):

$$\begin{gathered} MSE\left({\widehat{S}}_R^2\right)>MSE\left({\widehat{S}}_T^2\right)\mathrm{if} \\ \left(\eta -{\eta }^{\prime }\right)\left[{\Delta }_{040}^{*}-2{\Delta }_{022}^{*}-{\displaystyle \frac{1}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right]>0 \end{gathered}$$

For $\eta -{\eta }^{\prime }<0,$ that is, $\eta <{\eta }^{\prime }$

$$\left[{\Delta }_{040}^{*}-2{\Delta }_{022}^{*}-{\displaystyle \frac{1}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right]>0$$

(25)

Similarly, $\eta -{\eta }^{\prime }>0,$ that is, $\eta >{\eta }^{\prime }$

$$\left[{\Delta }_{040}^{*}-2{\Delta }_{022}^{*}-{\displaystyle \frac{1}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right]<0$$

(26)

If Conditions (25) or (26) hold true, the suggested estimator ${\widehat{S}}_T^2$ demonstrates a higher efficiency in comparison to $MSE\left({\widehat{S}}_R^2\right)$.

(iii)

Comparison of the estimators given in Equations (6) and (22):

$$\begin{gathered} MSE\left({\widehat{S}}_{lr}^2\right)>MSE\left({\widehat{S}}_T^2\right)\mathrm{if} \\ \left(\eta -{\eta }^{\prime }\right)\left[{\rho }_{yx}^{*2}+{\displaystyle \frac{1}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right]<0 \end{gathered}$$

For $\eta -{\eta }^{\prime }>0,$ that is, $\eta >{\eta }^{\prime }$

$$\left[{\rho }_{yx}^{*2}+{\displaystyle \frac{1}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right]<0$$

(27)

Similarly, $\eta -{\eta }^{\prime }<0,$ that is, $\eta <{\eta }^{\prime }$

$$\left[{\rho }_{yx}^{*2}+{\displaystyle \frac{1}{4}}\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)\right]>0$$

(28)

If Conditions (27) or (28) hold true, the suggested estimator ${\widehat{S}}_T^2$ demonstrates a higher efficiency in comparison to $MSE\left({\widehat{S}}_{lr}^2\right)$.

(iv)

Comparison of the estimators given in Equations (9) and (22):

$$\begin{gathered} MSE\left({\widehat{S}}_{bt}^2\right)>MSE\left({\widehat{S}}_T^2\right)\mathrm{if} \\ \left(\eta -{\eta }^{\prime }\right)\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left({\Delta }_{040}^{*}-4{\Delta }_{220}^{*}\right)\right]<0 \end{gathered}$$

For $\theta -{\theta }^{\prime }>0,$ that is, $\eta >{\eta }^{\prime }$

$$\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left({\Delta }_{040}^{*}-4{\Delta }_{220}^{*}\right)\right]<0$$

(29)

Similarly, $\eta -{\eta }^{\prime }<0,$ that is, $\eta <{\eta }^{\prime }$

$$\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left({\Delta }_{040}^{*}-4{\Delta }_{220}^{*}\right)\right]>0$$

(30)

If Conditions (29) or (30) hold true, the suggested estimator ${\widehat{S}}_T^2$ demonstrates a higher efficiency in comparison to $MSE\left({\widehat{S}}_{bt}^2\right)$.

(v)

Comparison of the estimators given in Equations (12) and (22):

$$\begin{gathered} MSE\left({\widehat{S}}_{us}^2\right)>MSE\left({\widehat{S}}_T^2\right)\mathrm{if} \\ \left(\eta -{\eta }^{\prime }\right)\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left(4g^2{\Delta }_{040}^{*}-4g{\Delta }_{220}^{*}\right)\right]<0 \end{gathered}$$

For $\eta -{\eta }^{\prime }>0,$ that is, $\eta >{\eta }^{\prime }$

$$\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left(4g^2{\Delta }_{040}^{*}-4g{\Delta }_{220}^{*}\right)\right]<0$$

(31)

Similarly, $\eta -{\eta }^{\prime }<0,$ that is, $\eta <{\eta }^{\prime }$

$$\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left(4g^2{\Delta }_{040}^{*}-4g{\Delta }_{220}^{*}\right)\right]>0$$

(32)

If Conditions (31) or (32) hold true, the suggested estimator ${\widehat{S}}_T^2$ demonstrates a higher efficiency in comparison to $MSE\left({\widehat{S}}_{us}^2\right)$.

(vi)

Comparison of the estimators given in Equations (17) and (22):

$$\begin{gathered} MSE\left({\widehat{S}}_{c{k}_i}^2\right)>MSE\left({\widehat{S}}_T^2\right)\mathrm{if} \\ \left(\eta -{\eta }^{\prime }\right)\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left(4t_i^2{\Delta }_{040}^{*}-4t_i{\Delta }_{220}^{*}\right)\right]<0 \end{gathered}$$

For $\eta -{\eta }^{\prime }>0,$ that is, $\eta >{\eta }^{\prime }$

$$\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left(4t_i^2{\Delta }_{040}^{*}-4t_i{\Delta }_{220}^{*}\right)\right]<0$$

(33)

Similarly, $\eta -{\eta }^{\prime }<0,$ that is, $\eta <{\eta }^{\prime }$

$$\left[\left(b_1^2{\Delta }_{040}^{*}+b_2^2{\Delta }_{004}^{*}-4b_1{\Delta }_{220}^{*}-4b_2{\Delta }_{202}^{*}+2b_1{b}_2{\Delta }_{022}^{*}\right)-\left(4t_i^2{\Delta }_{040}^{*}-4t_i{\Delta }_{220}^{*}\right)\right]>0$$

(34)

If Conditions (33) or (34) hold true, the suggested estimator ${\widehat{S}}_T^2$ demonstrates a higher efficiency in comparison to $MSE\left({\widehat{S}}_{cki}^2\right)$.

6. Numerical Comparison

In this section, we assess the effectiveness of the proposed class of estimators as compared to other existing estimators through the percent relative efficiency (PREs). This evaluation is carried out using both simulated data sets and three distinct real data sets.

6.1. Simulation Study

In order to validate the theoretical results presented in Section 5, we employ the methodology suggested by [27,29] to conduct a simulation analysis. The objective is to assess the performance of the proposed class of estimators based on the known minimum and maximum values of the auxiliary variable, as well as its ranks within the framework of two-phase sampling. By employing the following probability distributions, it is possible to carefully produce six different populations for the auxiliary variable X.

• Population 1: $X\sim Uniform(2,4).$
• Population 2: $X\sim Uniform(0,1).$
• Population 3: $X\sim Gamma(3,6).$
• Population 4: $X\sim Gamma(5,10).$
• Population 5: $X\sim Exponential\left(4\right).$
• Population 6: $X\sim Exponential\left(6\right).$

Subsequently, the dependent variable Y is measured as

$$Y=r_{yx}\times X+e,$$

where

$$r_{yx}=0.78$$

signifies the correlation coefficient between the dependent and independent variables, while

$$e\sim N(0,1)$$

indicates the error term. The selected value of $r_{yx}$ might reflect the quality and consistency of the data. A correlation coefficient $r_{yx}=0.78$ suggests that there is relatively low noise and the relationship between x and y is consistent across the data set.

To calculate the percent relative efficiencies (PREs), we adopted the following procedures in R-Software (latest v. 4.4.0).

Step 1: 

Firstly, we make use of particular probability distributions to obtain a population of 1500.

Step 2: 

We apply the simple random sampling without replacement (SRSWOR) approach to obtain a first phase sample of size $\acute{n}$ from a population of size N.

Step 3: 

Using the SRSWOR approach again, we obtain the second phase sample size n from the first phase sample.

Step 4: 

We calculate the population total and the minimum and maximum values of the auxiliary variable from the above steps.

Step 5: 

For each population, we generate samples of different sizes using SRSWOR.

Step 6: 

For each sample size, we find the $PREs$ values of all the estimators discussed in this article.

Step 7: 

We executed Steps 5 and 6 a total of 65,000 times. The outcomes for artificial populations are detailed in Table 2, and the results for real data sets are summarized in Table 6.

Table 2. Percent relative efficiency $\left(PRE\right)$ of the estimators based on artificial populations.

Estimator	$\mathit{Uni}(2,4)$	$\mathit{Uni}(0,1)$	$\mathit{Gam}(3,6)$	$\mathit{Gam}(5,10)$	$\mathit{Exp}\left(4\right)$	$\mathit{Exp}\left(6\right)$
${\widehat{S}}_y^2$	100	100	100	100	100	100
${\widehat{S}}_R^2$	130.789	119.760	123.025	118.247	130.126	117.587
${\widehat{S}}_{lr}^2$	140.970	133.638	145.188	132.315	132.505	118.765
${\widehat{S}}_{tb}^2$	146.486	137.946	138.344	134.200	131.425	116.078
${\widehat{S}}_{su}^2$	153.145	1 36.108	142.670	136.526	139.772	119.589
${\widehat{S}}_{c{k}_1}^2$	155.980	145.964	146.520	138.789	134.405	117.164
${\widehat{S}}_{c{k}_2}^2$	155.112	1 45.123	145.345	138.002	133.408	117.664
${\widehat{S}}_{c{k}_3}^2$	154.304	144.245	143.156	137.328	131.528	117.589
${\widehat{S}}_{T_1}^2$	214.668	210.356	180.712	190.139	177. 547	183.329
${\widehat{S}}_{T_2}^2$	383.129	415.467	350.225	363.167	324.724	310.289
${\widehat{S}}_{T_3}^2$	289.189	286.578	300.667	289.369	250.949	244.345
${\widehat{S}}_{T_4}^2$	192.707	198.689	210.576	195.508	187.333	172.148
${\widehat{S}}_{T_5}^2$	172.837	178.790	190.031	173.625	160.495	156.279
${\widehat{S}}_{T_6}^2$	220.456	232.098	230.321	212.353	202.132	192.399
${\widehat{S}}_{T_7}^2$	200.065	222.987	210.401	202.323	188.369	188.950
${\widehat{S}}_{T_8}^2$	226.825	240.876	246.677	250.688	230. 712	222.952

Finally, we use the following formulas to obtain the MSEs and PREs of each estimator across all replications:

$$MSE{\left({\widehat{S}}_i^2\right)}_{min}={\displaystyle \frac{{\sum }_{j=1}^{\mathrm{65,000}}{\left({\widehat{S}}_i^2-S_y^2\right)}^2}{\mathrm{65,000}}}$$

and

$$PRE={\displaystyle \frac{V\left({\widehat{S}}_{yst}^2\right)}{MSE{\left({\widehat{S}}_i^2\right)}_{min}}}\times 100,$$

where i is one of $R,lr,bt,us,c{k}_1,c{k}_2,c{k}_3,T_k(k=1,2,\dots ,8).$

6.2. Numerical Examples

We compared the percent relative efficiencies $\left(PREs\right)$ of different estimators using three real data sets in order to assess the performances of the proposed estimators. The descriptions of the data sets are defined below, while summary statistics of the data sets are given in Table 3, Table 4 and Table 5.

Table 3. Summary statistics for Data 1.

$N=36$	$\acute{n}=15$	$n=6$	$\overline{X}=1659.58$
$\overline{Y}=4015.22$	$\overline{R}=18.50$	$X_M=2379$	$X_m=49$
$R_M=36$	$R_m=1$	$S_x=3410$	$S_y=8512$
$S_r=10.53$	$C_x=2.06$	$C_y=2.12$	$C_r=0.57$
${\rho }_{yx}=0.65$	${\rho }_{yr}=0.36$	${\rho }_{xr}=0.75$	${\Delta }_{400}=2051$
${\Delta }_{040}=2237$	${\Delta }_{004}=3297$	${\Delta }_{220}=1542$	${\Delta }_{202}=2698$
${\Delta }_{022}=3698$	$\eta =0.14$	${\eta }^{\prime }=0.04$	${\eta }^{″}=0.10$

Table 4. Summary statistics for Data 2.

$N=33$	$\acute{n}=15$	$n=6$	$\overline{X}=72.55$
$\overline{Y}=27.49$	$\overline{R}=17$	$X_M=95$	$X_m=58$
$R_M=33$	$R_m=1$	$S_x=10.58$	$S_y=10.13$
$S_r=9.64$	$C_x=0.14$	$C_y=0.37$	$C_r=0.57$
${\rho }_{yx}=0.25$	${\rho }_{yr}=0.20$	${\rho }_{xr}=0.98$	${\Delta }_{400}=5.55$
${\Delta }_{040}=2.08$	${\Delta }_{004}=1.10$	${\Delta }_{220}=2.22$	${\Delta }_{202}=0.94$
${\Delta }_{022}=1.54$	$\eta =0.14$	${\eta }^{\prime }=0.04$	${\eta }^{″}=0.10$

Table 5. Summary statistics for Data 3.

$N=36$	$\acute{n}=15$	$n=6$	$\overline{X}=1054.39$
$\overline{Y}=14818.70$	$\overline{R}=18.50$	$X_M=2370$	$X_m=39$
$R_M=36$	$R_m=1$	$S_x=2214.22$	$S_y=32601.14$
$S_r=10.544$	$C_x=2.10$	$C_y=22.07$	$C_r=0.56$
${\rho }_{yx}=0.69$	${\rho }_{yr}=0.39$	${\rho }_{xr}=0.84$	${\Delta }_{400}=236.50$
${\Delta }_{040}=209.80$	${\Delta }_{004}=329.70$	${\Delta }_{220}=397.50$	${\Delta }_{202}=365.80$
${\Delta }_{022}=469.80$	$\eta =0.14$	${\eta }^{\prime }=0.04$	${\eta }^{″}=0.10$

• Data 1. This data set was selected from Bureau of Statistics page 226 [34] and was conducted in Pakistan during the year 2012, which comprised 33 divisions. The data set can be downloaded from the Pakistan Bureau of Statistics web page via the following link: https://www.pbs.gov.pk/content/microdata (accessed on 5 August 2024).
  Y: Departmental employment levels in 2012.
  X: Number of factories the departments registered in 2012.
  R: Ranks the number of factories the departments registered in 2012.
• Data 2. This data set was selected from Cochran page 23 [35], comprising 33 units of food cost and weekly income of families.
  Y: Food expenses related to the families’ employment.
  X: Families’ weekly income.
  R: Ranks the families’ weekly income.
• Data 3. Another data set was selected from Bureau of Statistics page 126 [34], conducted in Pakistan during the year 2012, which comprised 33 divisions. The data set can be downloaded from the Pakistan Bureau of Statistics web page via the following link: https://www.pbs.gov.pk/content/microdata (accessed on 5 August 2024).
  Y: The total enrollment of students in 2012.
  X: Government elementary and secondary schools in 2012.
  R: Ranks the government elementary and secondary schools in 2012.

Finally, we use the following formula to calculate the percent relative efficiencies (PREs):

$$PRE={\displaystyle \frac{V\left({\widehat{S}}_{yst}^2\right)=MSE\left({\widehat{S}}_{yst}^2\right)}{MSE\left({\widehat{S}}_l^2\right)}}\times 100,$$

where l is one of $R,lr,bt,us,c{k}_1,c{k}_2,c{k}_3,T_k(k=1,2,\dots ,8).$

We used simulation studies and three real data sets in order to determine the performance of the proposed class of estimators. The $PRE$ criterion was used for the comparisons between different estimators. The $PRE$ values of the proposed and existing estimators obtained from the simulation study are given in Table 2, while the outcomes for real data sets are presented in Table 6, respectively. The following are some general findings:

Table 6. Percent relative efficiency using empirical data sets.

Estimator	Data 1	Data 2	Data 3
${\widehat{S}}_y^2$	100	100	100
${\widehat{S}}_R^2$	141.830	143.664	121.469
${\widehat{S}}_{lr}^2$	143.225	104.673	125.532
${\widehat{S}}_{tb}^2$	152.034	128.010	115.634
${\widehat{S}}_{su}^2$	141.858	143.509	121.380
${\widehat{S}}_{c{k}_1}^2$	141.839	143.655	121.460
${\widehat{S}}_{c{k}_2}^2$	141.839	143.660	121.460
${\widehat{S}}_{c{k}_3}^2$	141.849	142.044	121.469
${\widehat{S}}_{T_1}^2$	147.360	154.388	133.017
${\widehat{S}}_{T_2}^2$	276.108	254.005	290.116
${\widehat{S}}_{T_3}^2$	235.023	220.660	212.978
${\widehat{S}}_{T_4}^2$	162.723	179.478	186.215
${\widehat{S}}_{T_5}^2$	152.411	160.308	170.165
${\widehat{S}}_{T_6}^2$	212.907	192.115	208.864
${\widehat{S}}_{T_7}^2$	188.606	181.529	198.220
${\widehat{S}}_{T_8}^2$	226.560	196.409	263.810

• For all simulated scenarios and real data sets, Table 2 and Table 6 illustrate that the $MSE$ values of each proposed estimate are smaller than those of the current estimators defined in the literature, confirming the higher accuracy of the recommended estimators over the existing estimators.
• Furthermore, all of the proposed estimators have $PRE$ values that are higher than those of the existing estimators, as shown in Table 2 and Table 6. This indicates that the performance of the proposed class of estimators is preferred to that of the existing estimators.

7. Conclusions

In order to estimate finite population variance, this study presented a new family of efficient estimators. These estimators considered the extreme values of the auxiliary variable, alongside its ranks. The suggested class of estimators is shown to be more efficient under the theoretical assumptions given in Section 5, enabling a comparative analysis compared to the existing ones. To investigate these constraints, we conducted a simulation study and examined multiple empirical data sets. The results show that the proposed class of estimators regularly outperforms the existing ones in terms of $PREs$. The results are shown in Table 2. The results presented in Table 6 provide additional evidence for this conclusion, which is consistent with the theoretical understandings presented in Section 5. We conclude that the suggested estimators ${\widehat{S}}_{T_i}^2$$(i=1,2,3,\dots ,8)$ are more efficient than the other estimators taken into consideration based on both the simulation and empirical results. Among these, ${\widehat{S}}_{T_2}^2$ is particularly preferred due to its minimal $MSE$.

However, we examined the characteristics of the proposed efficient class of estimators within a two-phase sampling framework. Additionally, it is feasible to develop new estimators using the two-phase stratified sampling method, and our findings may assist in identifying more efficient estimators with lower $MSE$s. It is also a good topic for future research work.