Finite Population Variance Estimation Using Monte Carlo Simulation and Real Life Application

Abstract

In this paper, we introduce a new, improved family of estimators aimed at improving the accuracy of finite population variance estimation. The proposed class of estimators is enhanced in terms of overall estimating performance and robustness by using the maximum and minimum values as supplementary information to improve precision and reliability. The bias and mean square error are theoretically obtained to the first order of approximation. We carried out a simulation study to evaluate the performance of the proposed estimators and validate the theoretical results. The findings show that the proposed estimators have higher percent relative efficiencies than the existing estimators across all of the simulated scenarios. Additionally, three independent symmetric and asymmetric datasets are analyzed to further confirm the better efficiency of proposed estimators compared with traditional estimators.

1. Introduction

In sampling theory, it is customary to make use of use auxiliary variables (independent variables) in association with study variables (dependent variables) to improve the sampling design. These auxiliary variables are used to enhance the efficiency of estimators. For example, when estimating average household incomes, auxiliary variables such as the number of household members can be used, as they are easier to measure and are correlated with income. By utilizing these auxiliary variables, we can improve the precision of our estimates of the population mean, making the process more cost-effective and efficient. There are situations where data on multiple auxiliary variables may be easily obtainable. For example, in order to examine the public health and welfare situation in an entire country or state, it may be necessary to have information on the number of beds in various hospitals, the number of physicians and other staff members, and the overall total funds available for medical treatment. In such cases, where this kind of information is unavailable, two-phase sampling is preferable because it aims to obtain a large preliminary sample from which the auxiliary variable is determined. This method is commonly used when it is more cost-effective or efficient to conduct a preliminary phase before the main phase. In two-phase sampling, the first and second samples are called the first-phase sample and the second-phase sample.

In sample surveys, two-phase sampling plays an important role due to the fact that it is a cost-effective sampling technique that is frequently employed when supplementary data is not available before conducting the survey. Two-phase sampling was first discussed by [1]. Until the works of [2], such works were not analyzed further. Further discussion on two-phase sampling included the new different estimators proposed by [3,4,5,6]. The ability of two-phase sampling to efficiently choose the variables while minimizing costs has attracted a lot of interest in recent years. For further details and examples of two-phase sampling studies, readers can refer to the following [7,8,9,10,11,12].

It can be challenging to control variability in applications, and it is a major concern to precisely estimate the variance of a finite population. In biological and agricultural research, this problem arises, which causes the intended results to be unpredictable. It is possible to enhance the estimators more precisely by using additional information in the proper way. The very first overview about population variance estimation was given by [4]. Exponential estimators based on ratio-product types were presented by [13] to estimate population variance. In order to discuss the improvement in the estimation of population variance, ref. [14] provided various estimators. Several researchers have introduced different estimators for calculating population variance. For more details, see [15,16,17,18,19,20,21].

Many populations contain extreme values, which can significantly affect the accuracy of estimates for unknown population characteristics if not properly considered. Ignoring this data may lead to either inflated or underestimated results. The study by [22] introduced two estimators that use the upper and lower values of the auxiliary variable. However, this approach was not explored further until research by [23], who applied the concept of maximum and minimum data points to different estimators. A stratification sampling approach including extreme values considerably improved the estimated values of the population mean by [24]. A generalized class of ratio estimators based on different regression techniques to handle the outlier values for the estimation of finite population mean was proposed by [25]. Some new family of estimators for variance by utilizing the idea of extreme values introduced by [26]. Recently, [27,28] developed several efficient estimator classes by applying transformations to extreme values for estimating population variance. For more details, see [29,30,31,32], and the references therein.

The mean square error (MSE) performance of classical estimators, which are important for statistical analysis, frequently declines when using datasets with extreme values. Researchers may be tempted to exclude outlier data points from their samples in an effort to improve analyses and enhance outcomes due to this decrease in accuracy. However, this technique can result in biased or insufficient estimates of population parameters. These extreme observations must be included so that the estimations can accurately represent the variability and characteristics of the population. To solve the issue, this article provides a family of different improved estimators for finite population variance that includes the upper and lower values of an auxiliary variable under two-phase sampling. The new family of estimators is enhanced in terms of overall estimating performance and robustness by using these extreme values as supplementary information to improve their precision and reliability. The following major factors led to the formulation of a new approach for computing the population variance:

• For finite population variance, classical estimators usually ignore extreme values, which can distort the results and inflate mean squared errors (MSEs). Two-phase sampling designs are inefficient, which emphasizes the need for an improved approach that addresses these challenges.
• The complicated data structure of two-phase sampling makes it challenging for existing estimators to handle. These issues show that reliable and effective estimators are required.
• In most cases, the two-phase method is more economical than the one-phase method, particularly when dealing with large samples. It reduces overall costs by allowing researchers to collect initial data using a small sample before choosing a second sample.
• Two-phase sampling is an effective technique in various research contexts, as it provides better representation and control of variability, along with cost efficiency, improved precision, and greater flexibility.
• This paper introduces a new class of estimators for estimating the finite population variance. Unlike traditional estimators that overlook extreme values, the proposed estimators incorporate the maximum and minimum values of auxiliary variables to enhance precision and robustness, particularly in the presence of outliers.
• The new estimators offer an improved bias and mean squared error (MSE) performance over existing methods. This is achieved by using extreme values, thus providing a more accurate estimate of the finite population variance, especially when the data contain extreme outliers or skewed distributions.
• In addition to the theoretical derivation of the estimators’ biases and MSE, the paper includes extensive Monte Carlo simulations and real-life applications. These demonstrate that the proposed estimators outperform traditional methods in terms of percent relative efficiency (PRE), validating the theoretical improvements with empirical evidence.
• By utilizing the maximum and minimum values of the auxiliary variable, the proposed estimators increase the robustness of the variance estimation, addressing the issue of extreme observations that often distort classical estimators.
• The estimators proposed in this study are not only mathematically robust, but also highly practical, providing a solution that is beneficial in real-life applications such as public health studies, educational research, and economic data analysis, where auxiliary information is often available.

The upcoming parts of this article are ordered as follows: The methods and notations used in the study are described completely in Section 2. Section 3 introduces some existing estimators. For the suggested class of estimators, we provide a detailed discussion in Section 4. A comprehensive mathematical comparison of these estimators is presented in Section 5. The simulation study described in Section 6 verifies the theoretical results defined in Section 5 by generating different suitable artificial populations. Numerical examples based on symmetric and asymmetric datasets are also given in this section in order to highlight how our theoretical results are applied in practical situations. Finally, Section 7 discusses the main conclusions and proposes directions for future research.

2. Notations and Existing Estimators

Consider a finite population $\delta =\left({\delta }_1,{\delta }_2,\dots ,{\delta }_N\right)$ consisting of N units. For each unit i, the realized values associated with the study variable Y and the auxiliary variable X are denoted by $y_i$ and $x_i$, respectively. For these variables, we define the population variances (without replacement sampling) as follows:

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

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

where ($\overline{Y},\overline{X}$) denote the population means, defined as follows:

$$\overline{Y}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{Y}_i$$

and

$$\overline{X}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{X}_i.$$

Let $C_y$ and $C_x$ be the coefficients of variation for Y and X, respectively, and ${\rho }_{yx}$ be the population correlation coefficient between them, which are defined as follows:

$$C_y={\displaystyle \frac{S_y}{\overline{Y}}},$$

$$C_x={\displaystyle \frac{S_x}{\overline{X}}},$$

$${\rho }_{yx}={\displaystyle \frac{S_{yx}}{S_y{S}_x}},$$

where

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

is the covariance between Y and X.

Consider the sample variances

$$s_y^2={\widehat{S}}_y^2={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{\left(Y_i-\overline{y}\right)}^2,$$

$$s_x^2={\widehat{S}}_x^2={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{\left(X_i-\overline{x}\right)}^2,$$

where

$$\overline{y}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{Y}_i$$

and

$$\overline{x}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{X}_i$$

are the sample means of $Y,$ and X. Further, the sample coefficients of variation are defined as follows:

$$c_y={\displaystyle \frac{s_y}{\overline{y}}}$$

and

$$c_x={\displaystyle \frac{s_x}{\overline{x}}}.$$

In this study, various transformations are utilized to develop some new estimators for estimating the finite population variance based on a two-phase sampling approach. The two-phase sampling is defined as follows:

• To estimate the population variance $S_x^2$, a sample of size $\left(m<N\right)$ is taken in the first phase.
• The second phase involves selecting a sample of size $\left(n<m\right)$ to measure x and y, respectively.

Moreover, let ${\acute{s}}_x^2$ represent the sample variance obtained from the first phase sample with a size of m, while $s_y^2$ and $s_x^2$ denote the sample variances derived from the second phase sample with a size of n.

3. Existing Estimators

In this section, we discuss the biases and mean squared errors of different existing estimators for estimating the finite population variances. For each estimator, the relative errors are used to compute the biases and mean square errors as

$$e_0=\left({\displaystyle \frac{s_y^2-S_y^2}{S_y^2}}\right),$$

$$e_1=\left({\displaystyle \frac{s_x^2-S_x^2}{S_x^2}}\right),$$

$$e_2=\left({\displaystyle \frac{{\acute{s}}_x^2-S_x^2}{S_x^2}}\right),$$

such that $E\left(e_i\right)=0$ for $i=0,1,2$. We know that

$$E\left(e_0^2\right)=\theta {\lambda }_{40}^{*},$$

$$E\left(e_1^2\right)=\theta {\lambda }_{04}^{*},$$

$$E\left(e_2^2\right)={\theta }^{\prime }{\lambda }_{04}^{*},$$

$$E\left(e_0{e}_1\right)=\theta {\lambda }_{22}^{*},$$

$$E\left(e_0{e}_2\right)={\theta }^{\prime }{\lambda }_{22}^{*},$$

$$E\left(e_1{e}_2\right)={\theta }^{\prime }{\lambda }_{04}^{*},$$

where

$${\lambda }_{40}^{*}=({\lambda }_{40}-1),$$

$${\lambda }_{04}^{*}=({\lambda }_{04}-1),$$

$${\lambda }_{22}^{*}=({\lambda }_{22}-1),$$

$$\theta =\left({\displaystyle \frac{1}{n}}-{\displaystyle \frac{1}{N}}\right),$$

$${\theta }^{\prime }=\left({\displaystyle \frac{1}{m}}-{\displaystyle \frac{1}{N}}\right)$$

and

$${\theta }^{\prime \prime }=\left({\displaystyle \frac{1}{n}}-{\displaystyle \frac{1}{m}}\right).$$

Also

$${\lambda }_{lq}={\displaystyle \frac{{\mu }_{lq}}{{\mu }_{20}^{l/2}{\mu }_{02}^{q/2}}},$$

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

where ${\lambda }_{lq}$ represents the moment ratio with orders $(l,q)$, and $({\mu }_{20},{\mu }_{02})$ are the second-order moments from means. Further, ${\lambda }_{400}={\beta }_{2\left(y\right)},$ and ${\lambda }_{040}={\beta }_{2\left(x\right)}$ are the population coefficients of kurtosis.

The well-know estimator of variance is defined as follows:

$${\widehat{S}}_y^2={\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{\left(y_i-\overline{Y}\right)}^2.$$

Its variance is given by:

$$Var\left({\widehat{S}}_y^2\right)=\theta S_y^4{\lambda }_{40}^{*}.$$

(1)

Using a ratio, [4] provided a ratio estimator for the first time in the literature, as follows:

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

The bias and MSE expressions for ${\widehat{S}}_R^2$ are given by the following equations:

$$Bias\left({\widehat{S}}_R^2\right)\cong {\theta }^{\prime \prime }{S}_y^4\left({\lambda }_{04}^{*}-{\lambda }_{22}^{*}\right)$$

(2)

and

$$MSE\left({\widehat{S}}_R^2\right)\cong S_y^4\left(\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }{\lambda }_{04}^{*}-2{\theta }^{\prime \prime }{\lambda }_{22}^{*}\right).$$

(3)

The linear regression estimator in terms of two-phase sampling by using the concept given in [33], is defined as follows:

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

where

$$b_{(s_y^2,s_x^2)}={\displaystyle \frac{s_y^2{\widehat{\lambda }}_{22}^{*}}{s_x^2{\widehat{\lambda }}_{04}^{*}}}$$

represents a sample regression coefficient.

The MSE expression for ${\widehat{S}}_{lr}^2$ is given by the following equation:

$$MSE\left({\widehat{S}}_{lr}^2\right)\cong S_y^4{\lambda }_{40}^{*}\left(\theta -{\theta }^{\prime \prime }{\rho }_{yx}^{*2}\right),$$

(4)

where ${\rho }_{yx}^{*}={\displaystyle \frac{{\lambda }_{22}^{*}}{\sqrt{{\lambda }_{40}^{*}}\sqrt{{\lambda }_{04}^{*}}}}$.

The exponential ratio type estimator ${\widehat{S}}_{bt}^2$ obtained by [13], is defined as follows:

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

The bias and MSE expressions for ${\widehat{S}}_{bt}^2$ are given by the following equations:

$$Bias\left({\widehat{S}}_{bt}^2\right)\cong {\displaystyle \frac{1}{2}}{\theta }^{\prime \prime }{S}_y^2\left({\displaystyle \frac{3{\lambda }_{04}^{*}}{4}}-{\lambda }_{22}^{*}\right)$$

(5)

and

$$MSE\left({\widehat{S}}_{bt}^2\right)\cong S_y^4\left[\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }\left({\displaystyle \frac{{\lambda }_{04}^{*}}{4}}-{\lambda }_{22}^{*}\right)\right].$$

(6)

An improved ratio-type estimator ${\widehat{S}}_{us}^2$ proposed by [14] through employing the kurtosis of an auxiliary variable, which is defined as follows:

$${\widehat{S}}_{us}^2=s_y^2\left({\displaystyle \frac{{\acute{s}}_x^2+{\lambda }_{04}}{s_x^2+{\lambda }_{04}}}\right).$$

The bias and MSE expressions for ${\widehat{S}}_{us}^2$ are given by the following equations:

$$Bias\left({\widehat{S}}_{us}^2\right)\cong {\theta }^{\prime \prime }g{S}_y^2\left(g{\lambda }_{04}^{*}-{\lambda }_{22}^{*}\right)$$

(7)

and

$$MSE\left({\widehat{S}}_{us}^2\right)\cong S_y^4\left[\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }\left(g^2{\lambda }_{04}^{*}-2g{\lambda }_{22}^{*}\right)\right],$$

(8)

where

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

Some ratio estimators suggested by [16], which are defined as follows:

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

$${\widehat{S}}_{c{k}_2}^2=s_y^2\left({\displaystyle \frac{{\lambda }_{04}{\acute{s}}_x^2+C_x}{{\lambda }_{04}{s}_x^2+C_x}}\right)$$

and

$${\widehat{S}}_{c{k}_3}^2=s_y^2\left({\displaystyle \frac{C_x{\acute{s}}_x^2+{\lambda }_{04}}{C_x{s}_x^2+{\lambda }_{04}}}\right).$$

The bias and MSE expressions for ${\widehat{S}}_{c{k}_i}^2(i=0,1,2)$ are given by the following equations:

$$Bias\left({\widehat{S}}_{c{k}_i}^2\right)\cong {\theta }^{\prime \prime }{t}_i{S}_y^2\left(t_i{\lambda }_{04}^{*}-{\lambda }_{22}^{*}\right),$$

(9)

and

$$MSE\left({\widehat{S}}_{c{k}_i}^2\right)\cong S_y^4\left[\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }\left(t_i^2{\lambda }_{04}^{*}-2t_i{\lambda }_{22}^{*}\right)\right],$$

(10)

where

$$t_0={\displaystyle \frac{S_x^2}{S_x^2+C_x}},$$

$$t_1={\displaystyle \frac{{\lambda }_{04}{S}_x^2}{{\lambda }_{04}{S}_x^2+C_x}}$$

and

$$t_2={\displaystyle \frac{C_x{S}_x^2}{C_x{S}_x^2+{\lambda }_{04}}}.$$

4. Proposed Family of Estimators

In this section, a new class of estimators is obtained by using the known smallest and largest observations of the auxiliary variable in a two-phase sampling scheme. The recommended estimator is defined below:

$${\widehat{S}}_{du}^2=\left[{\omega }_1{s}_y^2{\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)}^{{\gamma }_1}+{\omega }_2{\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)}^{{\gamma }_2}\right]exp\left({\displaystyle \frac{v_1({\acute{s}}_x^2-s_x^2)}{v_1({\acute{s}}_x^2+s_x^2)+2v_2}}\right),$$

(11)

where the scalar quantities (${\gamma }_1,{\gamma }_2$) are selected from the set $(0,-1,1)$, while $\left({\omega }_i,i=1,2\right)$ are unknown constants that must be determined to minimize the bias and mean squared errors. The auxiliary variable X has its maximum and minimum values represented by $X_{max}$ and $X_{min},$ respectively. From Equation (11), we derive different estimators, which are outlined in

Table 1. Some classes of the proposed estimator.

Subsets of the Proposed Estimator ${\widehat{\mathit{S}}}_{\mathbf{du}}^2$	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{v}}_1$	${\mathit{v}}_2$
${\widehat{S}}_{d{u}_1}^2=\left[{\omega }_1{s}_y^2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)+{\omega }_2\left({\displaystyle \frac{s_x^2}{{\acute{s}}_x^2}}\right)\right]H$	1	$-1$	${\rho }_{yx}$	$X_{max}-X_{min}$
${\widehat{S}}_{d{u}_2}^2=\left[{\omega }_1{s}_y^2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)+{\omega }_2\right]H$	1	0	$X_{max}-X_{min}$	$C_x$
${\widehat{S}}_{d{u}_3}^2=\left[{\omega }_1{s}_y^2\left({\displaystyle \frac{s_x^2}{{\acute{s}}_x^2}}\right)+{\omega }_2\left({\displaystyle \frac{s_x^2}{{\acute{s}}_x^2}}\right)\right]H$	$-1$	$-1$	$X_{max}-X_{min}$	1
${\widehat{S}}_{d{u}_4}^2=\left[{\omega }_1{s}_y^2+{\omega }_2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)\right]H$	0	1	$X_{max}-X_{min}$	${\beta }_{2\left(x\right)}$
${\widehat{S}}_{d{u}_5}^2=\left[{\omega }_1{s}_y^2+{\omega }_2\left({\displaystyle \frac{s_x^2}{{\acute{s}}_x^2}}\right)\right]H$	0	$-1$	${\beta }_{2\left(x\right)}$	$X_{max}-X_{min}$
${\widehat{S}}_{d{u}_6}^2=\left[{\omega }_1{s}_y^2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)+{\omega }_2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)\right]H$	1	1	$X_{max}-X_{min}$	${\rho }_{yx}$
${\widehat{S}}_{d{u}_7}^2=\left[{\omega }_1{s}_y^2\left({\displaystyle \frac{s_x^2}{{\acute{s}}_x^2}}\right)+{\omega }_2\left({\displaystyle \frac{{\acute{s}}_x^2}{s_x^2}}\right)\right]H$	$-1$	1	1	$X_{max}-X_{min}$
${\widehat{S}}_{d{u}_8}^2=\left[{\omega }_1{s}_y^2\left({\displaystyle \frac{s_x^2}{{\acute{s}}_x^2}}\right)+{\omega }_2\right]H$	$-1$	0	$C_x$	$X_{max}-X_{min}$

.

where

$$H=exp\left({\displaystyle \frac{v_1({\acute{s}}_x^2-s_x^2)}{v_1({\acute{s}}_x^2+s_x^2)+2v_2}}\right).$$

The proposed estimator is derived under a two-phase sampling design, where the maximum and minimum values of the auxiliary variable are utilized to enhance estimation accuracy. This method is shown to outperform traditional estimators in terms of bias and mean squared error, especially when dealing with extreme values. The following theorems provide the minimum bias and mean squared error of the proposed estimator ${\widehat{S}}_{du}^2.$

Theorem 1 

(Bias and MSE of the proposed estimator). Let ${\widehat{S}}_{du}^2$ be a proposed estimator of the finite population variance $S_y^2$ in a two-phase sampling scheme. The bias and mean squared error (MSE) of ${\widehat{S}}_{du}^2$ are given below:

$$Bias\left({\widehat{S}}_{du}^2\right)\cong \left[-S_y^2+{\omega }_1{S}_y^2{\psi }_3+{\omega }_2{\psi }_5\right]$$

and

$$MSE\left({\widehat{S}}_{du}^2\right)\cong \left[S_y^4+{\omega }_1^2{S}_y^4{\psi }_1+{\omega }_2^2{\psi }_2-2{\omega }_1{S}_y^4{\psi }_3-2{\omega }_2{S}_y^2{\psi }_5+2{\omega }_1{\omega }_2{S}_y^2{\psi }_4\right],$$

where ${\psi }_1,{\psi }_2,{\psi }_3,{\psi }_4,{\psi }_5$ are the functions of the auxiliary variables, sampling parameters, and transformation constants.

Proof. 

We know that

$$E\left(e_0\right)=E\left(e_1\right)=E\left(e_2\right)=0$$

$$E\left(e_0^2\right)=\theta {\lambda }_{40}^{*},E\left(e_1^2\right)=\theta {\lambda }_{04}^{*},E\left(e_2^2\right)={\theta }^{\prime }{\lambda }_{04}^{*},$$

$$E\left(e_0{e}_1\right)=\theta {\lambda }_{22}^{*},E\left(e_0{e}_2\right)={\theta }^{\prime }{\lambda }_{22}^{*},E\left(e_1{e}_2\right)={\theta }^{\prime }{\lambda }_{04}^{*}$$

where

$${\lambda }_{40}^{*}=({\lambda }_{40}-1),{\lambda }_{04}^{*}=({\lambda }_{04}-1),{\lambda }_{22}^{*}=({\lambda }_{22}-1).$$

To analyze the characteristics of the proposed estimator, we further simplify the Equation (11) in the form of relative errors to compute the bias and mean squared error (MSE) of ${\widehat{S}}_{du}^2$, i.e.,

$${\widehat{S}}_{du}^2=\left[{\omega }_1{S}_y^2\left(1+e_0\right){\left(1+e_1\right)}^{-{\gamma }_1}{\left(1+e_2\right)}^{{\gamma }_1}+{\omega }_2{\left(1+e_1\right)}^{-{\gamma }_2}{\left(1+e_2\right)}^{{\gamma }_2}\right]exp\left[{\displaystyle \frac{t_3\left(e_2-e_1\right)}{2}}{\left(1+{\displaystyle \frac{t_3}{2}}\left(e_2+e_1\right)\right)}^{-1}\right],$$

(12)

where

$$t_3={\displaystyle \frac{v_1{S}_x^2}{v_1{S}_x^2+v_2}}.$$

Expanding the right-hand sides of (12) and applying the Taylor series expansion up to the first order, while neglecting terms where $e_i>2$, we obtain

$$\begin{array}{c}{\widehat{S}}_{du}^2-S_y^2\cong -S_y^2+{\omega }_1{S}_y^2\left[1+e_0-e_1\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)+e_2\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)+e_1^2\left({\displaystyle \frac{{\gamma }_1{t}_3}{2}}+{\displaystyle \frac{3t_3^2}{8}}+{\displaystyle \frac{{\gamma }_1\left({\gamma }_1+1\right)}{2}}\right)\right.\hfill \\ \left.+e_2^2\left({\displaystyle \frac{{\gamma }_1{t}_3}{2}}-{\displaystyle \frac{t_3^2}{8}}+{\displaystyle \frac{{\gamma }_1\left({\gamma }_1-1\right)}{2}}\right)-e_0{e}_1\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)-e_0{e}_2\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)-e_1{e}_2{\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)}^2\right]\hfill \\ +{\omega }_2\left[1-e_1\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)+e_2\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)+e_1^2\left({\displaystyle \frac{{\gamma }_2{t}_3}{2}}+{\displaystyle \frac{3t_3^2}{8}}+{\displaystyle \frac{{\gamma }_2\left({\gamma }_2+1\right)}{2}}\right)\right.\hfill \\ \left.+e_2^2\left({\displaystyle \frac{{\gamma }_2{t}_3}{2}}-{\displaystyle \frac{t_3^2}{8}}+{\displaystyle \frac{{\gamma }_2\left({\gamma }_2-1\right)}{2}}\right)\left.-e_1{e}_2{\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)}^2\right]\right].\hfill \end{array}$$

(13)

Applying the expectation on both sides of Equation (13) and substituting the expected values of ($e_0,e_1,e_2,e_0^2,e_1^2,e_2^2,e_0{e}_1,e_0{e}_2,e_1{e}_2$), the bias of ${\widehat{S}}_{du}^2$ is obtained by

$$Bias\left({\widehat{S}}_{du}^2\right)\cong \left[-S_y^2+{\omega }_1{S}_y^2{\psi }_3+{\omega }_2{\psi }_5\right],$$

(14)

where

$$\begin{array}{c}{\psi }_3=\left[1+\theta \left\{{\lambda }_{04}^{*}\left({\displaystyle \frac{4{\gamma }_1\left({\gamma }_1+1+t_3\right)+3t_3^2}{8}}\right)-\left.{\lambda }_{22}^{*}\left({\displaystyle \frac{2{\gamma }_1+t_3}{2}}\right)\right\}\right.\right.\hfill \\ +{\theta }^{\prime }\left\{{\lambda }_{04}^{*}\left\{{\displaystyle \frac{-4{\gamma }_1\left({\gamma }_1+t_3+1\right)-3t_3^2}{2}}\right\}\right.+\left.\left.{\lambda }_{22}^{*}\left({\displaystyle \frac{2{\gamma }_1+t_3}{2}}\right)\right\}\right]\hfill \end{array}$$

and

$${\psi }_5=\left[1+\theta {\lambda }_{04}^{*}\left({\displaystyle \frac{4{\gamma }_2\left({\gamma }_2+t_3+1\right)+3t_3^2}{8}}\right)+{\theta }^{\prime }{\lambda }_{04}^{*}\left({\displaystyle \frac{-2{\gamma }_2\left({\gamma }_2+t_3-1\right)-3t_3^2}{4}}\right)\right].$$

We derived an MSE of ${\widehat{S}}_{du}^2$ by squaring both sides of (13) and applying the expectation, which is expressed by the following equation, i.e.,

$$MSE\left({\widehat{S}}_{du}^2\right)\cong \left[S_y^4+{\omega }_1^2{S}_y^4{\psi }_1+{\omega }_2^2{\psi }_2-2{\omega }_1{S}_y^4{\psi }_3-2{\omega }_2{S}_y^2{\psi }_5+2{\omega }_1{\omega }_2{S}_y^2{\psi }_4\right],$$

(15)

where

$$\begin{array}{c}{\psi }_1=\left[1+\theta \left\{{\lambda }_{40}^{*}+{\lambda }_{04}^{*}\left\{{\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)}^2+\left.\left({\gamma }_1{t}_3+{\displaystyle \frac{3t_3^2}{4}}+{\displaystyle \frac{{\gamma }_1\left({\gamma }_1+1\right)}{2}}\right)\right\}\right.\right.\right.-\left.4{\lambda }_{22}^{*}\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)\right\}\hfill \\ +{\theta }^{\prime }\left\{{\lambda }_{04}^{*}\left\{{\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)}^2\right.\right.+\left({\gamma }_1{t}_3-{\displaystyle \frac{t_3^2}{4}}+{\gamma }_1\left({\gamma }_1-1\right)\right)\left.-4\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)\right\}+\left.\left.4{\lambda }_{22}^{*}\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)\right\}\right],\hfill \end{array}$$

$$\begin{array}{c}{\psi }_2=\left[1+\theta {\lambda }_{04}^{*}\left\{{\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)}^2+\left.\left({\gamma }_2{t}_3+{\displaystyle \frac{3t_3^2}{4}}+{\gamma }_2\left({\gamma }_2+1\right)\right)\right\}\right.\right.+{\theta }^{\prime }{\lambda }_{04}^{*}\left\{{\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)}^2\right.\hfill \\ \left.\left.+\left({\gamma }_2{t}_3-{\displaystyle \frac{t_3^2}{4}}+{\gamma }_2\left({\gamma }_2-1\right)\right)-4{\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)}^2\right\}\right]\hfill \end{array}$$

and

$$\begin{array}{c}{\psi }_4=\left[1+\theta \left\{{\lambda }_{04}^{*}\left\{\left({\displaystyle \frac{{\gamma }_1{t}_3}{2}}+{\displaystyle \frac{3t_3^2}{8}}+{\displaystyle \frac{{\gamma }_1\left({\gamma }_1+1\right)}{2}}\right)+\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)+\left({\displaystyle \frac{{\gamma }_2{t}_3}{2}}+{\displaystyle \frac{3t_3^2}{8}}+{\displaystyle \frac{{\gamma }_2\left({\gamma }_2+1\right)}{2}}\right)\right.\right.\right.\hfill \\ -\left.{\lambda }_{22}^{*}\left({\gamma }_1+{\gamma }_2+t_3\right)\right\}+{\theta }^{\prime }\left\{{\lambda }_{04}^{*}\left\{\left({\displaystyle \frac{{\gamma }_1{t}_3}{2}}-{\displaystyle \frac{t_3^2}{8}}+{\displaystyle \frac{{\gamma }_1\left({\gamma }_1-1\right)}{2}}\right)-\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)\right.\right.\hfill \\ +\left({\displaystyle \frac{{\gamma }_2{t}_3}{2}}-{\displaystyle \frac{t_3^2}{8}}+{\displaystyle \frac{{\gamma }_2\left({\gamma }_2-1\right)}{2}}\right)\left.-{\left({\gamma }_1+{\displaystyle \frac{t_3}{2}}\right)}^2-{\left({\gamma }_2+{\displaystyle \frac{t_3}{2}}\right)}^2\right\}\left.\left.+{\lambda }_{22}^{*}({\gamma }_1+{\gamma }_2+t_3)\right\}\right].\hfill \end{array}$$

□

Theorem 2 

(Minimum bias and MSE of the proposed estimator). Let ${\widehat{S}}_{du}^2$ be a proposed estimator of the finite population variance $S_y^2$ in a two-phase sampling framework. The bias and mean squared error (MSE) of ${\widehat{S}}_{du}^2$ are minimized and can be expressed as follows:

$$\mathit{Bias}{\left({\widehat{S}}_{du}^2\right)}_{\mathit{min}}=-S_y^2\left[1-{\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right]$$

and

$$\mathit{MSE}{\left({\widehat{S}}_{du}^2\right)}_{\mathit{min}}=S_y^4\left[1-{\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right]$$

Proof. 

We minimize Equation (15) to find the optimal values for ${\omega }_1$ and ${\omega }_2$, which reduce the mean squared error and improve the accuracy of the proposed class of estimators, which can be expressed as follows

$${\omega }_{1\left(opt\right)}={\displaystyle \frac{{\psi }_2{\psi }_3-{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}$$

$${\omega }_{2\left(opt\right)}={\displaystyle \frac{S_y^2\left({\psi }_1{\psi }_5-{\psi }_3{\psi }_4\right)}{{\psi }_1{\psi }_2-{\psi }_4^2}}.$$

To achieve the minimum bias and MSE for ${\widehat{S}}_{du}^2,$ the optimum values of ${\omega }_1$ and ${\omega }_2$ are substituted into Equations (14) and (15), which are defined as follows

$$Bias{\left({\widehat{S}}_{du}^2\right)}_{min}\cong -S_y^2\left[1-{\displaystyle \frac{\left({\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5\right)}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right]$$

(16)

and

$$MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\cong S_y^4\left[1-{\displaystyle \frac{\left({\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5\right)}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right].$$

(17)

□

5. Mathematical Comparison

The suggested class of estimators ${\widehat{S}}_{du}^2$ is compared and analyzed with a number of other estimators in this section, including ${\widehat{S}}_y^2$, ${\widehat{S}}_R^2$, ${\widehat{S}}_{lr}^2$, ${\widehat{S}}_{bt}^2$, ${\widehat{S}}_{us}^2$, and ${\widehat{S}}_{c{k}_i}^2$.

Condition (i): A comparison between the estimators provided by Equations (1) and (17)

$$Var\left({\widehat{S}}_y^2\right)>MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\mathrm{if}$$

$$\theta {\lambda }_{40}^{*}+\left({\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right)>1.$$

Condition (ii): A comparison between the estimators provided by Equations (3) and (17)

$$MSE\left({\widehat{S}}_R^2\right)>MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\mathrm{if}$$

$$\left(\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }{\lambda }_{04}^{*}-2{\theta }^{\prime \prime }{\lambda }_{22}^{*}\right)+\left({\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right)>1.$$

Condition (iii): A comparison between the estimators provided by Equations (4) and (17)

$$MSE\left({\widehat{S}}_{lr}^2\right)>MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\mathrm{if}$$

$${\lambda }_{40}^{*}\left(\theta -{\theta }^{\prime \prime }{\rho }_{yx}^{*2}\right)+\left({\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right)>1.$$

Condition (iv): A comparison between the estimators provided by Equations (6) and (17)

$$MSE\left({\widehat{S}}_{bt}^2\right)>MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\mathrm{if}$$

$$\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }\left({\displaystyle \frac{{\lambda }_{04}^{*}}{4}}-{\lambda }_{22}^{*}\right)+\left({\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right)>1.$$

Condition (v): A comparison between the estimators provided by Equations (8) and (17)

$$MSE\left({\widehat{S}}_{us}^2\right)>MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\mathrm{if}$$

$$\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }\left(g^2{\lambda }_{04}^{*}-2g{\lambda }_{22}^{*}\right)+\left({\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right)>1.$$

Condition (vi): A comparison between the estimators provided by Equations (10) and (17)

$$MSE\left({\widehat{S}}_{c{k}_i}^2\right)>MSE{\left({\widehat{S}}_{du}^2\right)}_{min}\mathrm{if}$$

$$\theta {\lambda }_{40}^{*}+{\theta }^{\prime \prime }\left(t_i^2{\lambda }_{04}^{*}-2t_i{\lambda }_{22}^{*}\right)+\left({\displaystyle \frac{{\psi }_1{\psi }_5^2+{\psi }_2{\psi }_3^2-2{\psi }_3{\psi }_4{\psi }_5}{{\psi }_1{\psi }_2-{\psi }_4^2}}\right)>1.$$

6. Results and Discussion

This section evaluates the effectiveness of the proposed class of estimators by comparing their percent relative efficiency (PRE) against other estimators by using simulated and actual datasets.

6.1. Simulation Study

In this section, we perform a simulation analysis to assess the efficiency of the suggested estimators, which rely on the known maximum and minimum values of the auxiliary variable in a two-phase sampling setup. The following probability distributions are used to artificially generate the auxiliary variable X over six different populations:

• Simulated data 1: $X\sim Gamma(a_1=2,a_2=6)$
• Simulated data 2: $X\sim Gamma(a_1=6,a_2=10)$
• Simulated data 3: $X\sim Exponential(b_1=3)$
• Simulated data 4: $X\sim Exponential(b_1=7)$
• Simulated data 5: $X\sim Uniform(b_2=0,b_3=1)$
• Simulated data 6: $X\sim Uniform(b_2=2,b_3=4)$

In each distribution, the dependent variable Y is obtained by employing the following formula

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

where X is selected from each population distribution,

$$r_{yx}=0.80$$

is a correlation coefficient, and

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

denotes the random error term.

We used the following approaches in R-Software to calculate the PREs values of the proposed enhanced family of estimators and other existing methods:

• Step 1: Generate a population of size 2000 using specified probability distributions.
• Step 2: Use simple random sampling without replacement (SRSWOR) to choose a first-phase sample of size m from the population of size $N.$
• Step 3: Draw a second-phase sample of size n from the first-phase sample, again using SRSWOR.
• Step 4: From the above steps, obtain the population total and the largest and smallest observations of the auxiliary variable. Furthermore, we determine the values of the unknown constants in the proposed estimators.
• Step 5: Apply a two-phase sampling design with the first and second-phase as described in Step 2 and Step 3.
• Step 6: Compute the PRE values for all of the estimators discussed in this paper.
• Step 7: Repeat Step 5 and Step 6 a total of 70,000 times.

Finally, use the formulas given below to determine the MSEs and PREs for each estimator over all replications:

$$MSE{\left({\widehat{S}}_i^2\right)}_{min}=\sum _{j=1}^{70000}{\displaystyle \frac{{\left({\widehat{S}}_i^2-S_y^2\right)}^2}{70000}}$$

and

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

where i is one of $R,lr,bt,us,c{k}_1,c{k}_2,c{k}_3,d{u}_k(k=e,1,2,\dots ,8).$ The results for simulated populations are shown in

Table 2. PRE values using different artificial populations.

Estimator	Pop-I	Pop-II	Pop-III	Pop-IV	Pop-V	Pop-VI
${\widehat{S}}_y^2$	100	100	100	100	100	100
${\widehat{S}}_R^2$	146.523	129.587	133.126	128.545	140.545	127.760
${\widehat{S}}_{lr}^2$	155.645	143.155	155.302	142.315	142.765	128.505
${\widehat{S}}_{tb}^2$	162.448	147.505	148.140	144.200	141.404	126.454
${\widehat{S}}_{su}^2$	163.289	146.905	152.409	146.526	149.078	129.425
${\widehat{S}}_{c{k}_1}^2$	165.526	155.858	156.620	148.789	144.589	122.408
${\widehat{S}}_{c{k}_2}^2$	165.552	155.058	155.556	148.002	143.164	120.528
${\widehat{S}}_{c{k}_3}^2$	164.203	154.889	153.427	149.328	141.664	126.548
${\widehat{S}}_{d{u}_1}^2$	473.179	573.891	335.950	412.369	595.289	1101.333
${\widehat{S}}_{d{u}_2}^2$	1489.770	962.369	1383.356	1510.508	1891.345	1935.495
${\widehat{S}}_{d{u}_3}^2$	260.387	560.978	623.059	785.625	661.148	1120.132
${\widehat{S}}_{d{u}_4}^2$	405.654	505.576	423.498	589.353	610.279	810.369
${\widehat{S}}_{d{u}_5}^2$	319.560	422.616	270.104	300.323	525.399	995.712
${\widehat{S}}_{d{u}_6}^2$	330.228	230.448	210.102	327.688	339.950	568.259
${\widehat{S}}_{d{u}_7}^2$	783.921	883.149	543.125	583.167	889.329	1159.949
${\widehat{S}}_{d{u}_8}^2$	2084.370	1382.562	2148.014	2289.910	2410.460	2309.720

.

6.2. Numerical Examples

The objective is to evaluate the performance of the proposed family of estimators compared with existing methods. We compared their percentage relative efficiency values using three real-world datasets to highlight differences in performance. The choice to use three real-life datasets was because of their unique characteristics and significance to the study’s objectives. These datasets highlight the effectiveness and flexibility of the proposed class of estimators in various practical scenarios. Data 1, which includes departmental employment levels and the number of registered factories, illustrates how auxiliary information improves variance estimation. Data 2, which represents food expenses and weekly income, demonstrates the importance of improved estimator efficiency in economic data. Data 3, detailing student enrollment and school data, demonstrates the applicability of the proposed method in educational research. This diverse selection ensures comprehensive validation of the estimators across multiple fields, confirming their robustness and reliability.

Data 1. (Source: [34], p. 226)

Y: Employment levels in the departments for the year 2012.

X: Refers to the number of factories that were officially registered by the departments in 2012.

The descriptive statistics are:

$$\begin{array}{c}N=36,m=15,n=6,X_m=39,X_M=2379,\overline{Y}=4015.22,\overline{X}=1659.58,S_y=8512.266,\hfill \\ S_x=3410.428,C_x=2.06,C_y=2.12,{\rho }_{yx}=0.65,{\lambda }_{40}=2051,{\lambda }_{04}=2237,{\lambda }_{22}=1542,\hfill \\ \theta =0.14,{\theta }^{\prime }=0.04,{\theta }^{\prime \prime }=0.10.\hfill \end{array}$$

Data 2. (Source: [3], p. 24)

Y: Denotes the family’s food expenditure, which is connected to their working status and shows how food prices can change depending on their work environment,

X: Represents the family’s weekly income, which is an important reflection of their financial situation.

The summary and descriptive statistics are as follows:

$$\begin{array}{c}N=33,m=15,n=6,X_m=58,X_M=95,\overline{Y}=27.49,\overline{X}=72.55,S_y=10.13,S_x=10.58,\hfill \\ C_x=0.14,C_y=0.37,{\rho }_{yx}=0.25,{\lambda }_{40}=5.55,{\lambda }_{04}=2.08,{\lambda }_{22}=2.22,\theta =0.14,{\theta }^{\prime }=0.04,\hfill \\ {\theta }^{\prime \prime }=0.10.\hfill \end{array}$$

Data 3. (Source: [34], p. 135)

Y: Represents the total students enrolled in institutions overall in 2012. It records the total number of students enrolled at all schools and universities.

X: Represents the total number of public and private educational institutions in 2012.

The summary and descriptive statistics are given below:

$$\begin{array}{c}N=36,m=15,n=6,X_m=39,X_M=2370,\overline{Y}=14818.70,\overline{X}=1054.39,S_y=32601.14,\hfill \\ S_x=2214.219,C_x=2.10,C_y=22.07,{\rho }_{yx}=0.69,{\lambda }_{40}=236.50,{\lambda }_{04}=209.80,{\lambda }_{22}=397.50,\hfill \\ \theta =0.14,{\theta }^{\prime }=0.04,{\theta }^{\prime \prime }=0.10.\hfill \end{array}$$

Finally, we applied the formula given below to calculate the percent relative efficiency (PRE) across various datasets:

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

where K is one of $R,lr,bt,us,c{k}_1,c{k}_2,c{k}_3,d{u}_i(i=e,1,2,\dots ,8).$ The results obtained from real-life populations are shown in

Table 3. PRE values using real datasets.

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
${\widehat{S}}_{d{u}_1}^2$	435.023	270.345	342.290
${\widehat{S}}_{d{u}_2}^2$	202.327	179.708	252.155
${\widehat{S}}_{d{u}_3}^2$	182.114	160.539	190.650
${\widehat{S}}_{d{u}_4}^2$	254.709	192.105	228.598
${\widehat{S}}_{d{u}_5}^2$	228.806	181.103	262.206
${\widehat{S}}_{d{u}_6}^2$	336.560	216.005	293.180
${\widehat{S}}_{d{u}_7}^2$	376.826	294.409	310.216
${\widehat{S}}_{d{u}_8}^2$	1230.068	1080.120	1435.163

.

6.3. Discussion

To determine the usefulness of the suggested class of estimators, we performed simulations and examined three actual datasets. We used the percent relative efficiency criterion to compare different estimators. The simulation results are summarized in Table 2, which provides the percent relative efficiency values for the recommended and existing methods. The results from the real-life datasets are shown in Table 3. Our studies led us to the following general findings:

• As shown in Table 2 and Table 3, the new proposed class of estimators continuously shows higher PRE values compared with existing estimators in all simulated situations and real datasets. This highlights the enhanced performance of the proposed class of estimators over the existing methods.
• The proposed estimators demonstrate a remarkable ability to maintain stability when applied to datasets that contain extreme values or outliers. Outliers can significantly affect traditional variance estimators, but the use of auxiliary variables, specifically the maximum and minimum values, ensures the proposed estimators retain their accuracy and reliability across a wide range of data distributions, as demonstrated by both simulations and real-world data applications.
• The proposed class of estimators consistently shows higher PREs than existing estimators, as we can see in Figure 1 and Figure 2 across both simulation studies and real datasets. Therefore, the new classes of estimators appear to outperform the existing ones, indicating a direct relationship between their PRE values and enhanced performance.
• These figures show that the proposed estimators perform consistently better than existing methods across different data types, including Gamma, Exponential, and Uniform distributions. This versatility demonstrates the flexibility of the proposed approach, which adapts well to different underlying population structures, making it a useful tool for practical applications in fields like public health, economics, and education.

7. Conclusions

In this paper, we proposed an enhanced and efficient family of estimators based on the minimum and maximum values of the auxiliary variable for estimating the finite population variance. The first degree of approximation was used to obtain the biases, mean squared errors (MSEs), and minimum MSE of the new family of estimators. We conducted an extensive simulation study using different artificial populations, including Gamma, Exponential, and Uniform distributions, to evaluate the performance of the proposed estimators. In addition, three real datasets were analyzed to validate the practical applicability of the proposed method. The results, summarized in Table 2 and Table 3, clearly demonstrate that the proposed estimators consistently outperform traditional methods, particularly in terms of percent relative efficiency (PRE), across all simulated scenarios. Furthermore, the performance of the new estimators is visually represented in Figure 1 and Figure 2, which show a significant improvement in estimator efficiency when applied to both generated data (from various distributions) and real-world datasets. These findings highlight the robustness and effectiveness of the proposed estimators in handling different types of data, whether generated or observed in real-world applications.

Based on the simulation and empirical results, we can say that the suggested class of estimators ${\widehat{S}}_{d{u}_i}^2$$(i=e,1,2,3,\cdots ,8)$ is more efficient than the other estimators considered; ${\widehat{S}}_{d{u}_8}^2$ is especially useful because of its higher percent relative efficiency value.

However, we adopted a two-phase sampling methodology to investigate the properties of the proposed class of estimators. Furthermore, future research could explore incorporating robust measures of finite population variance estimation, particularly in the presence of outliers. Techniques such as the interquartile range (IQR), median absolute deviation (MAD), and biweight midvariance offer promising alternatives to traditional variance estimators. Additionally, Rousseeuw and Croux’s (1993) robust MAD variant could be considered for enhancing estimator stability. These methods may improve estimator precision by reducing sensitivity to extreme values. Future work should focus on integrating these robust techniques into existing sampling designs, assessing their theoretical efficiency, and comparing them with conventional methods to ensure more reliable population parameter estimates.