Modified Estimator of Finite Population Variance under Stratified Random Sampling ^†

Abstract

This paper proposes a generalized estimator of finite population variance using the auxiliary information under stratified random sampling. The expressions for bias and mean square error equations of the proposed estimator up to the first degree of approximation are derived. The theoretical efficiency conditions under which the proposed estimator is better than some existing estimators are obtained. The performances of the existing and proposed estimators were assessed using three real datasets based on the criteria of minimum mean square error and supreme percentage relative efficiency. Evidence from the study showed that the proposed estimator performed better and was more efficient than some existing estimators considered.

1. Introduction

Researchers have been considering the use of auxiliary information in various forms to construct more accurate estimators for population parameters in experimental surveys. This approach has been shown to significantly improve the accuracy of the population mean estimation, and it has been explored by several researchers, including [1,2,3]. The field of survey sampling has seen a significant amount of research devoted to developing estimators of population variance to measure variations that exist in real-world scenarios. These variations can arise in various industries, such as manufacturing, pharmaceuticals, agriculture, and biological experiments. Researchers like [4,5,6,7,8,9,10,11,12,13,14] have worked extensively in this area. In real-life situations, population units of variables of interest can have diverse features, which makes the population units heterogeneous. In such cases, estimators of finite population variance, based on simple random sampling, cannot be applied. Therefore, the stratification approach is used to provide precise results when population units are heterogeneous [5,15,16,17].

Most ratio-based and product-based estimators are not efficient in estimating population characteristics when there is a negative correlation between the study and auxiliary variables for ratio-based estimators or a positive correlation for product-based estimators. To address this issue under stratified random sampling, a new generalized ratio-product cum regression-type estimator is proposed. The generalized ratio-product cum regression-type estimator proposed in this study is flexible in providing better efficient estimates than existing estimators when the data exhibits either a negative or positive correlation between the study and auxiliary variables.

Following the introduction is Section 2, which contains the notations and the literature review, while Section 3 presents the methodology of the study. Section 4 presents the efficiency conditions of the proposed methodology. Section 5 discusses the results while the conclusion and recommendations are presented in Section 6.

2. Notations and Some Existing Estimators

Consider a finite population $U=\left\{U_1,\dots ,U_N\right\}$ of size N which is divided into L non-overlapping group or strata, with each stratum containing $N_h(h=\mathrm{1,2},\dots ,L)$ units, whose union gives N units, i.e., ${\sum }_{h=1}^G{N}_h=N.$A sample using simple random sample of size $n_h$ is drawn without replacement (SRSWOR) from the strata population $N_h$ such that${\sum }_{h=1}^G{n}_h=n$. Let ($y_{hi},x_{hi}$) be the value of the study variable Y and the auxiliary variable X on $ith$ unit$U_i,i=1,\dots N$. Let${\overline{Y}}_h$, and ${\overline{X}}_h$ be population means corresponding to the sample means ${\overline{y}}_h$and${\overline{x}}_h$, respectively, in each stratum. Let $s_{y\left(h\right)}^2$ and $s_{x\left(h\right)}^2$ be the sample mean squares for the study and auxiliary variables for the $hth$ strata, respectively, and let $S_{y\left(h\right)}^2$ and $S_{x\left(h\right)}^2$ be the population mean squares for the study and auxiliary variables for the $hth$ strata, respectively. Let $W_h=\frac{N_h}{N}$ be the proportion of the population units falling in the $hth$ stratum. Let the correlation coefficient between the study variable and the auxiliary variable for the $hth$ strata be${\rho }_{yx\left(h\right)}$. Also, let $C_{y\left(h\right)}=S_{y\left(h\right)}/{\overline{Y}}_h$ and $C_{x\left(h\right)}=S_{x\left(h\right)}/{\overline{X}}_h$ be the stratum coefficients of variation of the study variable Y and the auxiliary variable X. Let ${\beta }_{2\left(y\right)}={\psi }_{40}$ and ${\beta }_{2\left(x\right)}={\psi }_{04}$be the coefficients of kurtosis of y and x for the $hth$ strata, respectively.

$${\beta }_{2\left(y\right)}={\psi }_{40\left(h\right)}=\frac{{\mu }_{40\left(h\right)}}{{\mu }_{20\left(h\right)}^2},{\beta }_{2\left(x\right)}={\psi }_{04\left(h\right)}=\frac{{\mu }_{04\left(h\right)}}{{\mu }_{02\left(h\right)}^2},{\psi }_{22\left(h\right)}=\frac{{\mu }_{22\left(h\right)}}{{\mu }_{20\left(h\right)}{\mu }_{02\left(h\right)}},{\overline{Y}}_h=\frac{1}{N_h}{\sum }_{i=1}^{N_h}{Y}_{hi},$$

$${\overline{X}}_h=\frac{1}{N_h}{\sum }_{i=1}^{N_h}{X}_{hi},{\overline{y}}_h=\frac{1}{n_h}{\sum }_{i=1}^{n_h}{y}_{hi},{\overline{x}}_h=\frac{1}{n_h}{\sum }_{i=1}^{n_h}{x}_{hi},s_{yh}^2=\frac{1}{\left(n_h-1\right)}{\sum }_{i=1}^{n_h}{\left(y_{hi}-{\overline{y}}_h\right)}^2,$$

$$\begin{gathered} s_{xh}^2=\frac{1}{\left(n_h-1\right)}{\sum }_{i=1}^{n_h}{\left(x_{hi}-{\overline{x}}_h\right)}^2,S_{Y_h}^2=\frac{1}{\left(N_h-1\right)}{\sum }_{i=1}^{N_h}{\left(Y_{hi}-{\overline{Y}}_h\right)}^2,S_{X_h}^2=\frac{1}{\left(N_h-1\right)}{\sum }_{i=1}^{N_h}{\left(X_{hi}-{\overline{X}}_h\right)}^2,{\gamma }_h=\frac{1}{n_h}-\frac{1}{N_h}, \\ {\mu }_{rs\left(h\right)}=\frac{1}{N_h}{\sum }_{j=1}^{N_h}{\left(Y_{hj}-{\overline{Y}}_h\right)}^r{\left(X_{hj}-{\overline{X}}_h\right)}^s. \end{gathered}$$

The usual unbiased population variance estimator under stratified random sampling can be used to estimate the population variance and is given as:

$$s_{y_{st}}^2={\displaystyle \sum _{h=1}^L{W}_h^2{\gamma }_h}{s}_{y_h}^2$$

(1)

$$Var\left(s_{y_{st}}^2\right)={\displaystyle \sum _{h=1}^L{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left({\psi }_{40(h)}-1\right)$$

(2)

The usual ratio stratified estimator for estimating population variance proposed by [5] is given as:

$$T_{r(st)}={\displaystyle \sum _{h=1}^G{W}_h^2}{\gamma }_h\left(s_{y_h}^2\frac{S_{X_h}^2}{s_{x_h}^2}\right)$$

(3)

$$Bias(T_{r(st)})={\displaystyle \sum _{h=1}^G{W}_h^2}{\gamma }_h^2{S}_{y_h}^2\left[\left({\psi }_{40(h)}-1\right)-\left({\theta }_{22(h)}-1\right)\right]$$

(4)

$$MSE(T_{r(st)})\cong {\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left[\left({\psi }_{40(h)}-1\right)+\left({\psi }_{04(h)}-1\right)-2\left({\theta }_{22(h)}-1\right)\right]$$

(5)

The usual regression stratified estimator for estimating population variance proposed by [5] is given as:

$${\widehat{S}}_{reg(st)}^2={\displaystyle \sum _{h=1}^L{W}_h^2{\gamma }_h}\left[s_{y(h)}^2+b_h\left(S_{x(h)}^2-s_{x(h)}^2\right)\right]$$

(6)

$$Var({\widehat{S}}_{reg(st)}^2)={\displaystyle \sum _{h=1}^L{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left({\psi }_{40(h)}-1\right)\left(1-{\rho }_h^2\right)$$

(7)

Also, [15] proposed a difference-type estimator of population variance under stratified random sampling as follows:

$${\widehat{S}}_{D(st)}^2={\displaystyle \sum _{h=1}^G{W}_h^2}{\gamma }_h\left[k_{1h}{s}_{yh}^2+k_{2h}\left(S_{xh}^2-s_{xh}^2\right)\right]$$

(8)

$$Bias{\left({\widehat{S}}_{D(st)}^2\right)}_{\min }\cong {\displaystyle \sum _{h=1}^G{W}_h^2}{\gamma }_h^2{S}_{y_h}^2\left\{\frac{\left({\psi }_{04(h)}-1\right)}{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]}+\left({\psi }_{04(h)}-1\right)-1\right\}$$

(9)

$$MSE{\left({\widehat{S}}_{D(st)}^2\right)}_{\min }\cong {\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left\{\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-\frac{{\left({\theta }_{22(h)}-1\right)}^2}{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]}+\left({\psi }_{04(h)}-1\right)\right\}$$

(10)

where

$$k_{1h}^{(opt)}=\frac{\left({\psi }_{04(h)}-1\right)}{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]+\left({\psi }_{04(h)}-1\right)}$$

(11)

$$k_{2h}^{(opt)}=\frac{S_{y_h}^2\left({\theta }_{22(h)}-1\right)}{S_{x_h}^2\left\{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]+\left({\psi }_{04(h)}-1\right)\right\}}$$

(12)

The ratio-type and product-type estimators of finite population variance under stratified random sampling proposed by [17] is as follows:

$${\widehat{S}}_{SR(st)}^2={\displaystyle \sum _{h=1}^L{W}_h^2{\gamma }_h}{s}_{y_h}^2\exp \left(\frac{S_{x_h}^2-s_{x_h}^2}{S_{x_h}^2+s_{x_h}^2}\right)$$

(13)

$${\widehat{S}}_{SP(st)}^2={\displaystyle \sum _{h=1}^L{W}_h^2{\gamma }_h}{s}_{y_h}^2\exp \left(\frac{s_{x_h}^2-S_{x_h}^2}{S_{x_h}^2+s_{x_h}^2}\right)$$

(14)

$$Bias\left({\widehat{S}}_{SR(st)}^2\right)={\displaystyle \sum _{h=1}^L{W}_h^2{\gamma }_h^2}{S}_{y_h}^2\left[\frac{3}{8}\left({\psi }_{04(h)}-1\right)-\frac{1}{2}\left({\theta }_{22(h)}-1\right)\right]$$

(15)

$$Bias\left({\widehat{S}}_{SP(st)}^2\right)={\displaystyle \sum _{h=1}^L{W}_h^2{\gamma }_h^2}{S}_{y_h}^2\left[\frac{1}{2}\left({\theta }_{22(h)}-1\right)-\frac{1}{8}\left({\psi }_{04(h)}-1\right)\right]$$

(16)

$$MSE\left({\widehat{S}}_{SR(st)}^2\right)={\displaystyle \sum _{h=1}^L{W}_h^4{\gamma }_h^3}{S}_{y_h}^4\left[\left({\psi }_{40(h)}-1\right)+\frac{1}{4}\left({\psi }_{04(h)}-1\right)-\left({\theta }_{22(h)}-1\right)\right]$$

(17)

$$MSE\left({\widehat{S}}_{SP(st)}^2\right)={\displaystyle \sum _{h=1}^L{W}_h^4{\gamma }_h^3}{S}_{y_h}^4\left[\left({\psi }_{40(h)}-1\right)+\frac{1}{4}\left({\psi }_{04(h)}-1\right)+\left({\theta }_{22(h)}-1\right)\right]$$

(18)

An improved population variance estimator under stratified random sampling proposed by [16] is given as follows:

$${\widehat{S}}_{SG(st)}^2={\displaystyle \sum _{h=1}^G{W}_h^2}{\gamma }_h\left[{\omega }_{1h}{s}_{yh}^2+{\omega }_{2h}\left(S_{xh}^2-s_{xh}^2\right)\right]\exp \left(\frac{S_{x_h}^2-s_{x_h}^2}{S_{x_h}^2+s_{x_h}^2}\right)$$

(19)

$$Bias\left({\widehat{S}}_{SG(st)}^2\right)\cong {\displaystyle \sum _{h=1}^G{W}_h^2}{\gamma }_h^2\left[\begin{array}{l}{S}_{y_h}^2+{\omega }_{1h}{S}_{y_h}^2\left\{1+\frac{3}{8}\left({\psi }_{40(h)}-1\right)-\frac{1}{2}\left({\theta }_{22(h)}-1\right)\right\}\\ +\frac{1}{2}{\omega }_{2h}{S}_{x_h}^2{\gamma }_h\left({\psi }_{04(h)}-1\right)\end{array}\right]$$

(20)

$$MSE{\left({\widehat{S}}_{SG(st)}^2\right)}_{\min }\cong {\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left\{\frac{\begin{array}{l}64\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]-{\gamma }_h{\left({\psi }_{04(h)}-1\right)}^3\\ -16{\gamma }_h\left({\psi }_{04(h)}-1\right)\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]\end{array}}{64\left\{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]+\left({\psi }_{40(h)}-1\right)\right\}}+\left({\psi }_{04(h)}-1\right)\right\}$$

(21)

where ${\omega }_{1h}$ and ${\omega }_{2h}$ are suitable chosen constants with optimum values:

$${\omega }_{1h}^{(opt)}=\frac{{\gamma }_h\left({\psi }_{04(h)}-1\right)}{8}\left[\frac{8-{\gamma }_h\left({\psi }_{04(h)}-1\right)}{{\gamma }_h\left[\left({\psi }_{04(h)}-1\right)+\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]}\right]$$

(22)

$${\omega }_{2h}^{(opt)}=\frac{S_{y_h}^2}{8S_{x_h}^2}\left[\frac{\left({\psi }_{40(h)}-1\right)\left(4\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2-3\right)+4{\left({\theta }_{22(h)}-1\right)}^2}{\left[\left({\psi }_{04(h)}-1\right)+\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]}\right]$$

(23)

3. The Proposed Estimator

Motivated by [14] estimators and having studied some estimators under stratified random sampling, the following generalized estimator is proposed to estimate the population variance of the variable of interest by utilizing the auxiliary information.

$$G_{(st)}={\displaystyle \sum _{h=1}^L{W}_h^2}{\lambda }_h\left\{{\tau }_{1(h)}{s}_{y(h)}^2\left[\frac{1}{2}\left(\frac{S_{x(h)}^2}{s_{x(h)}^2}+\frac{s_{x(h)}^2}{S_{x(h)}^2}\right)\right]+{\tau }_{2(h)}\left(S_{x(h)}^2-s_{x(h)}^2\right)\right\}\exp \left\{\frac{S_{x(h)}^2-s_{x(h)}^2}{S_{x(h)}^2+s_{x(h)}^2}\right\}$$

(24)

where ${\tau }_{i\left(h\right)}(i=\mathrm{1,2})$ are unknown constants whose values to be determined later, and $\sigma$ is a suitably chosen constant.

In order to find the bias and MSE of the estimators, we define the following relative error terms:

$$e_{0(h)}=\frac{s_{y(h)}^2-S_{y(h)}^2}{S_{y(h)}^2}\mathrm{and}{e}_{1(h)}=\frac{s_{x(h)}^2-S_{x(h)}^2}{S_{x(h)}^2},\mathrm{where}{s}_{y(h)}^2=S_{y(h)}^2\left(1+e_{0(h)}\right)\mathrm{and}{s}_{x(h)}^2=S_{x(h)}^2\left(1+e_{1(h)}\right)$$

such that

$$E\left(e_{0(h)}\right)=E\left(e_{1(h)}\right)=0,E\left(e_{0(h)}^2\right)={\gamma }_h\left({\psi }_{40(h)}-1\right),E\left(e_{1(h)}^2\right)={\gamma }_h\left({\psi }_{04(h)}-1\right)\mathrm{and}E\left(e_{0(h)}{e}_{1(h)}\right)={\gamma }_h\left({\psi }_{22(h)}-1\right).$$

Expressing the estimator $G_{i\left(st\right)}$ in terms of $e_i(i=0,1)$ and by taking the expectation, the bias of the proposed generalized estimator, up to the first order of approximation, is given as:

$$Bias\left(G_{(st)}\right)={\displaystyle \sum _{h=1}^G\frac{W_h^2}{{\lambda }_h^{-1}}}\left[\begin{array}{l}\left({\tau }_{1(h)}-1\right)S_{y(h)}^2+{\tau }_{1(h)}{S}_{y(h)}^2\left(\frac{7}{8}{\lambda }_h\left({\psi }_{04(h)}-1\right)-\frac{1}{2}{\lambda }_h\left({\psi }_{22(h)}-1\right)\right)\\ +\frac{{\tau }_{2(h)}}{2}{S}_{x(h)}^2{\lambda }_h\left({\psi }_{04(h)}-1\right)\end{array}\right]$$

(25)

The mean square error of the proposed estimator, up to the first order of approximation, is given as:

$$MSE\left(G_{(st)}\right)={\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}\left\{\begin{array}{l}{\left({\tau }_{1(h)}-1\right)}^2{S}_{y(h)}^4+{\tau }_{1(h)}^2{S}_{y(h)}^4{\lambda }_h\left[\left({\psi }_{40(h)}-1\right)-2\left({\psi }_{22(h)}-1\right)+2\left({\psi }_{04(h)}-1\right)\right]\\ +{\tau }_{2(h)}^2{S}_{x(h)}^4{\lambda }_h\left({\psi }_{04(h)}-1\right)-{\tau }_{1(h)}{S}_{y(h)}^4{\lambda }_h\left(\frac{7}{4}\left({\psi }_{04(h)}-1\right)-\left({\psi }_{22(h)}-1\right)\right)\\ -2{\tau }_{1(h)}{\tau }_{2(h)}{S}_{y(h)}^2{S}_{x(h)}^2{\lambda }_h\left(\left({\psi }_{22(h)}-1\right)-\left({\psi }_{04(h)}-1\right)\right)-{\tau }_{2(h)}{S}_{y(h)}^2{S}_{x(h)}^2{\lambda }_h\left({\psi }_{04(h)}-1\right)\end{array}\right\}$$

(26)

Minimizing Equation (26) with respect to ${\tau }_{1(h)}$ and ${\tau }_{2(h)}$, we obtained the optimum values of ${\tau }_{1(h)}$ and ${\tau }_{2(h)}$ as:

$${\tau }_{1(h)}^{\left(opt\right)}=\frac{1+\frac{3}{8}{\lambda }_h\left({\psi }_{04(h)}-1\right)}{1+{\lambda }_h\left({\psi }_{04(h)}-1\right)+{\lambda }_h\left(\left({\psi }_{40(h)}-1\right)-\frac{{\left({\psi }_{22(h)}-1\right)}^2}{\left({\psi }_{04(h)}-1\right)}\right)}$$

(27)

and

$${\tau }_{2(h)}^{\left(opt\right)}=\frac{S_{y(h)}^2}{S_{x(h)}^2}\left\{\frac{1}{2}-{\tau }_{1(h)}^{\left(opt\right)}\left(1-\frac{\left({\psi }_{22(h)}-1\right)}{\left({\psi }_{04(h)}-1\right)}\right)\right\}$$

(28)

Substituting the optimum values of ${\tau }_{1(h)}$ and ${\tau }_{2(h)}$ into Equation (26), we obtained the minimum mean square error of $G_{i(st)}$ as:

$$MSE{\left(G_{(st)}\right)}_{\min }={\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{\left(1-\frac{{\lambda }_h\left({\psi }_{04(h)}-1\right)}{4}\right)-\frac{{\left[1+\frac{3}{8}{\lambda }_h\left({\psi }_{04(h)}-1\right)\right]}^2}{\left[1+{\lambda }_h\left({\psi }_{04(h)}-1\right)+{\lambda }_h\left(\left({\psi }_{40(h)}-1\right)-\frac{{\left({\psi }_{22(h)}-1\right)}^2}{\left({\psi }_{04(h)}-1\right)}\right)\right]}\right\}$$

(29)

4. Efficiency Comparisons

In this section, we obtain the theoretical efficiency conditions for the proposed estimator by comparing the mean square error (MSE) equation of the proposed estimator with the MSE equations of some existing estimators. Thus, conditions under which the proposed estimator is more efficient are given below:

• Comparing the proposed estimator’s MSE with that of the usual variance estimator, we have:

$$Var({\widehat{s}}_{yst}^2)-MSE{\left(G_{(st)}\right)}_{\min }>0$$

(30)

Substituting the variances of the usual variance estimator and proposed estimator into Equation (30) results in:

$${\displaystyle \sum _{h=1}^L\frac{W_h^4}{n_h^3}}{S}_{y_h}^4\left({\psi }_{40(h)}-1\right)-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{n_h^2}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(31)

Further simplifying Equation (31), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^L\frac{W_h^4}{n_h^3}}{S}_{y_h}^4\left({\psi }_{40(h)}-1\right)$$

(32)

where $B_{(h)}=\left(1-\frac{{\gamma }_h\left({\psi }_{04(h)}-1\right)}{4}\right)$, $C_{(h)}=1+\frac{3}{8}{\lambda }_h\left({\psi }_{04(h)}-1\right)$ and $D_{(h)}=1+{\lambda }_h\left({\psi }_{04(h)}-1\right)+{\lambda }_h\left(\left({\psi }_{40(h)}-1\right)-\frac{{\left({\theta }_{22(h)}-1\right)}^2}{\left({\psi }_{04(h)}-1\right)}\right)$

Since condition (32) is satisfied, the proposed estimators are more efficient than sample variance.

• Comparing the proposed estimator’s MSE with that of the usual ratio estimator defined by [5], we have:

$$MSE\left(T_{r(st)}\right)-MSE{\left(G_{(st)}\right)}_{\min }>0$$

(33)

Substituting the MSE expressions of the usual ratio estimator and proposed estimator into Equation (33) results in:

$${\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left[\begin{array}{l}\left({\psi }_{40(h)}-1\right)+\left({\psi }_{04(h)}-1\right)\\ -2\left({\theta }_{22(h)}-1\right)\end{array}\right]-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(34)

Further simplifying Equation (34), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left[\begin{array}{l}\left({\psi }_{40(h)}-1\right)+\left({\psi }_{04(h)}-1\right)\\ -2\left({\theta }_{22(h)}-1\right)\end{array}\right]$$

(35)

Since condition (35) is satisfied, the proposed estimators are more efficient than [5]’s stratified regression estimator.

• Comparing the proposed estimator’s MSE with that of the usual stratified regression estimator, we have:

$$Var({\widehat{S}}_{reg(st)}^2)-MSE{\left(G_{(st)}\right)}_{\min }>0$$

(36)

Substituting the MSE expressions of the usual ratio estimator and proposed estimator into Equation (36) results in:

$${\displaystyle \sum _{h=1}^L\frac{W_h^4}{n_h^3}}{S}_{y_h}^4\left({\psi }_{40(h)}-1\right)\left(1-{\rho }_h^2\right)-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(37)

Further simplifying Equation (37), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^L\frac{W_h^4{S}_{y_h}^4}{n_h^3}}\left({\psi }_{40(h)}-1\right)\left(1-{\rho }_h^2\right)$$

(38)

Since condition (38) is satisfied, the proposed estimators are more efficient than [5]’s stratified regression estimator.

• Comparing the proposed estimator’s MSE with that of the stratified estimator [15], we have:

$$MSE{\left({\widehat{S}}_{D(st)}^2\right)}_{\min }-MSE{\left(G_{i(st)}\right)}_{\min }>0$$

(39)

Substituting the MSE expressions of [15]’s estimator and the proposed estimator into Equation (39) results in:

$${\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left\{\begin{array}{l}\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)+\left({\psi }_{04(h)}-1\right)\\ -\frac{{\left({\theta }_{22(h)}-1\right)}^2}{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]}\end{array}\right\}-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(40)

Further simplifying Equation (40), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left\{\begin{array}{l}\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)+\left({\psi }_{04(h)}-1\right)\\ -\frac{{\left({\theta }_{22(h)}-1\right)}^2}{{\gamma }_h\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]}\end{array}\right\}$$

(41)

Since condition (41) is satisfied, the proposed estimators are more efficient than [15]’s stratified estimator.

• Comparing the proposed estimator’s MSE with that of [17]’s ratio-type and product-type estimators, respectively, we have:

$$MSE\left({\widehat{S}}_{SR(st)}^2\right)-MSE{\left(G_{(st)}\right)}_{\min }>0$$

(42)

Substituting the MSE expressions of [17]’s ratio-type estimator and proposed estimator into Equation (42) results in:

$${\displaystyle \sum _{h=1}^L{W}_h^4{\gamma }_h^3}{S}_{y_h}^4\left[\left({\psi }_{40(h)}-1\right)+\frac{1}{4}\left({\psi }_{04(h)}-1\right)-\left({\theta }_{22(h)}-1\right)\right]-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(43)

Further simplifying Equation (43), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^L{W}_h^4{\gamma }_h^3}{S}_{y_h}^4\left[\begin{array}{l}\left({\psi }_{40(h)}-1\right)+\frac{1}{4}\left({\psi }_{04(h)}-1\right)\\ -\left({\theta }_{22(h)}-1\right)\end{array}\right]$$

(44)

Similarly, for [17]’s product-type estimator, we have:

$$MSE\left({\widehat{S}}_{SP(st)}^2\right)-MSE{\left(G_{(st)}\right)}_{\min }>0$$

(45)

Substituting the MSE expressions of [17]’s product-type estimator and the proposed estimator into Equation (45) results in:

$${\displaystyle \sum _{h=1}^L{W}_h^4{\gamma }_h^3}{S}_{y_h}^4\left[\left({\psi }_{40(h)}-1\right)+\frac{1}{4}\left({\psi }_{04(h)}-1\right)+\left({\theta }_{22(h)}-1\right)\right]-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(46)

Further simplifying Equation (46), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^L{W}_h^4{\gamma }_h^3}{S}_{y_h}^4\left[\begin{array}{l}\left({\psi }_{40(h)}-1\right)+\frac{1}{4}\left({\psi }_{04(h)}-1\right)\\ +\left({\theta }_{22(h)}-1\right)\end{array}\right]$$

(47)

Since conditions (44) and (47) are satisfied, the proposed estimator is more efficient than [17]’s ratio-type and product-type stratified estimators, respectively.

• Comparing the proposed estimator’s MSE with that of [16]’s variance estimator, we have:

$$MSE\left({\widehat{S}}_{SG(st)}^2\right)-MSE{\left(G_{(st)}\right)}_{\min }>0$$

(48)

Substituting the MSE expressions of [16]’s estimator and proposed estimator into Equation (48) results in:

$${\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left\{\frac{64K-{\gamma }_h{\left({\psi }_{04(h)}-1\right)}^3-16{\gamma }_h\left({\psi }_{04(h)}-1\right)K}{64{\gamma }_{h}K+\left({\psi }_{40(h)}-1\right)}+\left({\psi }_{04(h)}-1\right)\right\}-{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\left\{B_{(h)}-\frac{C_{(h)}^2}{D_{(h)}}\right\}>0$$

(49)

Further simplifying Equation (49), we have:

$${\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4\frac{C_{(h)}^2}{D_{(h)}}>{\displaystyle \sum _{h=1}^G\frac{W_h^4}{{\lambda }_h^{-2}}}{S}_{y(h)}^4{B}_{(h)}-{\displaystyle \sum _{h=1}^G{W}_h^4}{\gamma }_h^3{S}_{y_h}^4\left\{\frac{\begin{array}{l}64K-{\gamma }_h{\left({\psi }_{04(h)}-1\right)}^3\\ -16{\gamma }_h\left({\psi }_{04(h)}-1\right)K\end{array}}{64{\gamma }_{h}K+\left({\psi }_{40(h)}-1\right)}+\left({\psi }_{04(h)}-1\right)\right\}$$

(50)

where $K=\left[\left({\psi }_{40(h)}-1\right)\left({\psi }_{04(h)}-1\right)-{\left({\theta }_{22(h)}-1\right)}^2\right]$

Since condition (50) is satisfied, the proposed estimator is more efficient than [16]’s variance estimator.

5. Empirical Results

To observe the performance of our proposed estimator with respect to other considered estimators, we use the following datasets for numerical comparisons, which were previously used by many authors in the literature (

Table 1. Source and description of datasets.

Data Set	Source	Y	X
1	[7]	Level of apple production	Rate of apple trees
2	[18]	Area under wheat in the region in 1974	Area under wheat in the region in 1973
3	[19]	Leaf area for the newly developed strain of wheat	Weight of leaves

,

Table 2. Parameters of Dataset 1.

Parameters	${\mathit{N}}_{\mathit{h}}$	${\mathit{n}}_{\mathit{h}}$	${\mathit{S}}_{{\mathit{y}}_{\mathit{h}}}^2$	${\mathit{S}}_{{\mathit{x}}_{\mathit{h}}}^2$	${\mathit{\beta }}_{2\left({\mathit{y}}_{\mathit{h}}\right)}$	${\mathit{\beta }}_{2\left({\mathit{x}}_{\mathit{h}}\right)}$	${\mathit{\lambda }}_{22\left(\mathit{h}\right)}$
Stratum 1	19	10	583,977.5	11.5715804	3.28	1.45	1.46
Stratum 2	32	16	456,563.3	8.139014	1.56	3.09	1.74
Stratum 3	14	7	195,208.8	12.4494617	1.62	1.62	1.80
Stratum 4	15	8	437,923.5	10.334267	2.22	1.90	2.02

,

Table 3. Parameters of Dataset 2.

Parameters	${\mathit{N}}_{\mathit{h}}$	${\mathit{n}}_{\mathit{h}}$	${\mathit{C}}_{\mathit{x}\mathit{h}}$	${\mathit{S}}_{{\mathit{y}}_{\mathit{h}}}^2$	${\mathit{S}}_{{\mathit{x}}_{\mathit{h}}}^2$	${\mathit{\beta }}_{2\left({\mathit{y}}_{\mathit{h}}\right)}$	${\mathit{\beta }}_{2\left({\mathit{x}}_{\mathit{h}}\right)}$	${\mathit{\lambda }}_{22\left(\mathit{h}\right)}$
Stratum 1	9	3	0.1118928	31,978.25	11.5715804	2.9286	2.07	2.38
Stratum 2	10	3	0.0733753	37,629.39	8.139014	1.511	1.42	1.42
Stratum 3	15	4	0.11937840	6893.067	12.4494617	2.20	2.42	2.28

and

Table 4. Parameters of Dataset 3.

Parameters	${\mathit{N}}_{\mathit{h}}$	${\mathit{n}}_{\mathit{h}}$	${\mathit{C}}_{\mathit{x}\mathit{h}}$	${\mathit{S}}_{{\mathit{y}}_{\mathit{h}}}^2$	${\mathit{S}}_{{\mathit{x}}_{\mathit{h}}}^2$	${\mathit{\beta }}_{2\left({\mathit{y}}_{\mathit{h}}\right)}$	${\mathit{\beta }}_{2\left({\mathit{x}}_{\mathit{h}}\right)}$	${\mathit{\lambda }}_{22\left(\mathit{h}\right)}$
Stratum 1	12	4	0.1118928	6.0664603	11.5715804	1.9394547	2.2748233	1.9123464
Stratum 2	13	5	0.0733753	5.2915229	8.139014	2.9819269	3.436904	2.970998
Stratum 3	14	5	0.1193784	6.4961301	12.4494617	2.3448986	2.8955496	2.5134376

).

Table 5. Bias estimates of the existing estimators and the proposed estimator.

	Dataset 1	Dataset 2	Dataset 3
Sample Variance	0	0	0
[5]’s Ratio Estimator	83.79038	−27.989	0.01521998
[5]’s Regression	0	0	0
[15]’s Estimator	21539.92	16941.3	1.338503
[17]’s Ratio	10.84317	−46.103	−0.00092405
[17]’s Product	51.26086	110.319	0.01799213
[16]’s	76.34123	154.855	0.1906794
Proposed Estimator	3.20301640	2.1320047	−0.00200567

and

Table 6. MSE and PRE values of proposed and existing estimators with respect to ${\widehat{S}}_y^2$.

	Dataset 1		Dataset 2		Dataset 3
	MSE	PRE	MSE	PRE	MSE	PRE
Sample Variance	419,607.1	100.00	188,354.1	100.00	0.004715021	100.00
[5]’s Ratio Estimator	220,271.5	190.50	21,269.09	885.58	0.0009897653	476.38
[5]’s Regression	254,877.6	164.63	20,959.36	898.66	0.0008908265	529.29
[15]’s Estimator	1,204,329	34.84	885,096.4	21.28	0.0036798567	128.13
[17]’s Ratio	298,307.4	140.66	62,669.71	300.55	0.0010637556	443.24
[17]’s Product	762,244.1	55.05	292,602.2	64.37	0.008128177	58.01
[16]’s	254,968	164.57	11,841.29	1590.66	0.002745902	171.71
Proposed Estimator	220,094.1	190.65	9616.144	1958.73	0.000173869	2711.82

showed the values of bias, mean square error (MSE), and percentage relative efficiency (PRE) of the proposed estimators and certain existing estimators considered using three datasets. Based on the results obtained from the three datasets, it can be observed that the proposed estimator has the minimum mean square error and the supreme percentage relative efficiency values, respectively, compared to the sample variance estimator [5], variance ratio and regression estimators [15], variance estimator [17], ratio-type and product-type exponential variance estimators, and [16] variance estimator. Therefore, based on the criteria of mean square error and percentage relative efficiency, the proposed estimator performed better and is more efficient than the existing estimators considered.

6. Conclusions

In this study, a new generalized estimator of finite population variance using the auxiliary information in stratified random sampling is proposed. The expressions for bias and mean square error equations of the proposed estimator up to first degree of approximation are derived. The theoretical efficiency conditions under which the proposed estimator is better than some existing estimators are obtained. The proposed estimator is thus empirically compared with certain existing estimators in the literature using real datasets. Evidence from the results revealed that the proposed estimator performs better than the existing estimators considered when compared using mean square error and percentage relative efficiency.