A family of estimators for variance estimation

Abstract

The potential of this study for estimating the finite population variance of the study variable of a class of estimators by utilizing an auxiliary variable in simple random sampling is enormous. The asymptotic properties of the proposed class of estimation procedure have been examined. The best fixed values are determined for which the mean squared error of the freshly suggested estimator is lowest. Several well-known existing estimators and class of suggested estimators are identified. To back up the theoretical results, a real population and an extensive simulation study using R software are performed, which demonstrate the dominance of the suggested estimator against all competitors. Appropriate suggestions have been made to the survey statisticians for their real-life implementation.

Keywords:

• Auxiliary variable
• study variable
• simple random sampling
• bias
• mean squared error
• class of estimators

1. Introduction

An appropriate utilization of auxiliary variables plays a key role in obtaining better estimates of population parameters, as identified in survey studies. It is supplementary information, usually referred to as a benchmark variable or auxiliary variable, and is frequently available through prior experience, census data, or organizational databases. For instance, to analyze the number of trees in a forest, the auxiliary variable will be the diameter of the tree; to study the sales of a book title at a book store in a certain city, the size of the book store is an auxiliary variable, among others. According to Cochran (2007), the inclusion of supplemental information offered by auxiliary variables is widely utilized to increase the precision of the estimators by taking advantage of the correlation between the study and auxiliary variable. In this regard, regression, ratio, and product estimators are excellent examples. Although Cochran (1940) discovered the ratio estimator to be best suited when the study and auxiliary variables are significantly positively correlated, the product estimator is preferable when the variables are highly negatively associated.

In numerous practical situations, survey practitioners are curious not only in the computation of population mean but also in the computation of population variance to enhance certain decision-making policies in the fields of agriculture, production planning in manufacturing industries, business, services industries, ecology, stock investments, seismology, and medical sciences. Thus, improved estimation of the population mean and variance are equally important for effective decision-making procedures. Plenty of situations rises in reality where the of population variance of the study variable is crucial. For these reasons, multiple authors such as Das and Tripathi (1978), Upadhyaya et al. (2004), Singh and Vishwakarma (2008), Gupta and Shabbir (2008), Shabbir and Gupta (2010), Singh et al. (2010), Singh and Solanki (2013), Sisodia and Dwivedi (1981), Kadilar and Cingi (2006), Yadav and Kadilar (2014), Yaqub and Shabbir (2016), Singh and Pal (2017), Cekim and Kadilar (2020), Singh et al. (2021), Singh and Usman (2021), Soni and Pandey (2022), Bhushan et al. (2023), Ahmad et al. (2023), Ali et al. (2024), Pandey et al. (2024a, 2024b) are investigated the population variance for conventional and modified estimators based on auxiliary information. Moreover, the extended exponential weighed moving average (EEWMA) statistic is a memory-type statistics that uses past observations along with current information for the estimation of population parameter to improve the efficiency of the estimators. Several authors, Qureshi et al. (2022, 2023, 2024), Zahid et al. (2023) and so on, have studied the memory type ratio and product estimators in different sampling schemes.

As we know variance estimation is a well-studied topic, so inspired by above-mentioned researchers, the main intent of this work is to propose a family of efficient estimators for the estimation of the population variance of the study variable by utilizing the knowledge of an auxiliary variable in simple random sampling. The leftovers of the article are intended in some sections. Section 2 described some relevant work of population variance estimation. The recommended family of estimators is addressed in Section 3. In Section 4, we have studied the problem of efficiency comparisons. To validate the theoretical findings, an empirical study is performed in Section 5. An ultimate conclusion is given in Section 6.

2. Some pertinent work

Observe that $U(=U_1,U_2,...,U_N)$ denotes the population, which is composed of N discrete units. Let (Y_i, X_i) be the study and the auxiliary variables (Y, X) on the i^th unit U_i; $i=1,2,...,N$ of the population U, respectively. Let us take a sample of size n from the population U and evaluate it using simple random sampling without replacement (SRSWOR).

Let ${\mu }_x={\sum }_{i=1}^N{X}_i/N$ and ${\mu }_y={\sum }_{i=1}^N{Y}_i/N$ be respectively the population means corresponding to the sample means $\overline{X}={\sum }_{i=1}^n{X}_i/n$ and $\overline{Y}={\sum }_{i=1}^n{Y}_i/n$ of variables X and Y. Also, consider $S_x^2={\sum }_{i=1}^N(X_i-{\mu }_x{)}^2/N$ and $S_y^2={\sum }_{i=1}^N(Y_i-{\mu }_y{)}^2/N$ be the population variances corresponding to the sample variances $s_x^2={\sum }_{i=1}^n(X_i-\overline{X}{)}^2/n$ and $s_y^2={\sum }_{i=1}^n(Y_i-\overline{Y}{)}^2/n$, respectively. The finite population correction (fpc) is responsible for changing the variance of sample mean when the sample is drawn from a finite population in comparison to an infinite population. Thus, we are omitting the finite population correction (fpc) from the beginning to the end of this article, i.e., we take n⁻¹ instead of $(n^{-1}-N^{-1})$. The purpose is to determine the population variance S_y of the study variable Y based on information from an auxiliary variable X that is highly correlated with the study variable Y.

To obtain the bias and mean squared error (MSE) of the considered estimators, we write

$s_y^2=S_y^2(1+e_0)$ and $s_x^2=S_x^2(1+e_1)$

such that $E(e_0)=E(e_1)=0$ and to the first degree of approximation,

$E(e_0^2)=\frac{1}{n}({\beta }_2(y)-1)={\delta }_{40}$, $E(e_1^2)=\frac{1}{n}({\beta }_2(x)-1)={\delta }_{04}$, $E(e_0{e}_1)=\frac{1}{n}({\lambda }_{22}-1)={\delta }_{22}$,

where ${\beta }_2(y)=\frac{{\mu }_{40}}{{\mu }_{20}^2}$, ${\beta }_2(x)=\frac{{\mu }_{04}}{{\mu }_{02}^2}$, ${\lambda }_{22}=\frac{{\mu }_{22}}{{\mu }_{20}{\mu }_{02}}$;

${\mu }_{\mathrm{pq}}=\frac{1}{N}{\sum }_{i=1}^N(y_i-{\mu }_y{)}^p(x_i-{\mu }_x{)}^q$; (p, q) are non-negative integers.

The list of existing estimator(s) along with their mean squared error up to first order of approximation, are given in Table A.1 of Appendix.

Table 1. Several members of the proposed family of estimator (T_HS) for different values of constant $(a,b,c,d)$.

Estimator	a	b	c	d
$T_{\mathrm{HS}}^{(1)}=s_y^2\{{\omega }_1(\frac{S_x^2-C_x{s}_x^2}{(1-C_x)S_x^2}{)}^{\gamma }+{\omega }_2\}[1+\log (\frac{s_x^2}{S_x^2}{)}^{\alpha }{]}^{\eta }$	1	0	1	Cx
$T_{\mathrm{HS}}^{(12)}=s_y^2\{{\omega }_1(\frac{s_x^2}{S_x^2}{)}^{\gamma }+{\omega }_2\}[1+\log (\frac{s_x^2}{S_x^2}{)}^{\alpha }{]}^{\eta }$	1	0	0	1
$T_{\mathrm{HS}}^{(3)}=s_y^2\{{\omega }_1(\frac{{\beta }_2(x)S_x^2-s_x^2}{({\beta }_2(x)-1)S_x^2}{)}^{\gamma }+{\omega }_2(\frac{(\overline{X}+1)S_x^2}{\overline{X}{S}_x^2+s_x^2}{)}^{\delta }\}[1+\log (\frac{\overline{X}{s}_x^2+1}{\overline{X}{S}_x^2+1}{)}^{\alpha }{]}^{\eta }$	$\overline{X}$	1	β2(x)	1
$T_{\mathrm{HS}}^{(4)}=s_y^2\{{\omega }_1(\frac{{\beta }_2(x)S_x^2-\overline{X}{s}_x^2}{({\beta }_2(x)-\overline{X})S_x^2}{)}^{\gamma }+{\omega }_2(\frac{(C_x+{\rho }_{\mathrm{yx}})S_x^2}{C_x{S}_x^2+{\rho }_{\mathrm{yx}}{s}_x^2}{)}^{\delta }\}[1+\log (\frac{C_x{s}_x^2+{\rho }_{\mathrm{yx}}}{C_x{S}_x^2+{\rho }_{\mathrm{yx}}}{)}^{\alpha }{]}^{\eta }$	Cx	ρyx	β2(x)	$\overline{X}$
$T_{\mathrm{HS}}^{(5)}=s_y^2\{{\omega }_1(\frac{S_x^2-\overline{X}{s}_x^2}{(1-\overline{X})S_x^2}{)}^{\gamma }+{\omega }_2\}[1+\log (\frac{s_x^2}{S_x^2}{)}^{\alpha }{]}^{\eta }$	1	0	1	$\overline{X}$
$T_{\mathrm{HS}}^{(6)}=s_y^2\{{\omega }_1(\frac{{\rho }_{\mathrm{yx}}{S}_x^2-C_x{s}_x^2}{({\rho }_{\mathrm{yx}}-C_x)S_x^2}{)}^{\gamma }+{\omega }_2(\frac{(\overline{X}+{\beta }_2(x))S_x^2}{\overline{X}{S}_x^2+{\beta }_2(x)s_x^2}{)}^{\delta }\}[1+\log (\frac{\overline{X}{s}_x^2+{\beta }_2(x)}{\overline{X}{S}_x^2+{\beta }_2(x)}{)}^{\alpha }{]}^{\eta }$	$\overline{X}$	β2(x)	ρyx	Cx

3. Recommended family of estimator

Enormous literature has been published on survey sampling from a finite population to tackle the challenges of efficient variance estimation of study variable when information on an auxiliary variable is available. Under certain efficiency conditions, exponential-type estimators are known to outperform the related customary ratio and product-type estimators in terms of lesser mean square errors. Therefore, the question arises as to what happens when the conditions that favor exponential-type estimators over the customary estimators are not readily met. The answer, of course, lies in the use of other efficient estimators that would perform better than both the existing exponential and customary estimators. So, we examine logarithmic-type estimator in our search for such efficient estimators because logarithm is the inverse operation to exponentiation. The logarithmic function has some helpful qualities and is used extensively in a variety of scientific and non-scientific domains. So, motivated by Singh and Solanki (2013) and Bhushan et al. (2023), we solely concentrate on establishing a generalized class of estimators for population variance that can be more efficient than the other conventional estimators for population variance that we have studied earlier in Section 2 only adopt simple random sampling without replacement (SRSWOR) and considers only a single auxiliary variable for convenience.

The suggested class of estimators for population variance $S_y^2$ of the study variable y as (1) $$\begin{array}{ll}{T}_{\mathrm{HS}}& =s_y^2[{\omega }_1\{\frac{c{S}_x^2-d{s}_x^2}{(c-d)S_x^2}{\}}^{\gamma }+{\omega }_2\{\frac{(a+b)S_x^2}{a{S}_x^2+b{s}_x^2}{\}}^{\delta }]\\ & \times [1+\log (\frac{a{s}_x^2+b}{a{S}_x^2+b}{)}^{\alpha }{]}^{\eta }\end{array}$$(1) where (ω₁, ω₂) are constants to be determined such that the mean squared error (MSE) of the proposed class of estimator T_HS is least, $(\gamma ,\delta ,\alpha ,\eta )$ being constants that take finite values for designing the various estimators and $(a,b,c,d)$ are either constants or functions of known parameters of the auxiliary variable x, such as $\overline{X}$, C_x, β₁(x), β₂(x) and ρ_yx.

Various additional estimators can be developed from the suggested estimator for distinct values of ($a,b,c,d$). Table 1 illustrates how the suggested family of estimators (T_HS) reduces to an extended form of existing estimators. Expressing (1) in terms of e₀ and e₁, we have

(2) $$\begin{array}{ll}{T}_{\mathrm{HS}}& =S_y^2(1+e_0)[{\omega }_1(1-{\eta }_2{e}_1{)}^{\gamma }+{\omega }_2(1+{\eta }_1{e}_1{)}^{-\delta }]\\ & \times [1+\log (1+\nu e_1{)}^{\alpha }{]}^{\eta },\end{array}$$(2)

where $\nu =\frac{S_x^2}{S_x^2+b}$, ${\eta }_1=\frac{b}{(a+b)}$, ${\eta }_2=\frac{d}{(c-d)}$.

We assume $|{\eta }_2{e}_1|\le 1$, $|{\eta }_1{e}_1|\le 1$ and $|\nu e_1|\le 1$; so that $(1-{\eta }_2{e}_1{)}^{\gamma }$, $(1-{\eta }_1{e}_1{)}^{-\delta }$ and $(1+\nu e_1{)}^{\alpha }$ are expandable. Further, we assume $|\mathrm{\alpha \nu }(e_1-\frac{\nu e_1^2}{2})\le 1|$ so that $(1+\mathrm{\alpha \nu }{e}_1-\frac{\alpha {\nu }^2{e}_1^2}{2}{)}^{\eta }$ is expandable.

Now, expanding the right hand side of (2), multiplying and neglecting terms of e^′s having power greater than two, we have (3) $$\begin{array}{ll}(T_{\mathrm{HS}}-S_y^2)& \cong S_y^2[{\omega }_1\{1+e_0+c_1(e_1+e_0{e}_1)+c_2{e}_1^2\}\\ & +{\omega }_2\{1+e_0+d_1(e_1+e_0{e}_1)+d_2{e}_1^2\}-1],\end{array}$$(3) where $c_1=(\mathrm{\alpha \eta \nu }-{\eta }_2\gamma )$, $c_2=\frac{1}{2}[(\mathrm{\alpha \eta \nu }-{\eta }_2\gamma {)}^2-\{{\eta }_2^2\gamma +\mathrm{\eta \alpha }(\alpha +1){\nu }^2\}{e}_1^2]$,

$d_1=(\mathrm{\alpha \eta \nu }-{\eta }_1\delta )$, $d_2=\frac{1}{2}[(\mathrm{\alpha \eta \nu }-{\eta }_1\delta {)}^2-\{{\eta }_1^2\delta +\mathrm{\eta \alpha }(\alpha +1){\nu }^2\}{e}_1^2]$.

Taking expectation of both sides of (3) we get, the bias of T_HS to the first degree of approximation, as (4) $$\begin{array}{l}\mathrm{Bias}(T_{\mathrm{HS}})=S_y^2[{\omega }_1{A}_4+{\omega }_2{A}_5-1]\end{array}$$(4) where $A_4=(1+c_1{\delta }_{22}+c_2{\delta }_{04})$, $A_5=(1+d_1{\delta }_{22}+d_2{\delta }_{04})$.

Squaring both sides of (3) and neglecting terms of e^′s having power greater than two we have (5) $$\begin{array}{ll}& (T_{\mathrm{HS}}-S_y^2{)}^2=S_y^4[1+{\omega }_1^2\{1+2e_0+2c_1{e}_1+e_0^2+c_1{e}_0{e}_1\\ & +(c_1^2+2c_2)e_1^2\}+{\omega }_2^2\{1+2e_0+2d_1{e}_1+e_0^2+d_1{e}_0{e}_1\\ & +(d_1^2+2d_2)e_1^2\}]+2{\omega }_1{\omega }_2\{1+2e_0+(c_1+d_1)(e_1+2e_0{e}_1)+e_0^2\\ & +(c_2+d_2+c_1{d}_1{e}_1^2)\}-2{\omega }_1\{1+e_0+c_1(e_1+e_0{e}_1)+c_2{e}_1^2\}\\ & -2{\omega }_2\{1+e_0+d_1(e_1+e_0{e}_1)+d_2{e}_1^2\}\end{array}$$(5)

Taking expectation of both sides of (5) we get, the MSE of T_HS to the first degree of approximation, as (6) $$\begin{array}{l}\mathrm{MSE}(T_{\mathrm{HS}})=S_y^4[1+{\omega }_1^2{A}_1+{\omega }_2^2{A}_2+2{\omega }_1{\omega }_2{A}_3-2{\omega }_1{A}_4-2{\omega }_2{A}_5],\end{array}$$(6) where $A_1=[1+{\delta }_{04}+4c_1{\delta }_{22}+(c_1^2+2c_2){\delta }_{04}]$, $A_2=[1+{\delta }_{04}+4d_1{\delta }_{22}+(d_1^2+2d_2){\delta }_{04}]$, $A_3=[1+{\delta }_{04}+2(c_1+d_1){\delta }_{22}+(c_2+d_2+c_1{d}_1){\delta }_{04}]$. The MSE(T_HS) is minimized for (7) $$\begin{array}{l}\begin{array}{c}{\omega }_1=\frac{(A_2{A}_4-A_3{A}_5)}{A_1{A}_2-A_3^2}={\omega }_{10}\text{(say)}\\ {\omega }_2=\frac{(A_1{A}_5-A_3{A}_4)}{A_1{A}_2-A_3^2}={\omega }_{20}\text{(say)}\end{array}\}\end{array}$$(7)

Thus, the resulting minimum MSE of T_HS is given by (8) $$\begin{array}{l}\mathrm{minMSE}(T_{\mathrm{HS}})=[1-\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]S_y^4\end{array}$$(8) which is true only when (9) $$\begin{array}{l}\begin{array}{c}0<\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}<1\\ \text{and}\\ A_1{A}_2-A_3^2>0\end{array}\}\end{array}$$(9)

For η = 0, the proposed class of estimator T_HS reduces to Singh and Solanki (2013) estimator, i.e., (10) $$\begin{array}{l}{T}_{\mathrm{HS}}=s_y^2[{\omega }_1\{\frac{c{S}_x^2-d{s}_x^2}{(c-d)S_x^2}{\}}^{\gamma }+{\omega }_2\{\frac{(a+b)S_x^2}{a{S}_x^2+b{s}_x^2}{\}}^{\delta }]\end{array}$$(10)

4. Efficiency comparisons

To evaluate the mean squared error of the suggested estimator T_HS over the mean squared error of competitors that are expressed in Table 1. So, in this portion of the article, we obtain the efficiency condition as follows

(i) $\min .\mathrm{MSE}(T_{\mathrm{HS}})<\mathrm{Var}(T_0)$ (11) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1-{\delta }_{40}\end{array}$$(11)

(ii) minMSE(T_HS) < minMSE(T_i); $i=1,2,4$ (12) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1-{\delta }_{40}+2\frac{{\delta }_{22}^2}{{\delta }_{04}}\end{array}$$(12)

(iii) minMSE(T_HS) < MSE(T₃)

(13) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1-{\delta }_{40}-{\delta }_{04}+2{\delta }_{22}\end{array}$$(13)

(iv) minMSE(T_HS) < minMSE(T₅)

(14) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1-\frac{\mathrm{MSE}(T_4)}{S^4+\mathrm{MSE}(T_4)}\end{array}$$(14)

(v) minMSE(T_HS) < MSE(T_i); $i=6,7,...,13$ (15) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1-{\delta }_{40}-{\psi }_i^2{\delta }_{04}+2{\psi }_i{\delta }_{22}\end{array}$$(15)

(vi) minMSE(T_HS) < minMSE(T₁₄); $i=6,7,...,13$ (16) $$\begin{array}{ll}& \mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1\\ & -\frac{1}{64}\{\frac{-{\delta }_{04}^2-16({\delta }_{40}-\frac{{\delta }_{22}^2}{{\delta }_{04}})({\delta }_{04}-4)}{1+{\delta }_{40}-\frac{{\delta }_{22}^2}{{\delta }_{04}}}\}\end{array}$$(16)

(vii) minMSE(T_HS) < minMSE(T₁₅)

(17) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>\{\frac{S_y^{-4}(A{E}^2+B{D}^2-\mathrm{CDE})}{\mathrm{AB}-C^2}\}\end{array}$$(17)

(viii) minMSE(T_HS) < minMSE(T₁₆)

(18) $$\begin{array}{ll}& \mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1\\ & -({\delta }_{40}+{\delta }_{04}-2{\delta }_{22}+16{\theta }_1{\beta }_1{\delta }_{04}(1-{\theta }_1-{\beta }_1+{\theta }_1{\beta }_1)\\ & +4{\delta }_{22}({\theta }_1-{\beta }_1{)}^2+4{\delta }_{04}({\theta }_1^2+{\beta }_1^2-{\theta }_1-{\beta }_1))\end{array}$$(18)

(ix) minMSE(T_HS) < minMSE(T₁₇)

(19) $$\begin{array}{ll}& \mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1\\ & -\frac{1}{16}\{\frac{64(1-{\delta }_{04})S_y^{-4}\mathrm{MSE}(T_4)-{\delta }_{04}^2}{{\delta }_{04}+4(1-{\delta }_{04})+4S_y^{-4}\mathrm{MSE}(T_4)}\}\end{array}$$(19)

(x) minMSE(T_HS) < minMSE(T₁₈)

(20) $$\begin{array}{l}\mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>\{\frac{(c_2-2c_3{c}_4+c_1{c}_4^2)}{c_1{c}_2-c_3^2}\}\end{array}$$(20)

(xi) minMSE(T_HS) < minMSE(T₁₉)

(21) $$\begin{array}{ll}& \mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]>1\\ & -\{(\log (2)-1{)}^2+(\log (2){)}^2{\delta }_{40}\}\end{array}$$(21)

(xi) minMSE(T_HS) < minMSE(T₂₀)

(22) $$\begin{array}{ll}& \mathrm{if}[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]\\ & >[\frac{(E_1{E}_5^2+E_2{E}_4^2-2E_3{E}_4{E}_5)}{(E_1{E}_2-E_3^2)}]\end{array}$$(22)

The suggested family of estimator outperforms the other conventional estimators listed in Table 1 when the conditions from (11-22) are met. In the next section, we conducted a numerical analysis to judge the theoretical results by using real and hypothetical population.

5. Empirical study

To test, the efficiency of the suggested family of estimators over other competing estimator(s), we conducted an empirical study by using real and simulated population. The percent relative efficiency (PRE) of all considered estimators is defined as (23) $$\begin{array}{l}\mathrm{PRE}(T_{\varphi },s_y^2)=\frac{\mathrm{Var}(T_0)}{\mathrm{MSE}(T_{\varphi })}\times 100\end{array}$$(23) where $\varphi =1,2,...,20$.

The PRE of the proposed estimator T_HS with respect to $s_y^2$ is defined as (24) $$\begin{array}{l}\mathrm{PRE}(T_{\mathrm{HS}},s_y^2)=\frac{{\delta }_{40}}{1-L}\times 100,\end{array}$$(24) where $L=[\frac{(A_2{A}_4^2-2A_3{A}_4{A}_5+A_1{A}_5^2)}{A_1{A}_2-A_3^2}]$.

The $\mathrm{PRE}(T_{\mathrm{HS}},s_y^2)$ is to be computed for different values of the constants ($\gamma ,\delta ,\alpha ,\eta ,a,b,c,d$).

5.1. Real dataset

To validate the theoretical results, we execute a numerical analysis with a real population, which is based on the literacy of persons in the village (refer to Mukherjee 1998). Let Y denotes the literate persons in the village and X denotes the number of workers in the village, respectively. The specifics of real population is defined in Tables 2 where population size N = 64 and sample size is 19.

Table 2. Data statistics.

N = 64	n = 19	μy = 5. 549	μx = 141. 500	λ22 = 1. 458
$S_y^2=2.277$	$S_x^2=5772.670$	${\rho }_{\mathrm{yx}}^2=0.2971$	β2(y) = 2. 773	β2(x) = 2. 341
δ40 = 0. 0656	δ04 = 0. 0496	δ22 = 0. 0169

Tables 3 and 4 show the bias and percent relative efficiency of the estimators using the real population. From Table 3, it is clear that the estimator $T_{\mathrm{HS}}^{(4)}$ which utilizes information on mean $\overline{X}$ and coefficient of kurtosis β₂(x) having minimum bias as compared to other family of suggested estimator. It is observed from Tables 4 that all estimators $T_{H{S}_j}$, ($j=1,2,...,6$) which are members of the suggested family of estimators T_HS perform better than the usual unbiased estimator, Das and Tripathi (1978), Isaki (1983), Singh et al. (1988), Upadhyaya and Singh (1999), Kadilar and Cingi (2006) ratio type estimators, Shabbir and Gupta (2007), Singh and Solanki (2013) class of estimators, Yadav and Kadilar (2014), Yaqub and Shabbir (2016), Singh and Pal (2017), Cekim and Kadilar (2020) and Bhushan et al. (2023) class of estimators. Besides this, among all the estimators and classes of estimators discussed here, estimator $T_{H{S}_1}$ which utilizes information on the population coefficient of variation C_x is the best in terms of having the highest percent relative efficiency.

Table 3. Bias of proposed family of estimators using real population.

Estimator(s)	$\mathrm{Bias}(T_{\mathrm{HS}}^{(j)})$; $j=1,2,...,6$
$T_{\mathrm{HS}}^{(1)}$	0.8458
$T_{\mathrm{HS}}^{(2)}$	0.7864
$T_{\mathrm{HS}}^{(3)}$	0.0052
$T_{\mathrm{HS}}^{(4)}$	0.8260
$T_{\mathrm{HS}}^{(5)}$	0.8031
$T_{\mathrm{HS}}^{(6)}$	0.8035

Table 4. Percent relative efficiency of distinct estimators using real population.

Estimator(s)	PRE	Estimator(s)	PRE
T0	100	$T_{15}^{(8)}$	3.1635
Ti; $i=1,2$	109.8362	T16	70.1811
T3	80.2746	T17	27.1718
T4	109.8362	T18	165.0968
T5	116.2662	T19	51.4187
T6	80.3012	$T_{20}^{(1)}$	157.0362
T7	79.8083	$T_{20}^{(2)}$	157.0333
T8	80.2480	$T_{20}^{(3)}$	156.9860
T9	80.0762	$T_{20}^{(4)}$	157.0361
T10	80.2740	$T_{20}^{(5)}$	157.1150
T11	80.7347	$T_{20}^{(6)}$	156.8070
T12	80.4719	$T_{20}^{(7)}$	157.2230
T13	80.3012	$T_{20}^{(8)}$	157.0361
T14	116.1683	$T_{20}^{(9)}$	157.0362
$T_{15}^{(1)}$	3.1625	$T_{20}^{(10)}$	156.9622
$T_{15}^{(2)}$	3.1678	$T_{\mathrm{HS}}^{(1)}$	294.6387
$T_{15}^{(3)}$	3.1802	$T_{\mathrm{HS}}^{(2)}$	211.8416
$T_{15}^{(4)}$	3.1797	$T_{\mathrm{HS}}^{(3)}$	186.8532
$T_{15}^{(5)}$	3.1626	$T_{\mathrm{HS}}^{(4)}$	248.6189
$T_{15}^{(6)}$	3.2154	$T_{\mathrm{HS}}^{(5)}$	224.9907
$T_{15}^{(7)}$	3.1657	$T_{\mathrm{HS}}^{(6)}$	219.9320

5.2. Hypothetical population

Making use of R software, a Monte Carlo simulation study is carried out to compare the percent relative efficiency (PRE) of the proposed class of estimator against other conventional estimator(s) that are considered in our article. We assume three finite populations of size N = 2000, N = 3000, and N = 4500 generated from a normal distribution, and from the population N = 2000 we take different samples of sizes, i.e., n = 200, n = 300, and n = 400. From N = 3000, we pick a sample of distinct sizes, i.e., n = 300, n = 400, and n = 500. Moreover, from a population N = 4500, we choose the sample of sizes n = 400, n = 500, and n = 600. The study and auxiliary variable are also generated from a normal distribution, i.e., $X=\mathrm{rnorm}(N,0,1)$ and $Y=X+\mathrm{rnorm}(N,0,1)$, respectively.

Table 5 represents the bias of the recommended family of estimator(s) with three hypothetical populations N and samples n. It is envisaged from Table 5 that for increase in the value of sample size from 200 to 400, the $\mathrm{Bias}(T_{\mathrm{HS}}^{(j)})$; $j=1,2,...,6$ of the family of recommended estimator(s) decreases. The estimator $T_{\mathrm{HS}}^{(5)}$ which utilizes information on mean $\overline{X}$ having minimum bias when population size i.e., N = 2000 and N = 4500 and when population size i.e., N = 3000, the estimator $T_{\mathrm{HS}}^{(4)}$ which utilizes information on coefficient of variation C_x, correlation coefficient ρ_yx, coefficient of kurtosis β₂(x) and mean $\overline{X}$ having minimum bias among all the family of estimators/classes of estimators addressed here.

Table 5. Bias of proposed family of estimators using simulated population with different values of N and n.

Estimator(s)	N = 2000			N = 3000			N = 4500
	n = 200	n = 300	n = 400	n = 200	n = 300	n = 400	n = 200	n = 300	n = 400
$T_{\mathrm{HS}}^{(1)}$	0.1089	0.0726	0.0545	0.1294	0.0863	0.0648	0.1069	0.0713	0.0535
$T_{\mathrm{HS}}^{(2)}$	0.1143	0.0763	0.0572	0.1066	0.0711	0.0533	0.1250	0.0834	0.0626
$T_{\mathrm{HS}}^{(3)}$	0.1402	0.0935	0.0702	0.1411	0.0941	0.0706	0.1817	0.1212	0.0909
$T_{\mathrm{HS}}^{(4)}$	0.1154	0.0769	0.0577	0.0501	0.0334	0.0251	0.1727	0.1152	0.0864
$T_{\mathrm{HS}}^{(5)}$	0.0756	0.0504	0.0378	0.0932	0.0621	0.0466	0.0788	0.0525	0.0394
$T_{\mathrm{HS}}^{(6)}$	0.1209	0.0807	0.0605	0.1074	0.0716	0.0537	0.1122	0.0748	0.0562

Tables 6–8 illustrates the percent relative efficiency of the proposed family of estimators over other competing estimators by using three hypothetical populations, i.e. $N=2000,3000,4500$. Following points are noted from Tables 6–8 that

Table 6. Percent relative efficiency of different estimators using simulated population when N = 2000.

Estimator(s)	PRE			Estimator(s)	PRE
	n = 200	n = 300	n = 400		n = 200	n = 300	n = 400
T0	100.0000	100.0000	100.0000	$T_{15}^{(8)}$	115.3206	113.8133	113.2202
Ti; $i=1,2$	136.8840	136.8840	136.8840	T16	106.2370	106.2370	106.2370
T3	106.2370	106.2370	106.2370	T17	4.3663	2.9100	2.1821
T4	136.8840	136.8840	136.8840	T18	148.4946	147.8175	147.4795
T5	137.8921	137.5560	137.3880	T19	10.1822	6.9006	5.2187
T6	98.7854	98.5765	98.3469	$T_{20}^{(1)}$	147.3214	146.7089	146.4031
T7	102.2971	100.1916	94.9884	$T_{20}^{(2)}$	147.4739	146.8129	146.4833
T8	112.6418	112.7956	112.9621	$T_{20}^{(3)}$	145.8407	145.7363	145.6953
T9	99.8262	99.9858	100.3537	$T_{20}^{(4)}$	147.3184	146.7087	146.4065
T10	106.0888	106.2243	106.5726	$T_{20}^{(5)}$	105.5298	105.1898	105.0182
T11	97.6023	99.8077	104.5916	$T_{20}^{(6)}$	104.0700	105.0710	107.7978
T12	100.1732	100.0142	99.6439	$T_{20}^{(7)}$	105.5213	105.1897	104.9913
T13	98.7854	98.5765	98.3469	$T_{20}^{(8)}$	118.5285	115.8823	109.3190
T14	138.1930	137.7559	137.5376	$T_{20}^{(9)}$	146.1251	145.7528	145.8251
$T_{15}^{(1)}$	115.3140	113.8105	113.2188	$T_{20}^{(10)}$	146.8949	146.6771	148.9262
$T_{15}^{(2)}$	115.4465	113.9390	113.3458	$T_{\mathrm{HS}}^{(1)}$	225.3202	225.3202	212.2328
$T_{15}^{(3)}$	115.3140	113.8105	113.2189	$T_{\mathrm{HS}}^{(2)}$	299.9002	226.0032	199.9410
$T_{15}^{(4)}$	116.6649	114.6776	113.8595	$T_{\mathrm{HS}}^{(3)}$	223.7854	189.6468	175.2578
$T_{15}^{(5)}$	115.3111	113.8102	113.2245	$T_{\mathrm{HS}}^{(4)}$	174.1813	156.5775	157.7046
$T_{15}^{(6)}$	115.3146	113.8105	113.2200	$T_{\mathrm{HS}}^{(5)}$	163.3070	165.6108	172.3192
$T_{15}^{(7)}$	115.3140	113.8105	113.2187	$T_{\mathrm{HS}}^{(6)}$	374.4094	273.3793	357.0680

Table 7. Percent relative efficiency of different estimators using simulated population when N = 3000.

Estimator(s)	PRE			Estimator(s)	PRE
	n = 300	n = 400	n = 500		n = 300	n = 400	n = 500
T0	100.0000	100.0000	100.0000	$T_{15}^{(8)}$	107.7489	107.1928	106.7398
Ti; $i=1,2$	134.3574	134.3574	134.3574	T16	102.5022	102.5022	102.5022
T3	102.5022	102.5022	102.5022	T17	2.9073	2.1801	1.7439
T4	134.3574	134.3574	134.3574	T18	140.8791	140.5591	140.3674
T5	135.0289	134.8610	134.7603	T19	6.8951	5.2145	4.1926
T6	98.2335	98.1355	98.0760	$T_{20}^{(1)}$	140.5372	140.2352	140.0541
T7	94.36931	106.5001	93.0440	$T_{20}^{(2)}$	140.5919	140.2770	140.0880
T8	106.4178	100.2674	106.5499	$T_{20}^{(3)}$	139.6413	139.5686	139.5211
T9	100.2191	102.7811	100.2677	$T_{20}^{(4)}$	140.5100	140.2378	140.0563
T10	102.7211	106.1166	102.7887	$T_{20}^{(5)}$	103.4022	103.2331	103.1319
T11	105.0712	99.7311	99.7309	$T_{20}^{(6)}$	106.4634	107.0218	107.0118
T12	99.77991	98.1355	98.0760	$T_{20}^{(7)}$	103.3675	103.1815	103.2977
T13	98.23352	135.0221	134.8890	$T_{20}^{(8)}$	98.8339	93.9970	94.2977
T14	135.2440	107.1914	106.7389	$T_{20}^{(9)}$	143.2984	142.7964	142.1807
$T_{15}^{(1)}$	107.7463	107.1910	106.8140	$T_{20}^{(10)}$	141.3621	139.4406	139.4041
$T_{15}^{(2)}$	107.8230	107.1917	106.7382	$T_{\mathrm{HS}}^{(1)}$	240.5403	209.8671	191.7023
$T_{15}^{(3)}$	107.7656	107.7466	107.1927	$T_{\mathrm{HS}}^{(2)}$	216.4664	191.5246	178.7711
$T_{15}^{(4)}$	108.5230	107.1964	106.7440	$T_{\mathrm{HS}}^{(3)}$	223.7854	169.4299	162.3677
$T_{15}^{(5)}$	107.7504	107.1939	106.7409	$T_{\mathrm{HS}}^{(4)}$	183.0113	146.5119	145.0218
$T_{15}^{(6)}$	107.7485	107.1914	106.7389	$T_{\mathrm{HS}}^{(5)}$	147.1966	167.4354	160.0327
$T_{15}^{(7)}$	107.7463	107.1928	106.7398	$T_{\mathrm{HS}}^{(6)}$	180.7464	171.5840	164.4006

Table 8. Percent relative efficiency of different estimators using simulated population when N = 4500.

Estimator(s)	PRE			Estimator(s)	PRE
	n = 700	n = 800	n = 900		n = 700	n = 800	n = 900
T0	100.0000	100.0000	100.0000	$T_{15}^{(8)}$	97.3566	97.0647	96.8689
Ti; $i=1,2$	126.7067	126.7076	126.7067	T16	93.5699	93.5699	93.5699
T3	93.5699	93.5699	93.5699	T17	2.2083	1.7665	1.4720
T4	126.7067	126.7067	126.7067	T18	131.4772	131.2900	131.1652
T5	127.2190	127.1165	127.0482	T19	5.3017	4.2631	3.5647
T6	93.3747	93.3810	93.3839	$T_{20}^{(1)}$	131.6219	131.4436	131.3247
T7	106.0193	104.1236	104.4207	$T_{20}^{(2)}$	131.6237	131.4449	131.3258
T8	93.7642	93.7579	99.9901	$T_{20}^{(3)}$	130.9674	130.9293	130.8951
T9	99.9862	99.9908	93.5605	$T_{20}^{(4)}$	131.6218	131.4435	131.3247
T10	93.5555	93.5609	95.0106	$T_{20}^{(5)}$	103.0035	102.9007	102.8322
T11	92.8748	95.3859	100.0098	$T_{20}^{(6)}$	97.9057	99.8824	99.6043
T12	100.0137	100.0091	93.3839	$T_{20}^{(7)}$	102.9366	102.8719	102.7986
T13	93.9747	93.3810	127.1518	$T_{20}^{(8)}$	130.0204	128.9885	129.0762
T14	127.3747	127.2409	96.8684	$T_{20}^{(9)}$	131.5990	131.4173	131.3047
$T_{15}^{(1)}$	97.35555	97.0640	96.9078	$T_{20}^{(10)}$	131.0128	130.9618	130.9226
$T_{15}^{(2)}$	97.3953	97.1032	96.8685	$T_{\mathrm{HS}}^{(1)}$	166.3721	160.9394	154.9402
$T_{15}^{(3)}$	97.3557	97.0641	96.1655	$T_{\mathrm{HS}}^{(2)}$	178.6470	167.0084	159.9187
$T_{15}^{(4)}$	97.8073	97.4225	96.1655	$T_{\mathrm{HS}}^{(3)}$	165.4187	157.4908	152.6833
$T_{15}^{(5)}$	97.3526	97.0621	96.8665	$T_{\mathrm{HS}}^{(4)}$	155.1422	151.1762	147.4334
$T_{15}^{(6)}$	97.3579	97.0648	96.8692	$T_{\mathrm{HS}}^{(5)}$	139.7097	138.7113	137.2701
$T_{15}^{(7)}$	97.3555	97.0640	96.8684	$T_{\mathrm{HS}}^{(6)}$	169.9884	163.1097	156.6713

1. From Table 6, the estimator $T_{\mathrm{HS}}^{(6)}$ which utilizes information on mean $\overline{X}$, coefficient of kurtosis β₂(x), coefficient of variation C_x, and correlation coefficient ρ_yx having the highest percent relative efficiency amongst all the estimators/classes of estimators addressed here.

2. From Table 7, the estimator $T_{\mathrm{HS}}^{(1)}$ that utilizes information on the population coefficient of variation C_x has the highest percent relative efficiency over the other considered estimators.

3. From Table 8, the estimator $T_{\mathrm{HS}}^{(2)}$ when the constants a = 1, b = 0, c = 0 and d = 1 having the highest percent relative efficiency among others.

According to the results from Tables 6–8, the PRE of the proposed family of estimators $T_{\mathrm{HS}}^{(j)}$; $j=1,2,...,6$ performs better than all the other estimators that are considered in our work. As a result, the proposed class of estimators explains its significance and is strongly advisable for survey practitioners.

6. Conclusion

For the population variance $S_y^2$ of a study variable, we have developed a generalized family of estimators where information on an auxiliary variable is available in this article. Additionally, some well-known estimators of population variance, such as the usual unbiased estimator, Das and Tripathi (1978), Isaki (1983), Singh et al. (1988), Upadhyaya and Singh (1999), Kadilar and Cingi (2006) ratio type estimators, Shabbir and Gupta (2007), Singh and Solanki (2013) class of estimators, Yadav and Kadilar (2014), Yaqub and Shabbir (2016), Singh and Pal (2017), Cekim and Kadilar (2020) and Bhushan et al. (2023) class of estimators are also advised. Several families of estimators are produced from the suggested estimator by using different parametric of $a,b,c,d$. Up to the first degree of approximation, we have established the bias and mean square error of the proposed estimator. A comparison study is conducted to determine the efficiency conditions under which the proposed family of estimators outperforms the other conventional estimators. A simulation study is also conducted to test the theoretical results using real and three hypothetical populations, and it is obvious from the simulation study that the PRE of the proposed family of estimators $T_{\mathrm{HS}}^{(j)}$; $j=1,2,...,6$ performs better than all the other estimators. As a result, in practice, we prefer the proposed class of estimators.

Disclosure statement

All authors have no conflict of interest to declare.

Appendix

Appendix

Table A.1. Existing estimators with their mean square errors under SRS.

Existing estimator(s)	Mean squared error (MSE)/Optimum mean squared error (OMSE)
Usual unbiased estimator	$\mathrm{Var}(T_0)=S_y^4{\delta }_{40}$
$T_0=s_y^2$
Das and Tripathi (1978)	$\mathrm{minMSE}(T_i)=S_y^4({\delta }_{40}-\frac{{\delta }_{22}^2}{{\delta }_{04}})$; $i=1,2$
$T_1=s_y^2(\frac{S_x^2}{s_x^2}{)}^{{\alpha }^{\ast }}$
$T_2=s_y^2(\frac{S_x^2}{S_x^2+{\alpha }^{\ast }(s_x^2-S_x^2)})$	at ${\alpha }_{\mathrm{opt}.}^{\ast }=-{\delta }_{22}/{\delta }_{04}$
where α^* is a duly opted scalar.
Isaki (1983)
Ratio: $T_3=s_y^2(\frac{S_x^2}{s_x^2})$	$\mathrm{MSE}(T_3)=S_y^4({\delta }_{40}+{\delta }_{04}-2{\delta }_{22})$
Regression: $T_4=s_y^2+{\beta }^{\ast }(S_x^2-s_x^2)$	$\mathrm{minMSE}(T_4)=S_y^4({\delta }_{40}-\frac{{\delta }_{22}^2}{{\delta }_{04}})$
where β^* is the regression coefficient	at ${\beta }_{\mathrm{opt}.}^{\ast }=S_y^2{\delta }_{22}/S_x^2{\delta }_{04}$
of Y and X.
Singh et al. (1988)	$\mathrm{minMSE}(T_5)=\frac{\mathrm{MSE}(T_4)}{1+S^{-4}\mathrm{MSE}(T_4)}$
$T_5=d_1{s}_y^2+d_2(S_x^2-s_x^2)$	at $d_{1(\mathrm{opt}.)}={\delta }_{04}/({\delta }_{04}+{\delta }_{04}{\delta }_{40}-{\delta }_{22}^2)$ and
where d1 and d2 are duly	$d_{2(\mathrm{opt}.)}=S_x^2{\delta }_{04}/(S_y^2({\delta }_{04}+{\delta }_{04}{\delta }_{40}-{\delta }_{22}^2))$
opted scalers.
Upadhyaya and Singh (1999)	$\mathrm{MSE}(T_i)=S_y^4({\delta }_{40}+{\psi }_i^2{\delta }_{04}-2{\psi }_i{\delta }_{22})$; $i=6,7,...,13$
$T_6=s_y^2(\frac{S_x^2+{\beta }_2(x)}{s_x^2+{\beta }_2(x)})$
Kadilar and Cingi (2006)	where ${\psi }_6=S_x^2/(S_x^2+{\beta }_2(x))$, ${\psi }_7=S_x^2/(S_x^2-C_x)$,
$T_7=s_y^2(\frac{S_x^2-C_x}{s_x^2-C_x})$	${\psi }_8=S_x^2/(S_x^2-{\beta }_2(x))$, ${\psi }_9={\beta }_2(x)S_x^2/({\beta }_2(x)S_x^2-C_x)$,
$T_8=s_y^2(\frac{S_x^2-{\beta }_2(x)}{s_x^2-{\beta }_2(x)})$	${\psi }_{10}=C_x{S}_x^2/(C_x{S}_x^2-{\beta }_2(x))$, ${\psi }_{11}=S_x^2/(S_x^2+C_x)$
$T_9=s_y^2(\frac{S_x^2{\beta }_2(x)-C_x}{s_x^2{\beta }_2(x)-C_x})$	${\psi }_{12}={\beta }_2(x)S_x^2/({\beta }_2(x)S_x^2+C_x)$ and
$T_{10}=s_y^2(\frac{S_x^2{C}_x-{\beta }_2(x)}{s_x^2{C}_x-{\beta }_2(x)})$	${\psi }_{13}=C_x{S}_x^2/(C_x{S}_x^2+{\beta }_2(x))$.
$T_{11}=s_y^2(\frac{S_x^2+C_x}{s_x^2+C_x})$
$T_{12}=s_y^2(\frac{S_x^2{\beta }_2(x)+C_x}{s_x^2{\beta }_2(x)+C_x})$
$T_{13}=s_y^2(\frac{S_x^2{C}_x+{\beta }_2(x)}{s_x^2{C}_x+{\beta }_2(x)})$
Shabbir and Gupta (2007)
$T_{14}=\{m_1{s}_y^2+m_2(S_y^2-s_y^2)\}\exp (\frac{S_y^2-s_y^2}{S_y^2+s_y^2})$	$\mathrm{minMSE}(T_{14})=\frac{S_y^4}{64}\{\frac{-{\delta }_{04}^2-16({\delta }_{40}-\frac{{\delta }_{22}^2}{{\delta }_{04}})({\delta }_{04}-4)}{1+{\delta }_{40}-\frac{{\delta }_{22}^2}{{\delta }_{04}}}\}$
where m1 and m2 are duly opted scalers.	at $m_{1(\mathrm{opt}.)}=\frac{{\delta }_{04}}{8}(\frac{8-{\delta }_{04}}{{\delta }_{04}+{\delta }_{04}{\delta }_{40}-{\delta }_{22}^2})$ and
	$m_{2(\mathrm{opt}.)}=\frac{S_y^2}{8S_x^2}(\frac{-4{\delta }_{04}+{\delta }_{04}+8{\delta }_{22}-{\delta }_{22}{\delta }_{04}+4{\delta }_{04}{\delta }_{40}-4{\delta }_{22}^2}{{\delta }_{04}+{\delta }_{04}{\delta }_{40}-{\delta }_{22}^2})$
Singh and Solanki (2013)	$\mathrm{minMSE}(T_{15})=\{S_y^4-\frac{A{E}^2+B{D}^2-\mathrm{CDE}}{\mathrm{AB}-C^2}\}$ when
$T_{15}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{a{S}_x^2+b}{a{s}_x^2+b})$
For a = 1, b = 0
$T_{15}^{(1)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{S_x^2}{s_x^2})$
For a = N, $b=-S_x^2$
$T_{15}^{(2)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{N{S}_x^2-S_x^2}{N{s}_x^2-S_x^2})$
For a = N, $b=-{\mu }_x^2$
$T_{15}^{(3)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{N{S}_x^2-{\mu }_x^2}{N{s}_x^2-{\mu }_x^2})$	${\phi }_{1(\mathrm{opt}.)}=(\mathrm{BD}+\mathrm{CE})/(\mathrm{AB}-C^2)$ and
For a = n, $b=-S_x^2$	${\phi }_{2(\mathrm{opt}.)}=(\mathrm{AE}+\mathrm{CD})/(\mathrm{AB}-C^2)$
$T_{15}^{(4)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{n{S}_x^2-S_x^2}{n{s}_x^2-S_x^2})$	where $A=S_y^4(1+{\delta }_{40}+3{\nu }^2{\delta }_{04}-4\nu {\delta }_{22})$,
For a = N, b = μx	$B=S_x^4{\delta }_{04}$, $C=S_x^2{S}_y^2({\delta }_{22}-2\nu {\delta }_{04})$,
$T_{15}^{(5)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{N{S}_x^2-{\mu }_x}{N{s}_x^2-{\mu }_x})$	$D=S_y^4(1+{\nu }^2{\delta }_{04}-\nu {\delta }_{22})$,
For a = n, $b=-{\mu }_x^2$	$E=S_x^2{S}_y^2\nu {\delta }_{04}$ and $\nu =a{S}_x^2/(a{S}_x^2+b)$.
$T_{15}^{(6)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{n{S}_x^2-{\mu }_x^2}{n{s}_x^2-{\mu }_x^2})$
For a = n^2, $b=-{\mu }_x^2$
$T_{15}^{(7)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{n^2{S}_x^2-{\mu }_x^2}{n^2{s}_x^2-{\mu }_x^2})$
For a = n^2, $b=-S_x^2$
$T_{15}^{(8)}=\{{\phi }_1{s}_y^2+{\phi }_2(S_y^2-s_y^2)\}(\frac{n^2{S}_x^2-S_x^2}{n^2{s}_x^2-S_x^2})$
where φ1 and φ2 are duly opted scalers.
And a and b are either real values or function of known
parameters of variable X namely, population mean μx,
population coefficient of variation Cx,
population coefficient of kurtosis β2(x) etc.
Yadav and Kadilar (2014)	$\mathrm{MSE}(T_{16})=S_y^4({\delta }_{40}+{\delta }_{04}-2{\delta }_{22}+16{\theta }_1{\beta }_1{\delta }_{04}(1-{\theta }_1-$
$T_{16}=s_y^4[{\theta }_1\{\frac{(1-{\beta }_1)s_x^2+{\beta }_1{S}_x^2}{{\beta }_1{s}_x^2+(1-{\beta }_1)S_x^2}\}$
${\beta }_1+{\theta }_1{\beta }_1)(1-{\theta }_1)\{\frac{{\beta }_1{s}_x^2+(1-{\beta }_1)S_x^2}{(1-{\beta }_1)s_x^2+{\beta }_1{S}_x^2}\}]$	$+4{\delta }_{22}({\theta }_1-{\beta }_1{)}^2+4{\delta }_{04}({\theta }_1^2+{\beta }_1^2-{\theta }_1-{\beta }_1))$
where θ1 and β1 are duly opted scalers.
Yaqub and Shabbir (2016)	$\mathrm{minMSE}(T_{17})=\frac{S_y^4}{16}\{\frac{64(1-{\delta }_{04})S_y^{-4}\mathrm{MSE}(T_4)-{\delta }_{04}^2}{{\delta }_{04}+4(1-{\delta }_{04})+4S_y^{-4}\mathrm{MSE}(T_4)}\}$
$T_{17}=s_y^2\{{\varsigma }_1+{\varsigma }_2(S_x^2-s_x^2)\}(\frac{a{S}_x^2+b}{a{s}_x^2+b})\times$	at ${\varsigma }_{1(\mathrm{opt}.)}=\frac{{\delta }_{04}(1+7(1-{\delta }_{04}))}{2({\delta }_{04}^2+4{\delta }_{04}(1-{\delta }_{04}+4{\delta }_{04}{\delta }_{40}-4{\delta }_{22}^2))}$
$[\frac{1}{2}\exp \{\frac{a(S_x^2-s_x^2)}{a(S_x^2+s_x^2)+2b}\}+\frac{1}{2}\exp \{\frac{a(s_x^2-S_x^2)}{a(s_x^2+S_x^2)+2b}\}]$
where ς1 and ς2 are duly opted scalers.	${\varsigma }_{2(\mathrm{opt}.)}=\frac{S_y^2({\delta }_{22}+7{\delta }_{22}(1-{\delta }_{04})-8{\delta }_{04}(1-{\delta }_{04})+8{\delta }_{04}{\delta }_{40}-8{\delta }_{22}^2)}{2S_x^2({\delta }_{04}^2+4{\delta }_{04}(1-{\delta }_{04}+4{\delta }_{04}{\delta }_{40}-4{\delta }_{22}^2))}$
Singh and Pal (2017)	$\mathrm{minMSE}(T_{18})=S_y^4\{1-\frac{(c_2-2c_3{c}_4+c_1{c}_4^2)}{c_1{c}_2-c_3^2}\}$
$T_{18}={\mathrm{\Theta }}_1{s}_y^2+{\mathrm{\Theta }}_2{s}_y^2(\frac{S_x^2}{s_x^2})\tau$	at ${\mathrm{\Theta }}_{1(\mathrm{opt}.)}=(c_2-c_3{c}_4)/(c-c_2-c_3^2)$ and
where $\tau =(1+{\gamma }^{\ast }{\rho }_{\mathrm{yx}}{C}_x{C}_y)/(+{\gamma }^{\ast }{C}_x^2)$; ${\gamma }^{\ast }=1/n$;	${\mathrm{\Theta }}_{2(\mathrm{opt}.)}=(c_1{c}_4-c_3)/\tau (c-c_2-c_3^2)$
Θ1 and Θ2 are duly opted scalers	where $c_1=1+{\delta }_{40}$, $c_2=1+{\delta }_{40}+3{\delta }_{04}-4{\delta }_{22}$,
whose sum need not be unity.	$c_3=1+{\delta }_{40}+{\delta }_{04}-2{\delta }_{22}$ and $c_4=1+{\delta }_{04}-{\delta }_{22}$.
Cekim and Kadilar (2020)	$\mathrm{MSE}(T_{19})=S_y^4\{(\log (2)-1{)}^2+(\log (2){)}^2{\delta }_{40}\}$
$T_{19}=s_y^2\log (\frac{a{s}_x^2+b}{a{S}_x^2+b}+\frac{a{S}_x^2+b}{a{s}_x^2+b})$
a and b are either real values or
function of known parameters.
Bhushan et al. (2023)	$\mathrm{minMSE}(T_{20})=S_y^4\{1-\frac{(E_1{E}_5^2+E_2{E}_4^2-2E_3{E}_4{E}_5)}{(E_1{E}_2-E_3^2)}\}$
$T_{20}=[w_1{s}_y^2+w_2{s}_y^2(\frac{a{S}_x^2+b}{\theta (a{s}_x^2+b)+(1-\theta )(a{S}_x^2+b)}{)}^g]$
$\times [1+\log (\frac{a{s}_x^2+b}{a{S}_x^2+b}){]}^{\eta }$
(g = 1 and θ = 1 are fixed)
For a = 1, b = 0
$T_{20}^{(1)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{S_x^2}{s_x^2})]\times [1+\log (\frac{s_x^2}{S_x^2}){]}^{\eta }$
For a = 1, b = β2(x)
$T_{20}^{(2)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{S_x^2+{\beta }_2(x)}{s_x^2+{\beta }_2(x)})]$
$\times [1+\log (\frac{s_x^2+{\beta }_2(x)}{S_x^2+{\beta }_2(x)}){]}^{\eta }$
For a = 1, b = Cx
$T_{20}^{(3)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{S_x^2+C_x}{s_x^2+C_x})]$
$\times [1+\log (\frac{s_x^2+C_x}{S_x^2+C_x}){]}^{\eta }$
For a = Cx, b = β2(x)
$T_{20}^{(4)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{C_x{S}_x^2+{\beta }_2(x)}{C_x{s}_x^2+{\beta }_2(x)})]$	at ${\eta }_{(\mathrm{opt}.)}=-\frac{{\delta }_{22}}{\nu {\delta }_{04}}$, $w_{1(\mathrm{opt}.)}=\frac{(E_2{E}_4-E_3{E}_5)}{(E_1{E}_2-E_3^2)}$
$\times [1+\log (\frac{C_x{s}_x^2+{\beta }_2(x)}{C_x{S}_x^2+{\beta }_2(x)}){]}^{\eta }$	and $w_{2(\mathrm{opt}.)}=\frac{(E_1{E}_5-E_3{E}_4)}{(E_1{E}_2-E_3^2)}$.
For a = β2(x), b = Cx	where $E_1=1+{\delta }_{40}+(2{\eta }^2{\nu }^2-2\eta {\nu }^2){\delta }_{04}+4\mathrm{\eta \nu }{\delta }_{22}$,
$T_{20}^{(5)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{{\beta }_2(x)S_x^2+C_x}{{\beta }_2(x)s_x^2+C_x})]$	$E_2=1+{\delta }_{40}+\{2{\eta }^2{\nu }^2-2\eta {\nu }^2+g^2{\theta }^2{\nu }^2-4\mathrm{\eta g\theta }{\nu }^2$
$\times [1+\log (\frac{{\beta }_2(x)s_x^2+C_x}{{\beta }_2(x)S_x^2+C_x}){]}^{\eta }$	$+g(g+1){\theta }^2{\nu }^2\}{\delta }_{04}+4\nu (\eta -\mathrm{g\theta }){\delta }_{22}$,
For a = μx, b = ρyx	$E_3=1+{\delta }_{40}+\{2{\eta }^2{\nu }^2-2\eta {\nu }^2-2\eta {\nu }^2+$
$T_{20}^{(6)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{{\mu }_x{S}_x^2+{\rho }_{\mathrm{yx}}}{{\mu }_x{s}_x^2+{\rho }_{\mathrm{yx}}})]$	$\frac{g(g+1)}{2}{\theta }^2{\nu }^2\}{\delta }_{04}2\nu (2\eta -\mathrm{g\theta }){\delta }_{22}$,
$\times [1+\log (\frac{{\mu }_x{s}_x^2+{\rho }_{\mathrm{yx}}}{{\mu }_x{S}_x^2+{\rho }_{\mathrm{yx}}}){]}^{\eta }$	$E_4=1+(\frac{{\eta }^2}{2}{\nu }^2-\eta {\nu }^2){\delta }_{04}+\mathrm{\eta \nu }{\delta }_{22},$
For a = μx, $b=S_x^2$	$E_5=1+(\frac{{\eta }^2}{2}{\nu }^2-\eta {\nu }^2-\mathrm{\eta g\theta }{\nu }^2+\frac{g(g+1)}{2}{\theta }^2{\nu }^2){\delta }_{04}$
$T_{20}^{(7)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{{\mu }_x{S}_x^2+S_x^2}{{\mu }_x{s}_x^2+S_x^2})]$	$+(\mathrm{\eta \nu }-\mathrm{g\theta \nu }){\delta }_{22}$.
$\times [1+\log (\frac{{\mu }_x{s}_x^2+S_x^2}{{\mu }_x{S}_x^2+S_x^2}){]}^{\eta }$
For $a=C_x^2$, b = ρyx
$T_{20}^{(8)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{C_x^2{S}_x^2+{\rho }_{\mathrm{yx}}}{C_x^2{s}_x^2+{\rho }_{\mathrm{yx}}})]$
$\times [1+\log (\frac{C_x^2{s}_x^2+{\rho }_{\mathrm{yx}}}{C_x^2{S}_x^2+{\rho }_{\mathrm{yx}}}){]}^{\eta }$
For a = μx, b = β2(x)
$T_{20}^{(9)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{{\mu }_x{S}_x^2+{\beta }_2(x)}{{\mu }_x{s}_x^2+{\beta }_2(x)})]$
$\times [1+\log (\frac{{\mu }_x{s}_x^2+{\beta }_2(x)}{{\mu }_x{S}_x^2+{\beta }_2(x)}){]}^{\eta }$
For a = β2(x), b = μx
$T_{20}^{(10)}=[w_1{s}_y^2+w_2{s}_y^2(\frac{{\beta }_2(x)S_x^2+{\mu }_x}{{\beta }_2(x)s_x^2+{\mu }_x})]$
$\times [1+\log (\frac{{\beta }_2(x)s_x^2+{\mu }_x}{{\beta }_2(x)S_x^2+{\mu }_x}){]}^{\eta }$
where θ, g are scalers which assume real values to design different estimators whereas w1, w2 and η are duly opted scalers.