Double Exponential Ratio Estimator of a Finite Population Variance under Extreme Values in Simple Random Sampling

Abstract

This article presents an improved class of efficient estimators aimed at estimating the finite population variance of the study variable. These estimators are especially useful when we have information about the minimum/maximum values of the auxiliary variable within a framework of simple random sampling. The characteristics of the proposed class of estimators, including bias and mean squared error ($MSE$) under simple random sampling are derived through a first-order approximation. To assess the performance and validate the theoretical outcomes, we conduct a simulation study. Results indicate that the proposed class of estimators has lower $MSEs$ as compared to other existing estimators across all simulation scenarios. Three datasets are used in the application section to emphasize the effectiveness of the proposed class of estimators over conventional unbiased variance estimators, ratio and regression estimators, and other existing estimators.

1. Introduction

The goal of survey sampling is to collect precise information on the characteristics of the population in order to maximize the effectiveness of the estimators under investigation while reducing expenses, time, and human efforts. There are a few extreme values in many populations, and it can be quite sensitive to estimate unknown population characteristics without taking these data into account. In some cases, the results can be inflated or underestimated. It is important to note that when dealing with extreme values in the dataset, the effectiveness of classical estimators tends to decrease in terms of mean square error ($MSE$). There may be a temptation to remove such data from the sample. To effectively confront this challenge, it is essential to incorporate this information into the process of estimating the population characteristics. By performing a linear transformation on the known minimum and maximum values of the auxiliary variable, ref. [1] provided two estimators. Such designs were not studied further after that, until the works of [2]. They employed the idea of using extreme values on various estimators of the finite population mean. Under the extreme values, ref. [3] improved the estimation of the finite population mean using a stratified random sampling strategy. For more details, see [4,5,6,7] and references therein.

It is difficult to control variability in applications, and the estimation problem of finite population variance is a significant concern. Researchers encounter this issue in biological and agricultural studies, which makes the desired outcomes appear unpredictable. A careful approach to utilizing auxiliary information can improve the estimator’s accuracy. For estimating the finite population variance, a number of researchers have proposed different kinds of estimators, including [8,9,10,11,12,13,14,15,16,17,18,19,20,21].

In this article, the extreme values of the auxiliary variable are retained in the data and used as auxiliary information. As discussed by [10], we propose an improved class of estimators in this article for estimating the finite population variance utilizing the known information on the extreme values of the auxiliary variable under a simple random sampling scheme for further improvement.

The next sections of this article are as follows. Appendix A introduces the concepts and notations. Some existing estimators are described in Section 2. In Section 3, we provide an in-depth discussion of our proposed class of estimators. Section 4 provides the mathematical comparison. Section 5 presents a simulation study to generate six different artificial populations by using different probability distributions to verify theoretical results discussed in Section 4. Some numerical examples are also listed in this section to illustrate our theoretical results. Finally, some conclusions and ideas for further research are discussed in Section 6.

2. Existing Estimators

In this section, we consider the existing estimators of the finite population variances and compare them with our proposed class of estimators.

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

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

(1)

For population variance ${\widehat{S}}_r^2$, [12] proposed a ratio estimator, which is given by:

$${\widehat{S}}_r^2=s_y^2\left(\frac{S_x^2}{s_x^2}\right).$$

(2)

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

$$Bias\left({\widehat{S}}_r^2\right)\cong \theta S_y^4\left({\lambda }_{04}^{\ast }-{\lambda }_{22}^{\ast }\right)$$

(3)

and

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

(4)

The linear regression estimator ${\widehat{S}}_{lr}^2,$ proposed by [22], is defined as:

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

(5)

where $b_{(s_y^2,s_x^2)}=\frac{s_y^2{\widehat{\lambda }}_{22}^{\ast }}{s_x^2{\widehat{\lambda }}_{04}^{\ast }}$ is the sample regression coefficient.

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

$$MSE\left({\widehat{S}}_{lr}^2\right)\cong \theta S_y^4{\lambda }_{40}^{\ast }\left(1-{\rho }^{\ast 2}\right),$$

(6)

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

For population variance under simple random sampling, [9] proposed an exponential ratio type estimator ${\widehat{S}}_{bt}^2$, which is defined as:

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

(7)

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

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

(8)

and

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

(9)

Using the kurtosis of an auxiliary variable in simple random sampling, ref. [18] suggested a ratio-type estimator ${\widehat{S}}_{us}^2$, which is defined as:

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

(10)

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

$$Bias\left({\widehat{S}}_{us}^2\right)\cong \theta S_y^2{g}_0\left(g_0{\lambda }_{04}^{\ast }-{\lambda }_{22}^{\ast }\right)$$

(11)

and

$$MSE\left({\widehat{S}}_{us}^2\right)\cong \theta S_y^4\left({\lambda }_{40}^{\ast }+g_0^2{\lambda }_{04}^{\ast }-2g_0{\lambda }_{22}^{\ast }\right),$$

(12)

where $g_0=\frac{S_x^2}{S_x^2+{\lambda }_{04}}$.

According to [13], some ratio estimators are defined as:

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

(13)

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

(14)

and

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

(15)

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

$$Bias\left({\widehat{S}}_{c{k}_i}^2\right)\cong \theta S_y^2{g}_i\left(g_i{\lambda }_{04}^{\ast }-{\lambda }_{22}^{\ast }\right)$$

(16)

and

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

(17)

where $g_1=\frac{S_x^2}{S_x^2+C_x},g_2=\frac{{\lambda }_{04}{S}_x^2}{{\lambda }_{04}{S}_x^2+C_x},g_3=\frac{C_x{S}_x^2}{C_x{S}_x^2+{\lambda }_{04}}$.

3. Proposed Estimator

In this section, motivated by [10], an improved class of estimators is introduced by utilizing the known minimum and maximum values of the auxiliary variable to estimate the finite population variance. The proposed estimators is defined as:

$${\widehat{S}}_{ed}^2=s_y^{2}exp\left[{\omega }_1\left\{\frac{\left(\overline{X}-\overline{x}\right)}{\left(\overline{X}+\overline{x}\right)+2a_1}\right\}\right]exp\left[{\omega }_2\left\{\frac{a_2\left(S_x^2-s_x^2\right)}{a_2\left(S_x^2+s_x^2\right)+2a_3}\right\}\right],$$

(18)

where $\left({\omega }_i,i=1,2\right)$ represent known constant values, whereas the auxiliary variables’ parameters are $a_1=X_M-X_m,$ $a_2,$ and $a_3$. We derive the various classes of the suggested estimator from (18), which are listed in

Table 1. Some classes of the proposed estimator.

Subsets of the proposed estimator ${\widehat{S}}_{ed}^2$	$a_2$	$a_3$
${\widehat{S}}_{e{d}_1}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{c_x\left(S_x^2-s_x^2\right)}{c_x\left(S_x^2+s_x^2\right)+2(x_M-x_m)}\right\}\right]$	$c_x$	$x_M-x_m$
${\widehat{S}}_{e{d}_2}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{-c_x\left(S_x^2-s_x^2\right)}{-c_x\left(S_x^2+s_x^2\right)+2(x_M-x_m)}\right\}\right]$	1	$x_M-x_m$
${\widehat{S}}_{e{d}_3}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{(x_M-x_m)\left(S_x^2-s_x^2\right)}{(x_M-x_m)\left(S_x^2+s_x^2\right)+2c_x}\right\}\right]$	$x_M-x_m$	$c_x$
${\widehat{S}}_{e{d}_4}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{(x_M-x_m)\left(S_x^2-s_x^2\right)}{(x_M-x_m)\left(S_x^2+s_x^2\right)-2c_x}\right\}\right]$	$x_M-x_m$	$-c_x$
${\widehat{S}}_{e{d}_5}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{(x_M-x_m)\left(S_x^2-s_x^2\right)}{(x_M-x_m)\left(S_x^2+s_x^2\right)+2{\beta }_{2\left(x\right)}}\right\}\right]$	$x_M-x_m$	${\beta }_{2\left(x\right)}$
${\widehat{S}}_{e{d}_6}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{{\beta }_{2\left(x\right)}\left(S_x^2-s_x^2\right)}{{\beta }_{2\left(x\right)}\left(S_x^2+s_x^2\right)+2(x_M-x_m)}\right\}\right]$	${\beta }_{2\left(x\right)}$	$x_M-x_m$
${\widehat{S}}_{e{d}_7}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{(x_M-x_m)\left(S_x^2-s_x^2\right)}{(x_M-x_m)\left(S_x^2+s_x^2\right)-2{\beta }_{2\left(x\right)}}\right\}\right]$	$x_M-x_m$	$-{\beta }_{2\left(x\right)}$
${\widehat{S}}_{e{d}_8}^2=s_y^{2}exp\left[{\omega }_1{L}_1\right]exp\left[{\omega }_2\left\{\frac{-{\beta }_{2\left(x\right)}\left(S_x^2-s_x^2\right)}{-{\beta }_{2\left(x\right)}\left(S_x^2+s_x^2\right)+2(x_M-x_m)}\right\}\right]$	$-{\beta }_{2\left(x\right)}$	$x_M-x_m$

, where $L_1=\left(\frac{\left(\overline{X}-\overline{x}\right)}{\left(\overline{X}+\overline{x}\right)+2(x_M-x_m)}\right).$

Now, we rewrite (18) in terms of errors to get the bias and the $MSE$ of the suggested estimator ${\widehat{S}}_{ed}^2$, that is:

$$\begin{array}{c}{\widehat{S}}_{ed}^2=S_y^2\left(1+e_0\right)exp{\left[\frac{-{\omega }_1{g}_4{e}_1}{2}\left(1+\frac{g_4{e}_1}{2}\right)\right]}^{-1}exp{\left[\frac{-{\omega }_2{g}_5{e}_2}{2}\left(1+\frac{g_5{e}_2}{2}\right)\right]}^{-1}\hfill \end{array}$$

(19)

where $g_4=\frac{\overline{X}}{\overline{X}+a_1}$ and $g_5=\frac{a_2{S}_x^2}{a_2{S}_x^2+a_3}$.

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

$$\begin{array}{c}{\widehat{S}}_{ed}^2-S_y^2\cong S_y^2\left[e_0-\frac{{\omega }_1{g}_4}{2}{e}_1-\frac{{\omega }_2{g}_{g_5}}{2}{e}_2+\left(\frac{{\omega }_1{g}_4^2}{4}+\frac{{\omega }_1^2{g}_4^2}{8}\right)e_1^2+\left(\frac{{\omega }_2{g}_5^2}{4}+\frac{{\omega }_2^2{g}_5^2}{8}\right)e_2^2\right.\hfill \\ \left.-\frac{{\omega }_1{g}_4}{2}{e}_0{e}_1-\frac{{\omega }_2{g}_5}{2}{e}_0{e}_2+\frac{{\omega }_1{\omega }_2{g}_4{g}_5}{2}{e}_1{e}_2\right].\hfill \end{array}$$

(20)

Using (20), the bias of ${\widehat{S}}_{ed}^2$ is given by:

$$\begin{array}{c}Bias\left({\widehat{S}}_{ed}^2\right)\cong \theta S_y^2\left[\left(\frac{{\omega }_1{g}_4^2}{4}+\frac{{\omega }_1^2{g}_4^2}{8}\right)C_x^2+\left(\frac{{\omega }_2{g}_5^2}{4}+\frac{{\omega }_2^2{g}_5^2}{8}\right){\lambda }_{04}^{\ast }-\frac{{\omega }_1{g}_4}{2}{C}_x{\lambda }_{21}\right.\hfill \\ \left.-\frac{{\omega }_2{g}_5}{2}{\lambda }_{22}^{\ast }+\frac{{\omega }_1{\omega }_{2h}{g}_4{g}_5}{2}{C}_x{\lambda }_{03}\right].\hfill \end{array}$$

(21)

By squaring both sides of (20) and taking the expected value, we obtained a first-order approximate $MSE$, which is given by the following equation:

$$\begin{array}{c}MSE\left({\widehat{S}}_{ed}^2\right)\cong \theta S_y^4\left[{\lambda }_{40}^{\ast }+\frac{{\omega }_1^2{g}_4^2}{4}{C}_x^2+\frac{{\omega }_2^2{g}_5^2}{4}{\lambda }_{04}^{\ast }-{\omega }_1{g}_4{C}_x{\lambda }_{21}-{\omega }_2{g}_5{\lambda }_{22}^{\ast }+\frac{{\omega }_1{\omega }_2{g}_4{g}_5}{2}{C}_x{\lambda }_{03}\right].\hfill \end{array}$$

(22)

The bias and $MSE$ for ${\widehat{S}}_{ed}^2$, can be rewritten by substituting the known constant values of $({\omega }_1={\omega }_2=1)$ into (21) and (22), and after the simple simplifications, we obtain:

$$\begin{array}{c}Bias\left({\widehat{S}}_{ed}^2\right)\cong \theta S_y^2\left[\frac{3}{8}\left(g_4^2{C}_x^2+g_5^2{\lambda }_{04}^{\ast }\right)-\frac{1}{2}\left(g_4{C}_x{\lambda }_{21}+g_5{\lambda }_{22}^{\ast }-g_4{g}_5{C}_x{\lambda }_{03}\right)\right]\hfill \end{array}$$

(23)

and

$$\begin{array}{c}MSE\left({\widehat{S}}_{ed}^2\right)\cong \theta S_y^4\left[{\lambda }_{40}^{\ast }+\frac{1}{4}\left(g_4^2{C}_x^2+g_5^2{\lambda }_{04}^{\ast }\right)-\frac{1}{2}\left(2g_4{C}_x{\lambda }_{21}+2g_5{\lambda }_{22}^{\ast }-g_4{g}_5{C}_x{\lambda }_{03}\right)\right].\hfill \end{array}$$

(24)

4. Mathematical Comparison

In this section, we discuss the comparisons between the proposed class of estimators ${\widehat{S}}_{ed}^2$, with other existing estimators ${\widehat{S}}_y^2,{\widehat{S}}_r^2,{\widehat{S}}_{lr}^2,{\widehat{S}}_{bt}^2,{\widehat{S}}_{us}^2$, and ${\widehat{S}}_{k{c}_i}^2$.

Condition (i): By (1) and (24):

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

$$\left(2g_4{C}_x{\lambda }_{21}+2g_5{\lambda }_{22}^{\ast }-g_4{g}_5{C}_x{\lambda }_{03}\right)>\frac{1}{2}\left(g_4^2{C}_x^2+g_5^2{\lambda }_{04}^{\ast }\right).$$

Condition (ii): By (4) and (24):

$$MSE\left({\widehat{S}}_r^2\right)>MSE\left({\widehat{S}}_{ed}^2\right) \mathrm{if}$$

$$\left[2g_4{C}_x{\lambda }_{21}+2{\lambda }_{22}^{\ast }\left(g_5-2\right)-g_4{g}_5{C}_x{\lambda }_{03}\right]>\frac{1}{2}\left[g_4^2{C}_x^2+{\lambda }_{04}^{\ast }(g_5^2-4)\right].$$

Condition (iii): By (6) and (24):

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

$$\left(2g_4{C}_x{\lambda }_{21}+2g_5{\lambda }_{22}^{\ast }-g_4{g}_5{C}_x{\lambda }_{03}\right)>\frac{1}{2}\left(4{\rho }_{yx}^{\ast 2}{\lambda }_{40}^{\ast }+g_4^2{C}_x^2+g_5^2{\lambda }_{04}^{\ast }\right).$$

Condition (iv): By (9) and (24):

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

$$\left[2g_4{C}_x{\lambda }_{21}+2{\lambda }_{22}^{\ast }(g_5-1)-g_4{g}_5{\lambda }_{03}\right]>\frac{1}{2}\left[g_4^2{C}_x^2+{\lambda }_{04}^{\ast }(g_5^2-1)\right].$$

Condition (v): By (12) and (24):

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

$$\left[g_4{C}_x{\lambda }_{21}+{\lambda }_{22}^{\ast }(g_5-4g_0)-g_4{g}_5{C}_x{\lambda }_{03}\right]>\frac{1}{2}\left[g_4^2{C}_x^2+{\lambda }_{04}^{\ast }(g_5^2-4g_0^2)\right].$$

Condition (vi): By (17) and (24):

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

$$\left[g_4{C}_x{\lambda }_{21}+{\lambda }_{22}^{\ast }(g_5-4g_i)-g_4{g}_5{C}_x{\lambda }_{03}\right]>\frac{1}{2}\left[g_4^2{C}_x^2+{\lambda }_{04}^{\ast }(g_5^2-4g_i^2)\right].$$

5. Numerical Comparison

In this section, to examine the performances of the proposed class of estimators, we compare the MSEs of different estimators using one simulated and three real datasets.

5.1. Simulation Study

To validate the theoretical findings presented in Section 4, we employ the concept from [10] to carry out a simulation study. The auxiliary variable X can be artificially generated into six distinct populations using the following probability distributions:

• Population 1: $X\sim Uniform({\alpha }_1=0,{\alpha }_2=3)$;
• Population 2: $X\sim Uniform({\alpha }_1=4,{\alpha }_2=6)$;
• Population 3: $X\sim Gamma({\gamma }_1=5,{\gamma }_2=6)$;
• Population 4: $X\sim Gamma({\gamma }_1=8,{\gamma }_2=10)$;
• Population 5: $X\sim Exponential(\mu =2)$;
• Population 6: $X\sim Exponential(\mu =6).$

Subsequently, the variable of interest Y is calculated as:

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

where

$$r_{yx}=0.80$$

represents the correlation coefficient between the study and auxiliary variables, and $e\sim N(0,1)$ denotes the error term.

In R Software (latest v. 4.4.0), we considered into account the subsequent processes to determine the mean squared errors ($MSEs$) of the suggested estimator:

• Step 1: First, we use certain types of probability distributions to get a population of size 1500.
• Step 2: From Step 1, we get the population total, as well as the minimum and maximum values of the auxiliary variable.
• Step 3: We use simple random sampling without replacement (SRSWOR) to generate different sizes of sample for each population.
• Step 4: Determine the $MSE$ values of all the estimators covered in this article for each sample size.
• Step 5: Steps 3 and 4 are performed 60,000 times, and the findings for artificial populations are presented in

  Table 2. $MSE$ of different estimators using the artificial populations.

  Estimator	Pop-I	Pop-II	Pop-III	Pop-IV	Pop-V	Pop-VI
  [1] ${\widehat{S}}_y^2$	0.98 $\times {10}^{-3}$	8.70 $\times {10}^{-4}$	8.00 $\times {10}^{-4}$	1.30 $\times {10}^{-3}$	2.46 $\times {10}^{-2}$	1.05 $\times {10}^{-3}$
  [2] ${\widehat{S}}_r^2$	6.98 $\times {10}^{-4}$	7.80 $\times {10}^{-4}$	7.02 $\times {10}^{-4}$	1.10 $\times {10}^{-3}$	2.09 $\times {10}^{-2}$	1.00 $\times {10}^{-3}$
  [3] ${\widehat{S}}_{lr}^2$	6.98 $\times {10}^{-4}$	6.70 $\times {10}^{-4}$	5.99 $\times {10}^{-4}$	1.05 $\times {10}^{-3}$	2.06 $\times {10}^{-2}$	1.00 $\times {10}^{-3}$
  [4] ${\widehat{S}}_{bt}^2$	6.20 $\times {10}^{-4}$	6.40 $\times {10}^{-4}$	5.99 $\times {10}^{-4}$	1.05 $\times {10}^{-3}$	2.06 $\times {10}^{-2}$	1.00 $\times {10}^{-3}$
  [5] ${\widehat{S}}_{us}^2$	6.20 $\times {10}^{-4}$	6.50 $\times {10}^{-4}$	5.99 $\times {10}^{-4}$	1.05 $\times {10}^{-3}$	2.10 $\times {10}^{-2}$	1.00 $\times {10}^{-3}$
  [6] ${\widehat{S}}_{c{k}_1}^2$	6.20 $\times {10}^{-4}$	6.09 $\times {10}^{-4}$	5.99 $\times {10}^{-4}$	1.00 $\times {10}^{-3}$	2.02 $\times {10}^{-2}$	1.00 $\times {10}^{-3}$
  [7] ${\widehat{S}}_{c{k}_2}^2$	6.20 $\times {10}^{-4}$	6.09 $\times {10}^{-4}$	5.99 $\times {10}^{-4}$	1.05 $\times {10}^{-3}$	2.04 $\times {10}^{-2}$	1.00 $\times {10}^{-3}$
  [8] ${\widehat{S}}_{c{k}_3}^2$	6.20 $\times {10}^{-4}$	6.20 $\times {10}^{-4}$	5.99 $\times {10}^{-4}$	1.05 $\times {10}^{-3}$	2.07 $\times {10}^{-2}$	8.50 $\times {10}^{-4}$
  [9] ${\widehat{S}}_{e{d}_1}^2$	1.19 $\times {10}^{-4}$	1.20 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	2.50 $\times {10}^{-4}$	3.60 $\times {10}^{-3}$	1.00 $\times {10}^{-4}$
  [10] ${\widehat{S}}_{e{d}_2}^2$	0.98 $\times {10}^{-4}$	0.98 $\times {10}^{-4}$	0.98 $\times {10}^{-4}$	1.00 $\times {10}^{-4}$	1.20 $\times {10}^{-3}$	4.50 $\times {10}^{-4}$
  [11] ${\widehat{S}}_{e{d}_3}^2$	1.39 $\times {10}^{-4}$	1.39 $\times {10}^{-4}$	1.10 $\times {10}^{-4}$	1.40 $\times {10}^{-4}$	3.40 $\times {10}^{-3}$	1.00 $\times {10}^{-4}$
  [12] ${\widehat{S}}_{e{d}_4}^2$	1.20 $\times {10}^{-4}$	1.45 $\times {10}^{-4}$	1.55 $\times {10}^{-4}$	180 $\times {10}^{-4}$	3.60 $\times {10}^{-3}$	1.00 $\times {10}^{-4}$
  [13] ${\widehat{S}}_{e{d}_5}^2$	1.50 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	2.80 $\times {10}^{-4}$	4.10 $\times {10}^{-3}$	1.00 $\times {10}^{-4}$
  [14] ${\widehat{S}}_{e{d}_6}^2$	1.50 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	2.60 $\times {10}^{-4}$	5.20 $\times {10}^{-3}$	1.50 $\times {10}^{-4}$
  [15] ${\widehat{S}}_{e{d}_7}^2$	1.20 $\times {10}^{-4}$	1.00 $\times {10}^{-4}$	1.30 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	2.50 $\times {10}^{-3}$	1.00 $\times {10}^{-4}$
  [16] ${\widehat{S}}_{e{d}_8}^2$	1.49 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$	2.45 $\times {10}^{-4}$	3.50 $\times {10}^{-4}$	5.50 $\times {10}^{-4}$	1.80 $\times {10}^{-4}$

  , while

  Table 3. MSEs using empirical datasets.

  Estimator	Data 1	Data 2	Data 3
  [1] ${\widehat{S}}_y^2$	1.02 $\times {10}^{23}$	1.32 $\times {10}^{23}$	5020.63
  [2] ${\widehat{S}}_r^2$	4.78 $\times {10}^{22}$	5.93 $\times {10}^{22}$	4663.93
  [3] ${\widehat{S}}_{lr}^2$	2.07 $\times {10}^{22}$	4.24 $\times {10}^{22}$	3070.74
  [4] ${\widehat{S}}_{bt}^2$	2.42 $\times {10}^{22}$	4.96 $\times {10}^{22}$	3091.4
  [5] ${\widehat{S}}_{us}^2$	4.38 $\times {10}^{22}$	5.69 $\times {10}^{22}$	4483.93
  [6] ${\widehat{S}}_{c{k}_1}^2$	4.78 $\times {10}^{22}$	5.93 $\times {10}^{22}$	4656.39
  [7] ${\widehat{S}}_{c{k}_2}^2$	4.78 $\times {10}^{22}$	5.93 $\times {10}^{22}$	4661.94
  [8] ${\widehat{S}}_{c{k}_3}^2$	3.83 $\times {10}^{22}$	5.75 $\times {10}^{22}$	3792.23
  [9] ${\widehat{S}}_{e{d}_1}^2$	1.55 $\times {10}^{22}$	3.59 $\times {10}^{22}$	2977.56
  [10] ${\widehat{S}}_{e{d}_2}^2$	1.52 $\times {10}^{22}$	3.60 $\times {10}^{22}$	2663.27
  [11] ${\widehat{S}}_{e{d}_3}^2$	1.49 $\times {10}^{22}$	3.58 $\times {10}^{22}$	2668.04
  [12] ${\widehat{S}}_{e{d}_4}^2$	1.49 $\times {10}^{22}$	3.58 $\times {10}^{22}$	2667.649
  [13] ${\widehat{S}}_{e{d}_5}^2$	1.49 $\times {10}^{22}$	3.58 $\times {10}^{22}$	2634.19
  [14] ${\widehat{S}}_{e{d}_6}^2$	1.498 $\times {10}^{22}$	3.58 $\times {10}^{22}$	2668.402
  [15] ${\widehat{S}}_{e{d}_7}^2$	1.49 $\times {10}^{22}$	3.58 $\times {10}^{22}$	2668.01
  [16] ${\widehat{S}}_{e{d}_8}^2$	1.40 $\times {10}^{22}$	3.57 $\times {10}^{22}$	2518.48

  summarizes the results for real data sets.
• Step 6: use the following formula to get the $MS{E}^{\prime }s$ of each estimator across all replications:

  $$MSE{\left({\widehat{S}}_k^2\right)}_{min}=\frac{{\sum }_{g=1}^{\mathrm{60,000}}{\left({\widehat{S}}_k^2-S_Y^2\right)}^2}{\mathrm{60,000}},k=r,lr,bt,us,c{k}_1,c{k}_2,c{k}_3,e{d}_i(i=1,2,\dots ,8).$$

5.2. Numerical Examples

In order to check the effectiveness of the proposed estimator, we employed three actual datasets to compare the Mean Squared Errors MSEs of various estimators. The descriptions and summary statistics of the datasets are provided below:

• Data 1. (Source: [10])
  Y: The total enrollment of students in 2012 and X: Government elementary and secondary schools in 2012. The following are the summary statistics:

  $$\begin{array}{c}N=36, n=15, \overline{X}=1054.39, \overline{Y}=\mathrm{148,718.70}, X_M=2370, X_m=388, S_x=402.61,\hfill \\ S_y=\mathrm{182,315.10}, C_x=0.38, C_y=1.23, {\rho }_{yx}=0.18, {\lambda }_{40}=2366, {\lambda }_{04}=4698, {\lambda }_{03}=3297,\hfill \\ {\lambda }_{21}=3298, {\lambda }_{22}=2976.\hfill \end{array}$$
• Data 2. (Source: [10])
  Y: Departmental employment levels in 2012 and X: Number of factories the departments registered in 2012. The following are the summary statistics:

  $$\begin{array}{c}N=36, n=15, \overline{X}=335.78, \overline{Y}=\mathrm{52,432.86}, X_M=2055, X_m=24, S_x=451.14,\hfill \\ S_y=\mathrm{178,201.10}, C_x=1.34, C_y=3.3986, {\rho }_{yx}=0.39, {\lambda }_{40}=3366, {\lambda }_{04}=4698,\hfill \\ {\lambda }_{03}=3697, {\lambda }_{21}=3698, {\lambda }_{22}=3276.\hfill \end{array}$$
• Data 3. (Source: [23], p. 24)
  Y: Food expenses related to the family’s employment and X: Families’ weekly income. The following are the summary statistics:

  $$\begin{array}{c}N=33, n=5, \overline{X}=72.55, \overline{Y}=27.49, X_M=95, X_m=58, S_x=10.58, S_y=10.13,\hfill \\ C_x=0.15, C_y=0.37, {\rho }_{yx}=0.25, {\lambda }_{40}=3.25, {\lambda }_{04}=3.80, {\lambda }_{03}=1.00, {\lambda }_{21}=2.91,\hfill \\ {\lambda }_{22}=2.22.\hfill \end{array}$$

To evaluate the performance of the proposed class of estimators, we employed simulation studies and three real datasets. In order to compare various estimators, the $MSE$ criterion is applied. Table 2 presents the $MSE$ values of the proposed and existing estimators derived from the simulation study, whereas Table 3 presents the results for real datasets. Furthermore, various diagrams are used to present the $MSEs$ of the proposed and existing estimators for both simulation studies and real datasets. The following are some general findings:

• Regarding all simulation scenarios and real datasets, Table 2 and Table 3 demonstrate that the $MSE$ values of each proposed estimator are lower when compared to the existing estimators described in the literature. This confirms the proposed estimators’ superior performance in comparison to existing estimators.
• The $MSEs$ of all proposed estimators are lower than those of existing estimators, as illustrated in Figure 1, Figure 2 and Figure 3 for simulation studies and real datasets because the lines in the graphs are going in the downward direction. Consequently, there exists an inverse relationship between the value of $MSEs$ for both the proposed and existing estimators.

6. Conclusions

In this article, we introduced a set of efficient estimators for estimating the finite population variance. These estimators utilize the known minimum and maximum values of the auxiliary variable. To compare the properties of the proposed estimators with existing ones, we presented theoretical conditions in Section 4 that demonstrate the superior efficiency of the proposed estimators. To validate these conditions, we conducted a simulation study and analyzed several empirical datasets. The results, as shown in Table 2, indicate that the proposed estimators consistently outperform the existing estimators in terms of $\left(MSEs\right)$. This observation is further supported by the empirical data presented in Table 3, which confirms the theoretical findings in Section 4. Based on both the simulation and empirical results, we conclude that the proposed estimators ${\widehat{S}}_{e{d}_i}^2$$(i=1,2,3,\dots ,8)$ exhibit greater efficiency compared to the other considered estimators. Among these proposed estimators, ${\widehat{S}}_{e{d}_8}^2$ is particularly preferred due to its lowest $MSE$.

However, we analyzed the properties of the proposed efficient class of estimators under a simple random sampling scheme. It is further possible to provide some new estimators in two-phase sampling technique, and our results can be helpful for finding more efficient estimators that can provide the least $MSEs$. Further research on this topic can help improve the estimators’ efficiency.