Improved family of estimators for the population mean using supplementary variables under PPS sampling

Abstract

In this article, we suggest an enhanced family of estimators for estimation of population mean employing the supplementary variables under probability proportional to size sampling. Up to the first order of approximation, numerical formulations of the bias and mean square error of estimators are obtained. From our suggested improved family of estimators, we give sixteen different members. The recommended family of estimators has specifically been used to derive the characteristics of sixteen estimators based on the known population parameters of the study as well as auxiliary variables. The performances of the suggested estimators have been assessed using three actual data. Furthermore, a simulation investigation is also accompanied to evaluate the effectiveness of estimators. The proposed estimators have a smaller MSE and an advanced PRE when linked to existing estimators, which are based on actual data sets and simulation studies. Theoretically and empirically studies also reveal that the suggested estimators accomplish well than the usual estimators.

Introduction

In survey sampling, the proper usage of the supplementary variable may increase estimator’s accuracy during both the construction and estimation phases. Supplementary variables are frequently used to enhance accuracy of the estimators. This information may be used at the design stage or estimation stage, or at both stages. A wide range of strategies for employing the supplementary information using ratio, product, and regression methods are described in the survey sampling literature. Many various types of estimators have been proposed, each one taking benefit of the connection between the study and the supporting variable by combining ratio, product, or regression estimators.

Many researchers have suggested various estimators by adequately modifying the supplementary variables including Singh and Espejo, ¹ Grover and Kaur, ² Shabbir et al., ³ Muneer et al., ⁴ Muili et al., ⁵ Grover and Kaur, ⁶ Singh and Usman, ⁷ Zaman and Kadilar, ⁸ Yadav and Zaman, ⁹ Zaman et al. ¹⁰

In some cases, when sampling elements vary significantly in size, e.g. in a health survey, related to a number of patients having a precise disease, the size of health units may differ, correspondingly survey connected to the income of the household, a household may have the different number of relations, then in such circumstances, it is important to use PPS sampling scheme. Numerous researchers have suggested various estimators by adequately modifying the supplementary variables under PPS. The researcher can investigate this research by Rao, ¹¹ Srivenkataramana and Tracy, ¹² Agarwal and Kumar, ¹³ Panday and Singh, ¹⁴ Ahmad and Shabbir, ¹⁵ Al-Marzouki et al. ¹⁶ and Singh et al. ¹⁷

Sampling methodology

Let a population B = {B₁, B₂,…, B_N} contain N identifiable units. Suppose $Y_i$ and { $X_{1i}$ , $X_{2i}$ } be the features of the study variable Y and the supplementary variables ( $X_1$ and $X_2$ ) respectively. The rank of the supplementary variables is denoted by $R_{xi}$ . Suppose a sample of size n is chosen by PPS with replacement. Let

${\mathrm{Q}}_i=\frac{X_{2i}}{{\sum }_{i=1}^n{X}_{2i}}$ , be the PPS sampling for obtaining the units. We take a sample of size n by adopting the PPS sampling with replacement.

Define

${\mathrm{u}}_{\mathrm{i}}=\frac{Y_i}{N{Q}_i},{\mathrm{v}}_{1\mathrm{i}}=\frac{X_{1i}}{N{Q}_i},{\mathrm{v}}_{2\mathrm{i}}=\frac{X_{2i}}{N{Q}_i},$ $\overline{u}=\frac{{\sum }_{i=1}^n{u}_i}{n}={\overline{y}}_{pps},{\overline{v}}_1=\frac{{\sum }_{i=1}^n{v}_{1i}}{n}={\overline{x}}_{1pps},{\overline{v}}_2=\frac{{\sum }_{i=1}^n{v}_{2i}}{n}={\overline{x}}_{2pps}$ , be the sample mean conforming to population mean $\overline{Y}$ , ${\overline{X}}_1$ and ${\overline{X}}_2$ .

As ${\overline{X}}_1$ and ${\overline{X}}_2$ are the supplementary variables, and ${\overline{R}}_x$ is the rank of the first supplementary variable.

Let

$${\epsilon }_0{\displaystyle \frac{\overline{u}-\overline{Y}}{\overline{Y}},{\epsilon }_1{\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{X}}_1},{\epsilon }_2={\displaystyle \frac{{\overline{v}}_1-{\overline{R}}_x}{{\overline{R}}_x},}}}$$

E( ${\epsilon }_i$ ) = 0, as i = 01,2

E( ${\epsilon }_0^2$ ) =  $\frac{C_u^2}{n}$ , E( ${\epsilon }_1^2$ ) =  $\frac{C_{v1}^2}{n}$ , E( ${\epsilon }_2^2$ ) =  $\frac{C_{v2}^2}{n}$ , E( ${\epsilon }_0{\epsilon }_1$ ) = ʎ ${\rho }_{uv1}{C}_u{C}_{v1}$ , E( ${\epsilon }_0{\epsilon }_1$ ) = ʎ ${\rho }_{uv2}{C}_u{C}_{v2}$ ,

E( ${\epsilon }_1{\epsilon }_2$ ) = λ ${\rho }_{v1v2}{C}_{v1}{C}_{v2}$ , $C_u^2$  =  $\frac{S_u^2}{\overline{u}},C_{v1}^2$  =  $\frac{S_{v1}^2}{{\overline{v}}_1},C_{v2}^2$  =  $\frac{S_{v2}^2}{{\overline{v}}_2}$ .

$$S_u^2=\sum _{i=1}^N{Q}_i(u_i-\overline{Y}{)}^2,S_{v1}^2=\sum _{i=1}^N{Q}_i(v_{1i}-{\overline{X}}_1{)}^2,S_{v2}^2=\sum _{i=1}^N{Q}_i(v_{2i}-{\overline{R}}_x{)}^2,$$

$$\begin{array}{rl}{S}_{uv1}& =\sum _{i=1}^N{Q}_i({\mathrm{u}}_{\mathrm{i}}-\overline{Y})({\mathrm{v}}_{1\mathrm{i}}-{\overline{X}}_1),S_{uv2}=\sum _{i=1}^N{Q}_i({\mathrm{u}}_{\mathrm{i}}-\overline{Y})({\mathrm{v}}_{2\mathrm{i}}-{\overline{R}}_x),\\ & S_{v1v2}=\sum _{i=1}^N{Q}_i({\mathrm{v}}_{1\mathrm{i}}-{\overline{X}}_1)({\mathrm{v}}_{2\mathrm{i}}-{\overline{R}}_x),\end{array}$$

$$\begin{array}{rl}{\rho }_{uv1}=& {\displaystyle \frac{{\sum }_{i=1}^N{Q}_i(u_i-\overline{Y})(v_{1i}-{\overline{X}}_1)}{S_u{S}_{v1}},{\rho }_{uv2}{\displaystyle \frac{{\sum }_{i=1}^N{Q}_i(u_i-\overline{Y})(v_{2i}-{\overline{R}}_x)}{S_u{S}_{v2}},}}\\ & {\rho }_{v1v2}{\displaystyle \frac{{\sum }_{i=1}^N{Q}_i(v_{1i}-{\overline{X}}_1)(v_{2i}-{\overline{R}}_x)}{S_{v1}{S}_{v2}},\text{Where}}\end{array}$$

Where ʎ =  $\frac{1}{n}$ .

Some of the existing estimation of mean

In this section, we have studied various adopted estimators that are available in the literature:

(i) Singh et al. ¹⁷ recommended the following estimator:

$$\begin{array}{rl}\breve{\overline{Y}}{}_{PPS}=& {\displaystyle \frac{{\displaystyle \frac{1}{n}{\sum }_{i=1}^n(y_i+\gamma )}}{{\mathbf{J}}_{i(n)}}-N\gamma }\\ =& {\displaystyle \frac{D_x+N\beta }{n}\sum _{i=1}^n{\displaystyle \frac{y_i+\gamma }{C_{xi}+\beta }-N\gamma ,}}\end{array}$$

(1)

where (C, $\beta$ , $\gamma$ ) are constants.

$${\mathbf{J}}_{i(n)}={\displaystyle \frac{C_{xi}+\beta }{C_{xi}+N\beta }}$$

We can write the above estimators as:

$$\breve{\overline{Y}}{}_{PPS}={\displaystyle \frac{1}{n}\sum _{i=1}^n\left\{{\displaystyle \frac{y_i+\gamma }{C_{xi}+\beta }-\gamma }\right\},}$$

$$\left({\displaystyle \frac{C_{xi}+N\beta }{Nn}}\right)\sum _{i=1}^n\left({\displaystyle \frac{y_i+\gamma }{C_{xi}+\beta }-\gamma }\right)$$

For C = 1, $\beta =0$ , $\gamma =0$ ,

We have the following traditional estimator:

$$\breve{\overline{Y}}{}_{PPS}={\displaystyle \frac{1}{n}\sum _{i=1}^n\left\{{\displaystyle \frac{y_i}{N{\mathbf{J}}_{i(n)}}}\right\}=ʎ\sum _{i=1}^n{u}_i=\overline{u}={\overline{y}}_{pps}}$$

$$\mathrm{Var}(\breve{\overline{Y}}{}_{PPS})=ʎ{\overline{Y}}^2{C}_u^2$$

(2)

(ii) Cochran ¹⁸ suggested the following estimator:

$$\breve{\overline{Y}}{}_{R(PPS)}=\overline{u}\left({\displaystyle \frac{{\overline{X}}_1}{{\overline{v}}_1}}\right)$$

(3)

The properties of $\breve{\overline{Y}}{}_{R(PPS)}$ :

$$\mathrm{Bias}(\breve{\overline{Y}}{}_{R(PPS)})\cong \overline{Y}ʎ[C_u^2-{\rho }_{uv1}{C}_u{C}_{v1}],$$

and

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{R(PPS)})\cong {\overline{Y}}^2ʎ[C_u^2+C_{v1}^2-2{\rho }_{uv1}{C}_u{C}_{v1}]$$

(4)

(iii) Murthy, ¹⁹ suggested the following estimator:

$$\breve{\overline{Y}}{}_{P(PPS)}=\overline{u}\left({\displaystyle \frac{{\overline{v}}_1}{{\overline{X}}_1}}\right)$$

(5)

The properties of $\breve{\overline{Y}}{}_{P(PPS)}$ :

$$\mathrm{Bias}(\breve{\overline{Y}}{}_{P(PPS)})\cong \overline{Y}ʎ{\rho }_{uv1}{C}_u{C}_{v1},$$

and

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{P(PPS)})\cong {\overline{Y}}^2ʎ[C_u^2+C_{v1}^2+2{\rho }_{uv1}{C}_u{C}_{v1}]$$

(6)

(iv) Bahl and Tuteja ²⁰ given by:

$$\breve{\overline{Y}}{}_{BT,R(PPS)}=\overline{y}\exp \left({\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right),$$

(7)

$$\breve{\overline{Y}}{}_{BT,P(PPS)}=\overline{y}\exp \left({\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}}\right).$$

(8)

The properties of $\breve{\overline{Y}}{}_{BT,R(PPS)}$ , $\breve{\overline{Y}}{}_{BT,P(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{BT,R(PPS)}\right)\cong ʎ\overline{Y}\left({\displaystyle \frac{3}{8}{C}_{v1}^2-{\displaystyle \frac{1}{2}{\rho }_{uv1}{C}_u{C}_{v1}}}\right).$$

and

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{BT,R(PPS)}\right)\cong ʎ{\overline{Y}}^2\left(C_u^2+{\displaystyle \frac{1}{4}{C}_{v1}^2-{\rho }_{uv1}{C}_u{C}_{v1}}\right).$$

(9)

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{BT,P(PPS)}\right)\cong ʎ\overline{Y}\left({\rho }_{uv1}{C}_u{C}_{v1}-{\displaystyle \frac{1}{4}{C}_{v1}^2}\right),$$

and

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{BT,P(PPS)}\right)\cong ʎ{\overline{Y}}^2\left[C_u^2+{\displaystyle \frac{1}{4}{C}_{v1}^2+{\rho }_{uv1}{C}_u{C}_{v1}}\right]$$

(10)

(v) Kumar and Bhougal, ²¹ suggested the following estimators:

$$\breve{\overline{Y}}{}_{KB(PPS)}=\overline{u}\left[\sigma \exp \left({\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right)+(1-\sigma )exp\left({\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}}\right)\right],$$

(11)

where $\sigma$ is a constant. The ideal value of $\sigma$ is given by:

$$\sigma ={\displaystyle \frac{1}{2}+{\rho }_{uv1}{\displaystyle \frac{C_u}{C_{v1}},}}$$

$$\mathrm{Bias}(\breve{\overline{Y}}{}_{KB(PPS)})\cong \overline{Y}ʎ\left[{\displaystyle \frac{1}{8}(4\sigma -1)C_{v1}^2-\left(\sigma -{\displaystyle \frac{1}{2}}\right){\rho }_{uv1}{C}_u{C}_{v1}}\right],$$

The least mean square error at the optimal value of $\sigma$ , is given by:

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{KB(PPS)})\cong ʎ{\overline{Y}}^2{C}_u^2(1-{\rho }_{uv1}^2)$$

(12)

v) Singh and Kumar ²² suggested the following estimators:

$$\breve{\overline{Y}}{}_{SKRP(PPS)}=\overline{u}\left[\left({\displaystyle \frac{{\overline{X}}_1}{{\overline{v}}_1}}\right)\left({\displaystyle \frac{{\overline{v}}_1}{{\overline{X}}_1}}\right)\right]$$

(13)

$$\breve{\overline{Y}}{}_{SKP(PPS)}=\overline{u}\left[\left({\displaystyle \frac{{\overline{v}}_1}{{\overline{X}}_1}}\right)\left({\displaystyle \frac{{\overline{X}}_1}{{\overline{v}}_1}}\right)\right]$$

(14)

The biases and MSE of $\breve{\overline{Y}}{}_{SKR(PPS)}$ and $\breve{\overline{Y}}{}_{SKP(PPS)}$ , are given by:

$$\mathrm{Bias}(\breve{\overline{Y}}{}_{SKRP(PPS)})\cong \overline{Y}ʎ[C_{v1}^2+C_{v2}^2+{\rho }_{v1v2}{C}_{v1}{C}_{v2}-{\rho }_{uv1}{C}_u{C}_{v1}-{\rho }_{uv2}{C}_u{C}_{v2}],$$

and

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{SKRP(PPS)})\cong {\overline{Y}}^2ʎ[C_u^2+C_{v1}^2+C_{v2}^2-2({\rho }_{uv1}{C}_u{C}_{v1}-{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv2}{C}_u{C}_{v2})]$$

(15)

$$\mathrm{Bias}(\breve{\overline{Y}}{}_{SKP(PPS)})\cong \overline{Y}ʎ[{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv1}{C}_u{C}_{v1}+{\rho }_{uv2}{C}_u{C}_{v2}]$$

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{SKP(PPS)})\cong {\overline{Y}}^2ʎ[C_u^2+C_{v1}^2+C_{v2}^2-2({\rho }_{uv1}{C}_u{C}_{v1}+{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv2}{C}_u{C}_{v2})]$$

(16)

Suggested efficient estimator for mean

An appropriate usage of the supplementary variables may help in improving the exactness of an estimator both during the design stage and at the estimation stage. Enchanting inspiration from Ahmad et al., ²³ we suggested a family of estimators that includes many more effective estimators expending two supplementary variables. The main advantages of our suggested estimator is that it is further elastic, effective than the existing estimators, which is given by:

$$\breve{\overline{Y}}{}_{Prop(PPS)}=\overline{u}\left[\exp \left({\displaystyle \frac{{\kappa }_u-{\kappa }_{\mathrm{u}}}{{\kappa }_{\mathrm{u}}+{\kappa }_{\mathrm{u}}}}\right)\exp \left({\displaystyle \frac{{\kappa }_{\mathrm{v}}-{\kappa }_{\mathrm{v}}}{{\kappa }_{\mathrm{v}}+{\kappa }_{\mathrm{v}}}}\right)\right],$$

(17)

where

$${\kappa }_{\mathrm{u}}=\left({\mathrm{F}}_1+{\dot{G}}_1\right){\overline{H}}_1+\tilde{I}{M}_1{h}_1,$$

$${\kappa }_{\mathrm{v}}=\left({\mathrm{F}}_2+{\dot{G}}_2\right){\overline{H}}_2+\tilde{I}{M}_2{h}_2,$$

$${\tau }_{\mathrm{u}}=\left({\mathrm{F}}_1+\tilde{I}{\mathrm{M}}_1\right){\overline{H}}_1+{\dot{G}}_1{h}_1,$$

$${\tau }_{\mathrm{v}}=\left({\mathrm{F}}_2+\tilde{I}{M}_2\right){\overline{H}}_2+{\dot{G}}_2{h}_2,$$

$$F_i=\left(S_i-1\right)\left(S_i-2\right),$$

$$M_i=\left(S_i-1\right)\left(S_i-4\right),$$

$${\dot{G}}_i=\left(S_i-2\right)\left(S_i-3\right)\left(S_i-4\right)$$

$$\tilde{\mathrm{I}}=n/N$$

Putting values of $S_i$ , where i  =  1,2,3,4, in (17)

By solving $\breve{\overline{Y}}{}_{Prop(PPS)}$ given in (17), we have

$$\breve{\overline{Y}}{}_{Prop(PPS)}=\overline{Y}(1+{\epsilon }_0)\left\{\left(1+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_1{\epsilon }_1-{\displaystyle \frac{1}{4}{\mathrm{\Phi }}_1{\mathrm{\Psi }}_1{\epsilon }_1^2+{\displaystyle \frac{1}{8}{\mathrm{\Phi }}_1^2{\epsilon }_1^2}}}\right)\left(1+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_2{\epsilon }_2-{\displaystyle \frac{1}{4}{\mathrm{\Phi }}_2{\mathrm{\Psi }}_2{\epsilon }_2^2+{\displaystyle \frac{1}{8}{\mathrm{\Phi }}_2^2{\epsilon }_2^2}}}\right)\right\}$$

(18)

where

$${\mathrm{\Phi }}_1={\displaystyle \frac{\tilde{\mathrm{I}}{M}_1-{\dot{\mathrm{G}}}_1}{A_1+\tilde{\mathrm{I}}{M}_1+{\dot{\mathrm{G}}}_1},}$$

$${\mathrm{\Psi }}_1={\displaystyle \frac{\tilde{\mathrm{I}}{M}_1+{\dot{\mathrm{G}}}_1}{A_1+\tilde{\mathrm{I}}{M}_1+{\dot{\mathrm{G}}}_1},}$$

$${\mathrm{\Phi }}_2={\displaystyle \frac{\tilde{\mathrm{I}}{M}_2-{\dot{\mathrm{G}}}_2}{A_2+\tilde{\mathrm{I}}{M}_2+{\dot{\mathrm{G}}}_2}}$$

$${\mathrm{\Psi }}_2={\displaystyle \frac{\tilde{\mathrm{I}}{M}_2+{\dot{\mathrm{G}}}_2}{A_2+\tilde{\mathrm{I}}{M}_2+{\dot{\mathrm{G}}}_2}.}$$

$$\left({\breve{\overline{Y}}}_{Prop(PPS)}-\overline{Y}\right)=\overline{Y}\left[\begin{array}{cc}& {\epsilon }_0+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_1{\epsilon }_1+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_2{\epsilon }_2+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_1{\epsilon }_0{\epsilon }_1+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_2{\epsilon }_0{\epsilon }_2}}}}\\ & -{\displaystyle \frac{1}{4}{\mathrm{\Phi }}_1{\mathrm{\Psi }}_1{\epsilon }_1^2-{\displaystyle \frac{1}{4}{\mathrm{\Phi }}_2{\mathrm{\Psi }}_2{\epsilon }_2^2+{\displaystyle \frac{1}{8}{\mathrm{\Phi }}_1^2{\epsilon }_1^2+{\displaystyle \frac{1}{8}{\mathrm{\Phi }}_2^2{\epsilon }_2^2}}}}\end{array}\right]$$

(19)

Using (19), the properties of $\breve{\overline{Y}}{}_{Prop(PPS)}$ , are given by:

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop(PPS)}\right)\cong \lambda \overline{Y}\left[\begin{array}{cc}& {\displaystyle \frac{1}{2}{\mathrm{\Phi }}_1{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_2{\rho }_{uv2}{C}_u{C}_{v2}+{\displaystyle \frac{1}{8}{C}_{\mathrm{v}1}^2({\mathrm{\Phi }}_1^2-2{\mathrm{\Phi }}_1{\mathrm{\Psi }}_1)}}}\\ & +{\displaystyle \frac{1}{8}{C}_{\mathrm{v}2}^2({\mathrm{\Phi }}_2^2-2{\mathrm{\Phi }}_2{\mathrm{\Psi }}_2)}\end{array}\right]$$

and

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop(PPS)}\right)\cong {\overline{Y}}^2\lambda \left[\begin{array}{cc}& C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{\mathrm{\Phi }}_1^2{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{\mathrm{\Phi }}_2^2{C}_{\mathrm{v}2}^2+{\mathrm{\Phi }}_1{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}}\\ & +{\mathrm{\Phi }}_2{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}+{\displaystyle \frac{1}{2}{\mathrm{\Phi }}_1{\mathrm{\Phi }}_2{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}\end{array}\right].$$

(20)

Differentiate Equation (20) with respect to ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$ , we have

$${\mathrm{\Phi }}_{1(opt)}={\displaystyle \frac{2C_{\mathrm{u}}({\rho }_{\mathrm{uv}1}-{\rho }_{\mathrm{v}1\mathrm{v}2}{\rho }_{\mathrm{uv}2})}{C_{\mathrm{v}1}({\rho }_{\mathrm{v}1\mathrm{v}2}^2-1)},}$$

$$\mathrm{\Phi }{}_{2(opt)}={\displaystyle \frac{2C_{\mathrm{u}}({\rho }_{\mathrm{uv}2}-{\rho }_{\mathrm{v}1\mathrm{v}2}{\rho }_{\mathrm{uv}1})}{C_3({\rho }_{23}^2-1)}.}$$

Putting values of ${\mathrm{\Phi }}_{1(opt)}$ and ${\mathrm{\Phi }}_{2(opt)}$ in (20), we get minimum MSE of $\breve{\overline{Y}}{}_{Prop(PPS)}$ and is given by:

$$\mathrm{MSE}{\left({\breve{\overline{Y}}}_{Prop(PPS)}\right)}_{min}\cong {\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2],$$

(21)

where

$${\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2={\displaystyle \frac{{\rho }_{\mathrm{uv}1}^2+{\rho }_{\mathrm{uv}2}^2-2{\rho }_{\mathrm{uv}1}{\rho }_{\mathrm{uv}2}{\rho }_{\mathrm{v}1\mathrm{v}2}}{1-{\rho }_{\mathrm{v}1\mathrm{v}2}^2},}$$

is the coefficient of multiple determination of u on $v_1$ and $v_2$ .

Now placing changed values of $S_i$ in equation (17), we get:

1. As $S_1=1$ , $S_2=1$

$$\breve{\overline{Y}}{}_{Prop1(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right\}\exp \left\{{\displaystyle \frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop1(PPS)}$ , are given by:

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop1(PPS)}\right)\cong \lambda \overline{Y}\left\{{\displaystyle \frac{3}{8}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{3}{8}{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\}$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop1(PPS)}\right)\cong \lambda {\overline{Y}}^2\left\{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2-{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}}}\right\}.$$

2. As $S_1=1$ , $S_2=2$

$$\breve{\overline{Y}}{}_{Prop2(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right\}\exp \left\{{\displaystyle \frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop2(PPS)}$ , are given by:

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop2(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}1}^2-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop2(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2-{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}+{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}}}\right\}.$$

3. As $S_1=1$ , $S_2=3$

$$\breve{\overline{Y}}{}_{Prop3(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right\}\exp \left\{{\displaystyle \frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-(n({\overline{v}}_2+{\overline{X}}_2)}}\right\}$$

The Properties of $\breve{\overline{Y}}{}_{Prop3(PPS)}$ , are given by:

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop3(PPS)}\right)\cong \lambda \overline{Y}\left\{{\displaystyle \frac{3}{8}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{8}{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\displaystyle \frac{1}{2}\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}}\right\}$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop3(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{\begin{array}{cc}& C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}2}^2-{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}\\ & +{\displaystyle \frac{1}{2}\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}\end{array}\right\}$$

4. As $S_1=1$ , $S_2=4$

$$\breve{\overline{Y}}{}_{Prop4(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop4(PPS)}$ , are given by:

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop4(PPS)}\right)\cong \lambda \overline{Y}\left\{{\displaystyle \frac{3C_{\mathrm{v}1}^2}{8}-{\displaystyle \frac{{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}{2}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop4(PPS)}\right)\cong {\displaystyle \frac{{\overline{Y}}^2\lambda }{4}\{4C_{\mathrm{u}}^2+C_{\mathrm{v}1}^2-4{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}\}}$$

5. As $S_1=2$ , $S_2=1$

$$\breve{\overline{Y}}{}_{Prop5(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}}\right\}\exp \left\{{\displaystyle \frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop5(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop5(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{3}{8}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop5(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2+{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\rho }_{\mathrm{uv}3}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}}}\right\}$$

6. As $S_1=2$ , $S_2=2$

$$\breve{\overline{Y}}{}_{Prop6(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}}\right\}\exp \left\{{\displaystyle \frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop6(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop6(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}1}^2-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}2}^2+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop6(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2+{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}+{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}}}\right\}$$

7. As $S_1=2$ , $S_2=3$

$$\breve{\overline{Y}}{}_{Prop7(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}}\right\}\exp \left\{{\displaystyle \frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-(n({\overline{v}}_2+{\overline{X}}_2)}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop7(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop7(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}1}^2-{\displaystyle \frac{1}{8}\left({\displaystyle \frac{f}{1-f}}\right)C_{\mathrm{v}2}^2+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop7(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{\begin{array}{cc}& C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}2}^2+{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}\\ & -{\displaystyle \frac{1}{2}\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}\end{array}\right\}$$

8. As $S_1=2,S_2=4$

$$\breve{\overline{Y}}{}_{Prop8(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop8(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop8(PPS)}\right)\cong \lambda \overline{Y}\left\{{\displaystyle \frac{{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}{2}-{\displaystyle \frac{C_{\mathrm{v}1}^2}{8}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop8(PPS)}\right)\cong {\displaystyle \frac{{\overline{Y}}^2\lambda }{4}\{4C_{\mathrm{u}}^2+C_{\mathrm{v}1}^2+4{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}\}.}$$

9. As $S_1=3$ , $S_2=1$

$$\breve{\overline{Y}}{}_{Prop9(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-(n({\overline{v}}_1+{\overline{X}}_1)}}\right\}\exp \left\{{\displaystyle \frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop9(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop9(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}1}^2+{\displaystyle \frac{3}{8}{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop9(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{\begin{array}{cc}& C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2-\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}\\ & +{\displaystyle \frac{1}{2}\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}\end{array}\right\}$$

10. As $S_1=3$ , $S_2=2$

$$\breve{\overline{Y}}{}_{Prop10(PPS)}=\overline{u}\exp \left({\displaystyle \frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-n({\overline{v}}_1+{\overline{X}}_1)}}\right)\exp \left({\displaystyle \frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}}\right)$$

The properties of $\breve{\overline{Y}}{}_{Prop10(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop10(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{\left({\displaystyle \frac{f}{1-f}}\right)}^2{C}_{\mathrm{v}1}^2-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}+{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\},$$

$$\begin{array}{rl}\mathrm{MSE}\left({\stackrel{⌣}{\overline{Y}}}_{Prop10(PPS)}\right)\cong {\overline{Y}}^2\lambda & \{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{\left({\displaystyle \frac{f}{1-f}}\right)}^2{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2-\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}}\\ +{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}-{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}\}\end{array}$$

11. As $S_1=3$ , $S_2=3$

$$\breve{\overline{Y}}{}_{Prop11(PPS)}=\overline{u}\exp \left({\displaystyle \frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-n({\overline{v}}_1+{\overline{X}}_1)}}\right)\exp \left({\displaystyle \frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-\mathrm{n}({\overline{v}}_2+{\overline{X}}_2)}}\right)$$

The properties of $\breve{\overline{Y}}{}_{Prop11(PPS)}$ :

$$\begin{array}{rl}& \mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop11(PPS)}\right)\\ & =\lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{\left({\displaystyle \frac{f}{1-f}}\right)}^2{C}_{\mathrm{v}1}^2-{\displaystyle \frac{1}{8}{\left({\displaystyle \frac{f}{1-f}}\right)}^2{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}-{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}}}\right\},\end{array}$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop11(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{\begin{array}{cc}& C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}1}^2+{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}2}^2-\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}}\\ & -\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}+{\displaystyle \frac{1}{2}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{\rho }_{\mathrm{v}1\mathrm{v}2}{C}_{\mathrm{v}1}{C}_{\mathrm{v}2}}\end{array}\right\}$$

12. As $S_1=3$ , $S_2=4$

$$\breve{\overline{Y}}{}_{Prop12(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-n({\overline{v}}_1+{\overline{X}}_1)}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop12(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop12(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}1}^2-{\displaystyle \frac{1}{2}\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop12(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{\left\{{\displaystyle \frac{f}{1-f}}\right\}}^2{C}_{\mathrm{v}1}^2-\left\{{\displaystyle \frac{f}{1-f}}\right\}{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}1}}\right\}$$

13. As $S_1=4$ , $S_2=1$

$$\breve{\overline{Y}}{}_{Prop13(PPS)}=\overline{u}\exp \left({\displaystyle \frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}}\right)$$

The properties of $\breve{\overline{Y}}{}_{Prop13(PPS)}$ , are given by:

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop13(PPS)}\right)\cong \lambda \overline{Y}\left[{\displaystyle \frac{3}{8}{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}\right],$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop13(PPS)}\right)\cong {\overline{Y}}^2\lambda \left[C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2-{\rho }_{\mathrm{uv}1}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}\right]$$

14. As $S_1=4$ , $S_2=2$

$$\breve{\overline{Y}}{}_{Prop14(PPS)}=\overline{u}\exp \left({\displaystyle \frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}}\right)$$

The properties of $\breve{\overline{Y}}{}_{Prop14(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop14(PPS)}\right)\cong \lambda \overline{Y}\left[-{\displaystyle \frac{1}{8}{C}_{\mathrm{v}2}^2+{\displaystyle \frac{1}{2}{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}\right],$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop14(PPS)}\right)\cong {\overline{Y}}^2\lambda \left[C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{C}_{\mathrm{v}2}^2+{\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}\right]$$

15. As $S_1=4$ , $S_2=3$

$$\breve{\overline{Y}}{}_{Prop15(PPS)}=\overline{u}\exp \left\{{\displaystyle \frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-\mathrm{n}({\overline{v}}_2+{\overline{X}}_2)}}\right\}$$

The properties of $\breve{\overline{Y}}{}_{Prop15(PPS)}$ :

$$\mathrm{Bias}\left({\breve{\overline{Y}}}_{Prop15(PPS)}\right)\cong \lambda \overline{Y}\left\{-{\displaystyle \frac{1}{8}{\left({\displaystyle \frac{f}{1-f}}\right)}^2{C}_{\mathrm{v}2}^2-{\displaystyle \frac{1}{2}\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}}\right\},$$

$$\mathrm{MSE}\left({\breve{\overline{Y}}}_{Prop15(PPS)}\right)\cong {\overline{Y}}^2\lambda \left\{C_{\mathrm{u}}^2+{\displaystyle \frac{1}{4}{\left({\displaystyle \frac{f}{1-f}}\right)}^2{C}_{\mathrm{v}2}^2-\left({\displaystyle \frac{f}{1-f}}\right){\rho }_{\mathrm{uv}2}{C}_{\mathrm{u}}{C}_{\mathrm{v}2}}\right\}$$

16. As $S_1=4$ , $S_2=4$

$$\breve{\overline{Y}}{}_{Prop16(PPS)}=\overline{u}$$

The variance of $\breve{\overline{Y}}{}_{Prop16(PPS)}$ :

$$\mathrm{Var}(\breve{\overline{Y}}{}_{Prop16(PPS)})={\overline{Y}}^2\lambda C_{\mathrm{u}}^2$$

Theoretic assessment

In this unit, we compared the adopted and suggested estimators in terms of MSE.

From (2) and (21)

$$\mathrm{Var}(\breve{\overline{Y}}{}_{PPS})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$$ʎ{\overline{Y}}^2{C}_u^2-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$${\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2>0$$

(ii) From (4) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{R(PPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$${\overline{Y}}^2ʎ[C_u^2+C_{v1}^2-2{\rho }_{uv1}{C}_u{C}_{v1}]-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$[C_u^2-2{\rho }_{uv1}{C}_u{C}_{v1}+C_u^2{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]>0$$

(iii) From (6) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{P(PPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$${\overline{Y}}^2ʎ[C_u^2+C_{v1}^2+2{\rho }_{uv1}{C}_u{C}_{v1}]-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$[C_u^2+2{\rho }_{uv1}{C}_u{C}_{v1}+C_u^2{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]>0$$

(iv) From (9) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{BT,R(RPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$$ʎ{\overline{Y}}^2\left(C_u^2+{\displaystyle \frac{1}{4}{C}_{v1}^2-{\rho }_{uv1}{C}_u{C}_{v1}}\right)-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$\left[{\displaystyle \frac{1}{4}{C}_{v1}^2-{\rho }_{uv1}{C}_u{C}_{v1}+C_u^2{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2}\right]>0$$

(v) From (10) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{BT,P(PPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$$ʎ{\overline{Y}}^2\left(C_u^2+{\displaystyle \frac{1}{4}{C}_{v1}^2+{\rho }_{uv1}{C}_u{C}_{v1}}\right)-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$\left[{\displaystyle \frac{1}{4}{C}_{v1}^2+{\rho }_{uv1}{C}_u{C}_{v1}+C_u^2{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2}\right]>0$$

(vi) From (12) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{KB(PPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$$ʎ{\overline{Y}}^2{C}_u^2(1-{\rho }_{uv1}^2)-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$[{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2-{\rho }_{uv1}^2]>0$$

(vii) From (15) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{SKRP(PPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$${\overline{Y}}^2ʎ[C_u^2+C_{v1}^2+C_{v2}^2-2({\rho }_{uv1}{C}_u{C}_{v1}-{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv2}{C}_u{C}_{v2})]-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$[C_u^2+C_{v1}^2-2({\rho }_{uv1}{C}_u{C}_{v1}-{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv2}{C}_u{C}_{v2})+C_u^2{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]>0$$

(viii) From (16) and (21)

$$\mathrm{MSE}(\breve{\overline{Y}}{}_{SKP(PPS)})\le \mathrm{MSE}(\breve{\overline{Y}}{}_{Prop(PPS)})\mathrm{or}$$

$${\overline{Y}}^2ʎ[C_u^2+C_{v1}^2+C_{v2}^2+2({\rho }_{uv1}{C}_u{C}_{v1}-{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv2}{C}_u{C}_{v2})]-{\overline{Y}}^2\lambda C_{\mathrm{u}}^2[1-{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]\ge 0$$

$$[C_u^2+C_{v1}^2+2({\rho }_{uv1}{C}_u{C}_{v1}+{\rho }_{v1v2}{C}_{v1}{C}_{v2}+{\rho }_{uv2}{C}_u{C}_{v2})+C_u^2{\mathcal{R}}_{\mathrm{uv}1\mathrm{v}2}^2]>0$$

Numerical illustration

In this unit, we deliberate altered population data sets for mathematical evaluations of the suggested and existing estimators. Data descriptions of these data are given in Table 2. The presentation of the considered estimators is compared in terms of PRE. We obtain the competence of estimators with existing estimators with the help of following expressions.

$$\mathrm{PRE}={\displaystyle \frac{Var({\breve{\overline{Y}}}_{PPS})}{MSE\left({\breve{\overline{Y}}}_{u(PPS)}\right)}\times 100,}$$

where u  =   $\breve{\overline{Y}}{}_{R(PPS)}$ , $\breve{\overline{Y}}{}_{P(PPS)}$ , $\breve{\overline{Y}}{}_{BT,R(PPS)}$ , $\breve{\overline{Y}}{}_{BT,P(PPS)}$ , $\breve{\overline{Y}}{}_{KB(PPS)}$ , $\breve{\overline{Y}}{}_{SKRP(PPS)}$ , $\breve{\overline{Y}}{}_{SKP(PPS)}$ , $\breve{\overline{Y}}{}_{Prop(PPS)}$ .

Table 2.

Summary statistics using populations I–III.

Parameters	Population-I	Population-II	Population-II
N	80	67	34
$n$	15	15	10
ʎ	0.06666667	0.06666667	0.6666667
$\overline{Y}$	5182.637	23.634333	199.4412
${\overline{X}}_1$	1126.463	20.59851	208.8824
${\overline{X}}_2$	285.125	9.79253	747.5882
${\overline{R}}_x$	40.4875	34	17.47059
${\overline{v}}_1$	1338.756	22.75134	203.3169
${\overline{v}}_2$	51.94002	37.57293	17.8003
$C_u$	0.4758864	0.5356861	0.3630288
$C_{v1}$	0.2371875	0.6869816	0.3401158
$C_{v2}$	0.4163328	0.7442662	0.3598615
${\rho }_{uv1}$	0.8520118	0.8344614	0.8890212
${\rho }_{uv2}$	0.4794675	0.8235835	0.7029563
${\rho }_{v1v2}$	0.7103788	0.9655628	0.7909407
$S_{uv1}$	740038	71.13702	3387.898
$S_{uv2}$	3886.483	128.748	202.4327
$S_{v1v2}$	48099.24	180.1726	233.6354
$S_u^2$	10568817	93.8327	3825.127
$S_{v1}^2$	63257.12	107.6524	3689.709
$S_{v2}^2$	344.6741	336.2949	27.96725

Table 1.

Various members of our suggested estimator $\breve{\overline{Y}}{}_{Prop(PPS)}$ .

$S.No$	$S_1$	$S_2$	Members of suggested estimators
1	1	1	$\breve{\overline{Y}}{}_{Prop1(PPS)}=\overline{u}\exp \left(\frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}\right)\exp \left(\frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}\right)$
2	1	2	$\breve{\overline{Y}}{}_{Prop2(PPS)}=\overline{u}\exp \left(\frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}\right)\exp \left(\frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}\right)$
3	1	3	$\breve{\overline{Y}}{}_{Prop3(PPS)}=\overline{u}\exp \left({\displaystyle \frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}}\right)\exp \left({\displaystyle \frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-(n({\overline{v}}_2+{\overline{X}}_2)}}\right)$
4	1	4	$\breve{\overline{Y}}{}_{Prop4(PPS)}=\overline{u}\exp \left(\frac{{\overline{X}}_1-{\overline{v}}_1}{{\overline{X}}_1+{\overline{v}}_1}\right)$
5	2	1	$\breve{\overline{Y}}{}_{Prop5(PPS)}=\overline{u}\exp \left(\frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}\right)\exp \left(\frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}\right)$
6	2	2	$\breve{\overline{Y}}{}_{Prop6(PPS)}=\overline{u}\exp \left(\frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}\right)\exp \left(\frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}\right)$
7	2	3	$\breve{\overline{Y}}{}_{Prop7(PPS)}=\overline{u}\exp \left(\frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}\right)\exp \left(\frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-(n({\overline{v}}_2+{\overline{X}}_2)}\right)$
8	2	4	$\breve{\overline{Y}}{}_{Prop8(PPS)}=\overline{u}\exp \left(\frac{{\overline{v}}_1-{\overline{X}}_1}{{\overline{v}}_1+{\overline{X}}_1}\right)$
9	3	1	$\breve{\overline{Y}}{}_{Prop9(PPS)}=\overline{u}\exp \left(\frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-(n({\overline{v}}_1+{\overline{X}}_1)}\right)\exp \left(\frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}\right)$
10	3	2	$\breve{\overline{Y}}{}_{Prop10(PPS)}=\overline{u}\exp \left(\frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-n({\overline{v}}_1+{\overline{X}}_1)}\right)\exp \left(\frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}\right)$
11	3	3	$\breve{\overline{Y}}{}_{Prop11(PPS)}=\overline{u}\exp \left(\frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-n({\overline{v}}_1+{\overline{X}}_1)}\right)\exp \left(\frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-\mathrm{n}({\overline{v}}_2+{\overline{X}}_2)}\right)$
12	3	4	$\breve{\overline{Y}}{}_{Prop12(PPS)}=\overline{u}\exp \left(\frac{n({\overline{X}}_1-{\overline{v}}_1)}{2N{\overline{X}}_1-n({\overline{v}}_1+{\overline{X}}_1)}\right)$
13	4	1	$\breve{\overline{Y}}{}_{Prop13(PPS)}=\overline{u}\exp \left(\frac{{\overline{X}}_2-{\overline{v}}_2}{{\overline{X}}_2+{\overline{v}}_2}\right)$
14	4	2	$\breve{\overline{Y}}{}_{Prop14(PPS)}=\overline{u}\exp \left(\frac{{\overline{v}}_2-{\overline{X}}_2}{{\overline{v}}_2+{\overline{X}}_2}\right)$
15	4	3	$\breve{\overline{Y}}{}_{Prop15(PPS)}=\overline{u}\exp \left(\frac{n({\overline{X}}_2-{\overline{v}}_2)}{2N{\overline{X}}_2-\mathrm{n}({\overline{v}}_2+{\overline{X}}_2)}\right)$
16	4	4	$\breve{\overline{Y}}{}_{Prop16(PPS)}=\overline{u}$

The MSE and PRE using three actual populations are given in Tables 3 and 4.

Table 3.

MSE using populations I–III.

Estimators	Population-I	Population-II	Population-III
$\breve{\overline{Y}}{}_{(PPS)}$	405524.4	10.68602	524.2179
$\breve{\overline{Y}}{}_{R(PPS)}$	271801.7	19.56867	295.3003
$\breve{\overline{Y}}{}_{P(PPS)}$	740723.4	36.95258	1673.401
$\breve{\overline{Y}}{}_{BT,R(PPS)}$	313478.6	10.73369	294.7258
$\breve{\overline{Y}}{}_{BT,P(PPS)}$	547939.4	19.42565	983.7763
$\breve{\overline{Y}}{}_{KB(PPS)}$	269101.7	9.611316	266.254
$\breve{\overline{Y}}{}_{SKRP(PPS)}$	514784.1	45.94476	849.6745
$\breve{\overline{Y}}{}_{SKP(PPS)}$	1769273	82.37996	3168.316
$\breve{\overline{Y}}{}_{Prop(PPS)}$	268054.3	9.134157	260

Table 4.

PRE using populations I–III.

Estimators	Population-I	Population-II	Population-III
$\breve{\overline{Y}}{}_{R(PPS)}$	100	100	100
$\breve{\overline{Y}}{}_{R(PPS)}$	149.1986	54.60782	177.5203
$\breve{\overline{Y}}{}_{P(PPS)}$	54.74708	28.9182	31.32649
$\breve{\overline{Y}}{}_{BT,R(PPS)}$	129.3627	99.55586	177.8663
$\breve{\overline{Y}}{}_{BT,P(PPS)}$	74.009	55.00985	53.28628
$\breve{\overline{Y}}{}_{KB(PPS)}$	150.6956	111.1817	196.8863
$\breve{\overline{Y}}{}_{SKR(PPS)}$	78.77564	23.25841	61.69632
$\breve{\overline{Y}}{}_{SKRPPPS)}$	22.92039	12.97163	16.54563
$\breve{\overline{Y}}{}_{Prop(PPS)}$	151.2844	116.9897	200.8582

Population-I: [Source: Murthy ²⁴ ]

Y  =  Output for 80 factories in an area,

$X_1$  = fixed capital in the area,

$X_2$  = number of workers,

$R_{X1}$  = rank of $X_1$ .

Population-II: [Source: Herbert ²⁵ ]

Y  =  Aspartame.

$X_1$  = Lucien,

$X_2$  = isoleucine,

$R_{X1}$  = rank of $X_1$ .

Population-III: [Source: Murthy ²⁴ ]

Y  =  Farming maize in 1964,

$X_1$  = farming maize in 1963,

$X_2$  = farming in 1961,

$R_{X1}$  = rank of $X_1$ .

Simulation study

From a multivariate normal distribution with modified covariance matrices, we generated three groups with a combined size of 5000. Below are the population means and covariance matrices:

Population-I:

$$\mu =\left[\begin{array}{c}500\\ 500\\ 500\end{array}\right]$$

and

$$\sum =\left[\begin{array}{ccc}1000& 800& 810\\ 800& 850& 820\\ 810& 820& 840\end{array}\right]$$

${\rho }_{XY}$  = 0.8820, ${\rho }_{XZ}$  = 0.9722 and ${\rho }_{YZ}$  = 0.7884

Population-II:

$$\mu =\left[\begin{array}{c}500\\ 500\\ 500\end{array}\right]$$

and

$$\sum =\left[\begin{array}{ccc}1500& 870& 820\\ 870& 900& 800\\ 820& 800& 740\end{array}\right]$$

${\rho }_{XY}$  = 0.75290, ${\rho }_{XZ}$  = 0.8684 and ${\rho }_{YZ}$  = 0.73445

Population-II:

$$\mu =\left[\begin{array}{c}500\\ 500\\ 500\end{array}\right]$$

and

$$\sum =\left[\begin{array}{ccc}400& 270& 220\\ 270& 500& 300\\ 220& 300& 675\end{array}\right]$$

${\rho }_{XY}$  = 0.6197, ${\rho }_{XZ}$  = 0.5105 and ${\rho }_{YZ}$  = 0.4965

Discussion and findings

We used three actual data and a simulation to observe the MSE and PRE of the existing and the suggested estimators. The minimum MSE of the suggested estimator is pointed out in Equation (21). The suggested estimator and the existing estimators were linked in terms of PRE. Summary statistics are shown in Table 2. The results of MSE and PRE on the basis of real data sets are available in Tables 3–4. It is observed from the numerical results that our suggested estimator is best among all the existing counterparts. The improvement in proficiency in Data 3 is more as compared to Data 1 and 2. The MSE and PRE result using simulated data sets are given in Tables 5 and 6. The outcome of the simulation study clearly determines that for the simulated data sets 1–3, the PRE of the suggested estimator is better than the existing estimator. Thus, we applause emphatically, the use of our suggested estimator over the existing estimators are better as linked to other considered estimators.

Table 5.

MSE using simulation results of populations I–III.

Estimators	Population-I	Population-II	Population-III
$\breve{\overline{Y}}{}_{R(PPS)}$	1.784719	1.72736	1.69054
$\breve{\overline{Y}}{}_{R(PPS)}$	0.553815	0.62479	0.62596
$\breve{\overline{Y}}{}_{P(PPS)}$	7.683389	7.52173	7.36046
$\breve{\overline{Y}}{}_{BT,R(PPS)}$	0.5857964	0.58960	0.58258
$\breve{\overline{Y}}{}_{BT,P(PPS)}$	4.150583	4.03807	3.94983
$\breve{\overline{Y}}{}_{KB(PPS)}$	0.423499	0.46004	0.45953
$\breve{\overline{Y}}{}_{SKR(PPS)}$	27.41200	28.4530	28.1090
$\breve{\overline{Y}}{}_{SKRPPPS)}$	35.68900	36.1730	36.8870
$\breve{\overline{Y}}{}_{Prop(PPS)}$	0.292409	0.32070	0.33007

Table 6.

Percentage relative efficiency using simulation results of populations I–III.

Estimators	Population-I	Population-II	Population-III
$\breve{\overline{Y}}{}_{R(PPS)}$	100	100	100
$\breve{\overline{Y}}{}_{R(PPS)}$	322.259	276.469	270.0713
$\breve{\overline{Y}}{}_{P(PPS)}$	23.228	22.964	22.9678
$\breve{\overline{Y}}{}_{BT,R(PPS)}$	304.665	292.971	290.1805
$\breve{\overline{Y}}{}_{BT,P(PPS)}$	42.999	42.776	42.80028
$\breve{\overline{Y}}{}_{KB(PPS)}$	421.421	375.472	367.8788
$\breve{\overline{Y}}{}_{SKR(PPS)}$	6.5107	6.07092	6.014223
$\breve{\overline{Y}}{}_{SKRPPPS)}$	5.0007	4.7752	4.58302
$\breve{\overline{Y}}{}_{Prop(PPS)}$	610.350	538.6145	512.1712

Conclusion

In this article, we suggested a modified family of estimators under PPS sampling using two supplementary variables. From our suggested family of estimators, we generate sixteen new estimators which are shown in Table 1. According to results based on three actual data, it is emerged that the suggested estimator achieves fine as compared to its existing counterparts. A simulation study also gives the same reflective as observed in real data sets. In theoretical and empirical efficiency comparisons, it has been revealed that our suggested estimator proves more efficient than the usual estimators. Consequently, we acclaim the use of our suggested estimators for proficiently estimating the finite population mean under probability proportional to size using supplementary variables. The present idea can be protracted to advance an enhanced family of estimators based on stratified sampling, proportion, and systematic sampling.