On Some Improved Class of Estimators by Using Stratified Ranked Set Sampling

Abstract

In this manuscript, we propose the combined and separate difference and ratio type estimators of population mean using stratified ranked set sampling. Additionally, several well-known estimators are identified as the sub-class of the suggested estimators. The characteristics of the suggested estimators have been analyzed and their effective performances are compared with the prominent estimators existing till date. Moreover, to prove the credibility of the theoretical findings, an extensive empirical study is administered over some real and hypothetically yielded symmetric and asymmetric populations.

1. Introduction

In sample surveys, the principle focus of any survey practitioner is to provide an estimate of the population parameters with higher efficiency, which not only relies upon the size of the sample and sampling fraction but also upon the heterogeneity or variability of the population. The customary sampling procedure utilized to reduce the heterogeneity of the population is stratified simple random sampling (SSRS). Several estimation procedures have been developed by various prominent authors, namely, [1,2,3,4,5], to efficiently estimate population parameters under SSRS. Ref. [6] is known as the pioneer who acquainted the notion of ranked set sampling (RSS) as an efficient option of simple random sampling (SRS) scheme, whereas motivated by [6,7] originated the idea of stratified ranked set sampling (SRSS) as an alternative sampling scheme to stratified simple random sampling which merge the profits of stratification and ranked set sampling to determine an unbiased estimator of population mean having likely improvement in the efficiency. In counterpoint to SSRS, the quantified units in SRSS do not lend identically to draw inference because of the further structure that imposes ranking, permitting observations to tempt various dimensions of the population and to obtain possible advantages in efficiency.

Ranked set sampling has quite broad literature starting with [6] and extending to many variations till date. One of the variations of RSS is SRSS mooted by [7]. Later on, Ref. [8] envisaged the ratio estimator under SRSS. [9] analyzed some modified ratio estimators by using auxiliary information under SRSS. Ref. [10] suggested the regression estimator under SRSS. Ref. [11] considered the Hartley-Ross kind of unbiased estimators under RSS and SRSS. Ref. [12] developed naive and ratio estimators under bivariate SRSS with optimal allocation. Ref. [13] examined a regression estimator under different SRSS schemes. Ref. [14] introduced the ratio type estimator using quartiles as auxiliary information under SRSS. Ref. [15] considered estimating the population mean using stratified median RSS. Ref. [16] investigated stratified percentile RSS, whereas, [17] suggested ratio estimators of the population mean using auxiliary information in SRS and median RSS. Some novel log type estimators have been suggested by [18] for the estimation of population mean under RSS. Ref. [19] discussed an estimation procedure of population mean under different SRSS with simulation study application to body mass index data. The interested readers may also see the study of [20,21,22,23,24,25,26,27,28], and the references cited therein.

Methodology and Notations

In sampling theory, the major concern of any surveyor is to enhance the efficacy of the proposed estimators. The SRSS becomes a better alternative over the other sampling strategies provided the ranking of units is possible. The methodology of SRSS is consisting of drawing independently $m_h$ random samples of size $m_h$ units from the $h^{th}$ ($h=1,2,\dots ,L$) stratum of the population. Here, $m_h$ is a set size of stratum h that should be kept small to ward off the large ranking errors. Now, the ranking is performed according to the variable of interest over the observations of each sample and the procedure of RSS is utilized to obtain independent L ranked set samples each of size $m_h$ provided ${\sum }_{h=1}^L{m}_h=m$. The prior process consummates a cycle of SRSS. The entire cycle can be iterated r times to ascertain the required $n_h=m_{h}r$ sample size in stratum h.

To determine the estimate of the population mean $\overline{Y}$ of the study variable Y, we perform the ranking with the auxiliary variable X. For cycle r and stratum h, let $(X_{h\left(1\right)r},Y_{h\left[1\right]r})$, $(X_{h\left(2\right)r},Y_{h\left[2\right]r})$, …, $(X_{h\left(m_h\right)r},Y_{h\left[m_h\right]r})$ be the stratified ranked set sample having joint probability density function $f(x_h,y_h)$ and cumulative distribution function $F(x_h,y_h)$. We note that the parenthesis $\left(\right)$ used in the variable X symbolizes the perfect ranking of units, whereas the parenthesis $\left[\right]$ used in the variable Y symbolizes the imperfect ranking of units. Moreover, the notations used throughout this article are defined hereunder.

N: size of population;

$N_h$: size of population in stratum h;

n: size of sample;

$n_h$: size of sample in stratum h;

$W_h=N_h/N$: weight of stratum h;

${\overline{y}}_h={\sum }_{i=1}^{n_h}{y}_{h_i}/n_h$: sample mean of variable Y in stratum h;

${\overline{y}}_{st}={\sum }_{h=1}^L{W}_h{y}_h$: sample mean of variable Y;

${\overline{x}}_h={\sum }_{i=1}^{n_h}{x}_{h_i}/n_h$: sample mean of variable X in stratum h;

${\overline{x}}_{st}={\sum }_{h=1}^L{W}_h{x}_h$: sample mean of variable X;

${\overline{y}}_{\left[\mathrm{srss}\right]}={\sum }_{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}$: stratified ranked set sample mean of variable Y;

${\overline{x}}_{\left(\mathrm{srss}\right)}={\sum }_{h=1}^L{W}_h{\overline{x}}_{h\left(\mathrm{rss}\right)}$: stratified ranked set sample mean of variable X;

${\overline{y}}_{h\left[\mathrm{rss}\right]}={\sum }_{i=1}^{m_h}{\sum }_{j=1}^r{y}_{h\left[i\right]j}/m_{h}r$: ranked set sample mean of variable Y in stratum h;

${\overline{x}}_{h\left(rss\right)}={\sum }_{i=1}^{m_h}{\sum }_{j=1}^r{x}_{h\left(i\right)j}/m_{h}r$; ranked set sample mean of variable X in stratum h;

${\overline{Y}}_h={\sum }_{i=1}^{N_h}{y}_{h_i}/N_h$: population mean of variable Y in stratum h;

$\overline{Y}={\overline{Y}}_{st}={\sum }_{h=1}^L{W}_h{Y}_h$: population mean of variable Y;

${\overline{X}}_h={\sum }_{i=1}^{N_h}{x}_{h_i}/N_h$: population mean of variable X in stratum h;

$\overline{X}={\overline{X}}_{st}={\sum }_{h=1}^L{W}_h{X}_h$: population mean of variable X;

$R=\overline{Y}/\overline{X}$: population ratio;

$R_h={\overline{Y}}_h/{\overline{X}}_h$: population ratio in stratum h;

$S_{y_h}^2={(N_h-1)}^{-1}{\sum }_{h=1}^{N_h}{(y_{h_i}-{\overline{Y}}_h)}^2$: population variance of variable Y in stratum h;

$S_{x_h}^2={(N_h-1)}^{-1}{\sum }_{h=1}^{N_h}{(x_{h_i}-{\overline{X}}_h)}^2$: population variance of variable X in stratum h;

$S_{x{y}_h}={(N_h-1)}^{-1}{\sum }_{h=1}^{N_h}(x_{h_i}-{\overline{X}}_h)(y_{h_i}-{\overline{Y}}_h)$: population covariance between variables X and Y in stratum h;

${\rho }_{x{y}_h}=S_{x{y}_h}/S_{x_h}{S}_{y_h}$: population correlation coefficient between variables X and Y in stratum h;

$C_{y_h}$: population coefficient of variation for variable Y in stratum h;

$C_{x_h}$: population coefficient of variation for variable X in stratum h;

${\beta }_1\left(x_h\right)={(E{({\overline{x}}_h-{\overline{X}}_h)}^3)}^2/{(E{({\overline{x}}_h-{\overline{X}}_h)}^2)}^2$: population coefficient of skewness for variable X in stratum h;

${\beta }_2\left(x_h\right)={(E({\overline{x}}_h-{\overline{X}}_h))}^4/{(E{({\overline{x}}_h-{\overline{X}}_h)}^2)}^2$: population coefficient of kurtosis for variable X in stratum h.

To determine the MSE of the combined estimators, the following notations will be utilized throughout this paper. Suppose, ${\overline{y}}_{\left[\mathrm{srss}\right]}=\overline{Y}(1+{ϵ}_0)\mathrm{and}{\overline{x}}_{\left(\mathrm{srss}\right)}=\overline{X}(1+{ϵ}_1)\mathrm{such}\mathrm{that}E\left({ϵ}_0\right)=E\left({ϵ}_1\right)=0$ and

$$V_{r,s}=\sum _{h=1}^L{W}_h^{r+s}\frac{E{({\overline{x}}_{\left(\mathrm{srss}\right)}-\overline{X})}^r{({\overline{y}}_{\left[\mathrm{srss}\right]}-\overline{Y})}^s}{{\overline{X}}^r{\overline{Y}}^s}$$

(1)

Following (1), we can write

$$E({{ϵ}_0}^2)=\sum _{h=1}^L{W}_h^2({\gamma }_h{C}_{y_h}^2-D_{y_{h\left[i\right]}}^2)=V_{0,2}$$

(2)

$$E({{ϵ}_1}^2)=\sum _{h=1}^L{W}_h^2({\gamma }_h{C}_{x_h}^2-D_{x_{h\left(i\right)}}^2)=V_{2,0}$$

(3)

$$E({ϵ}_0,{ϵ}_1)=\sum _{h=1}^L{W}_h^2({\gamma }_h{\rho }_{x{y}_h}{C}_{x_h}{C}_{y_h}-D_{{x{y}_h}_{\left[i\right]}})=V_{1,1}$$

(4)

where

$$\begin{array}{cc}\hfill {\gamma }_h& =\frac{1}{m_{h}r}\hfill \\ \hfill C_{x_h}& =\frac{S_{x_h}}{\overline{X}}\hfill \\ \hfill C_{y_h}& =\frac{S_{y_h}}{\overline{Y}}\hfill \\ \hfill D_{x_{h\left(i\right)}}^2& =\frac{1}{m_h^{2}r}\sum _{i=1}^{m_h}\frac{{\tau }_{x_{h\left(i\right)}}^2}{{\overline{X}}^2}\hfill \\ \hfill D_{y_{h\left[i\right]}}^2& =\frac{1}{m_h^{2}r}\sum _{i=1}^{m_h}\frac{{\tau }_{y_{h\left[i\right]}}^2}{{\overline{Y}}^2}\hfill \\ \hfill D_{{x{y}_h}_{\left[i\right]}}& =\frac{1}{m_h^{2}r}\sum _{i=1}^{m_h}\frac{{\tau }_{{x{y}_h}_{\left[i\right]}}}{\overline{X}\overline{Y}}\hfill \\ \hfill {\tau }_{x_{{}_{h\left(i\right)}}}& =({\mu }_{x_{{}_{h\left(i\right)}}}-{\overline{X}}_h)\hfill \\ \hfill {\tau }_{y_{{}_{h\left[i\right]}}}& =({\mu }_{y_{{}_{h\left[i\right]}}}-{\overline{Y}}_h)\hfill \\ \hfill {\tau }_{{x{y}_h}_{{}_{\left[i\right]}}}& =({\mu }_{x_{{}_{h\left(i\right)}}}-{\overline{X}}_h)({\mu }_{y_{{}_{h\left[i\right]}}}-{\overline{Y}}_h)\hfill \end{array}$$

It is worth mentioning that the valuations ${\mu }_{x_{{}_{h\left(i\right)}}}$ and ${\mu }_{y_{{}_{h\left[i\right]}}}$ consist of order statistics with fixed distributions which can be obtained from [29], whereas the quantities ${\mu }_{x_{{}_{h\left(i\right)}}}$ and ${\mu }_{y_{{}_{h\left[i\right]}}}$ can be considered equal in the absence of judgement error given that both variables possess the identical distributions, refer to [30].

To determine the properties of separate estimators, the following notations will be used throughout this paper. Let ${\overline{y}}_{\left[\mathrm{rss}\right]}=\overline{Y}(1+{ϵ}_{0_h})$, ${\overline{x}}_{h\left(\mathrm{rss}\right)}=\overline{X}(1+{ϵ}_{1_h})$ given that $E\left({ϵ}_{0_h}\right)=E\left({ϵ}_{1_h}\right)=0$,

$$\begin{array}{cc}\hfill E\left({{ϵ}_{0_h}}^2\right)& =\left({\gamma }_h\frac{S_{y_h}^2}{{\overline{Y}}_h^2}-M_{y_{h\left[i\right]}}^2\right)=U_0\hfill \\ \hfill E\left({{ϵ}_{1_h}}^2\right)& =\left({\gamma }_h\frac{S_{x_h}^2}{{\overline{X}}_h^2}-M_{x_{h\left(i\right)}}^2\right)=U_1\hfill \\ \hfill E\left({ϵ}_{0_h},{ϵ}_{1_h}\right)& =\left({\gamma }_h{\rho }_{x{y}_h}\frac{S_{x_h}}{{\overline{X}}_h}\frac{S_{y_h}}{{\overline{Y}}_h}-M_{{x{y}_h}_{\left[i\right]}}\right)=U_{10}\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill M_{x_{h\left(i\right)}}^2& =\frac{1}{m_h^{2}r}\sum _{i=1}^{m_h}\frac{{\tau }_{x_{h\left(i\right)}}^2}{{\overline{X}}_h^2}\hfill \\ \hfill M_{y_{h\left[i\right]}}^2& =\frac{1}{m_h^{2}r}\sum _{i=1}^{m_h}\frac{{\tau }_{y_{h\left[i\right]}}^2}{{\overline{Y}}_h^2}\hfill \\ \hfill M_{{x{y}_h}_{\left[i\right]}}& =\frac{1}{m_h^{2}r}\sum _{i=1}^{m_h}\frac{{\tau }_{{x{y}_h}_{\left[i\right]}}}{{\overline{X}}_h{\overline{Y}}_h}\hfill \end{array}$$

The principle objective of this article is to study the performance of the proposed combined and separate classes of estimators under SRSS. The outline of the article is divided as follows. A detailed review of the combined and separate estimators is discussed in Section 2. Some improved class of combined and separate estimators along with their characteristics are introduced in Section 3. The theoretical comparison of the proposed combined and separate estimators with the existing estimators is shown in Section 4. In favour of the theoretical results, an empirical study has been accomplished in Section 5. The interpretation of the empirical findings is drawn in Section 6. Finally, the conclusion of this study is discussed in Section 7.

2. Review of Existing Estimators

2.1. Combined Estimators

The conventional mean estimator utilizing SRSS is prescribed as

$$T_m^c={\overline{y}}_{\left[\mathrm{srss}\right]}$$

(5)

where the superscript “c” stands for “combined” and the subscript “srss” stand for “stratified ranked set sampling”.

Ref. [8] suggested the combined ratio estimator as

$$T_r^c=\frac{{\overline{y}}_{\left[\mathrm{srss}\right]}}{{\overline{x}}_{\left(\mathrm{srss}\right)}}\overline{X}$$

(6)

Ref. [10] introduced the combined regression estimator utilizing $SRSS$ as

$$T_{\beta }^c={\overline{y}}_{\left[\mathrm{srss}\right]}+\beta (\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)})$$

(7)

where $\beta$ is the regression coefficient of Y on X.

Following [31], we define the combined regression cum ratio estimators under SRSS as

$$T_{s{g}_1}^c={\lambda }_s\left({\overline{y}}_{\left[\mathrm{srss}\right]}+\beta (\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)})\right)\left(\frac{{\overline{z}}_{\left(\mathrm{srss}\right)}}{Z}\right)$$

(8)

where ${\lambda }_s$ is characterizing scalar, ${\overline{z}}_{\left(\mathrm{srss}\right)}={\sum }_{h=1}^L{W}_h({\overline{x}}_h+X)$, $\overline{Z}={\sum }_{h=1}^L{W}_h({\overline{X}}_h+X)$, and X is the population total.

Following [32], one may introduce a combined family of estimator under SRSS as

$$\begin{array}{c}\hfill T_{s{v}_1}^c=\left({\lambda }_1{\overline{y}}_{\left[\mathrm{srss}\right]}+{\lambda }_{2}b(\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)})\right){\left(\frac{{\overline{z}}_{\left(\mathrm{srss}\right)}^{*}}{{\overline{Z}}^{*}}\right)}^{\alpha }\end{array}$$

(9)

where ${\lambda }_1$, ${\lambda }_2$, b, and $\alpha$ are characterizing scalars.

Motivated by [33], one may develop a combined class of estimators under SRSS as

$$T_{k{k}_1}^c={\lambda }_k{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{a\overline{X}+b}{\alpha (a{\overline{x}}_{\left(\mathrm{srss}\right)}+b)+(1-\alpha )(a\overline{X}+b)}\right)}^g$$

(10)

where $\alpha$ is a characterizing scalar and g is a constant that takes quantities 1 and −1 to produce ratio and product kind of estimators, respectively. Furthermore, $(a\ne 0)$ and b are real values generally taken to be a function of available parameters of the auxiliary variable X, such as population standard deviation $S_{x_h}$, population coefficient of kurtosis ${\beta }_2\left(x_h\right)$, population coefficient of variation $C_{x_h}$, and population coefficient of correlation ${\rho }_{x{y}_h}$.

Following [34], one may consider a combined new family of estimator as

$$\begin{array}{c}\hfill T_{k{k}_2}^c=\left(k_1{\overline{y}}_{\left[\mathrm{srss}\right]}+k_2(\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)})\right)\left(\frac{a\overline{X}+b}{a{\overline{x}}_{\left(\mathrm{srss}\right)}+b}\right)\end{array}$$

(11)

where $k_1$ and $k_2$ are characterizing scalars.

Motivated by [35], we investigate a combined regression cum exponential type estimator under SRSS as

$$T_{s{g}_2}^c=\left(d_1{\overline{y}}_{\left[\mathrm{srss}\right]}+d_2(\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)})\right)exp\left(\frac{\overline{Z}-{\overline{z}}_{\left(\mathrm{srss}\right)}}{\overline{Z}+{\overline{z}}_{\left(\mathrm{srss}\right)}}\right)$$

(12)

where $d_1$ and $d_2$ are characterizing scalars.

Following [36], one may consider a general combined estimation procedure of population mean $\overline{Y}$ as

$$\begin{array}{c}\hfill T_{s{v}_2}^c={\mathrm{\Lambda }}_1{\overline{y}}_{\left[\mathrm{srss}\right]}+{\mathrm{\Lambda }}_2{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{{\overline{x}}_{\left(\mathrm{srss}\right)}^{*}}{{\overline{X}}^{*}}\right)}^{\beta }\end{array}$$

(13)

where ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ are characterizing scalars. Additionally, ${\overline{x}}_{\left(\mathrm{srss}\right)}^{*}={\sum }_{h=1}^L{W}_h(a_h{\overline{x}}_h+b_h)$ and ${\overline{X}}^{*}={\sum }_{h=1}^L{W}_h(a_h{\overline{X}}_h+b_h)$.

On the lines of [1], one may consider a new combined family of estimators under SRSS as

$$\begin{array}{c}\hfill T_{s{s}_1}^c=\begin{array}{c}{\kappa }_1{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{\alpha (a{\overline{x}}_{\left(\mathrm{srss}\right)}+b)+(1-\alpha )(a\overline{X}+b)}{(a\overline{X}+b)}\right)}^{\delta }+{\psi }_1{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{(a\overline{X}+b)}{\alpha (a{\overline{x}}_{\left(\mathrm{srss}\right)}+b)+(1-\alpha )(a\overline{X}+b)}\right)}^g\hfill \end{array}\end{array}$$

(14)

where ${\kappa }_1$, ${\psi }_1$, $\delta$, g, and $\alpha$ are characterizing scalars.

Following [3], one may suggest a combined ratio exponential type estimator in SRSS as

$$\begin{array}{c}\hfill t_u^c={\overline{y}}_{\left[\mathrm{srss}\right]}exp\left(\frac{\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)}}{\overline{X}+(a-1){\overline{x}}_{\left(\mathrm{srss}\right)}}\right)\end{array}$$

(15)

where a is the positive real constant.

Following [2], we envisage a class of combined estimator under SRSS as

$$\begin{array}{c}\hfill T_{s{s}_2}^c={\kappa }_2{\overline{y}}_{\left[\mathrm{srss}\right]}\left(\frac{{\overline{X}}^{*}}{\alpha {\overline{x}}_{\left(\mathrm{srss}\right)}^{*}+(1-\alpha ){\overline{X}}^{*}}\right)+{\psi }_2{\overline{y}}_{\left[\mathrm{srss}\right]}exp\left(\frac{\delta ({\overline{X}}^{*}-{\overline{x}}_{\left(\mathrm{srss}\right)})}{({\overline{X}}^{*}+{\overline{x}}_{\left(\mathrm{srss}\right)})}\right)\end{array}$$

(16)

where $\alpha$ and $\delta$ are real constants. Furthermore, ${\kappa }_2$ and ${\psi }_2$ are characterizing scalars.

Ref. [9] developed some ratio estimators of [37] under SRSS as

$$\begin{array}{cc}\hfill T_{m{m}_1}^c& ={\overline{y}}_{\left[\mathrm{srss}\right]}\frac{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{X}}_h+C_{x_h})}{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{x}}_{h\left(r_h\right)}+C_{x_h})}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill T_{m{m}_2}^c& ={\overline{y}}_{\left[\mathrm{srss}\right]}\frac{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{X}}_h+{\beta }_2\left(x_h\right))}{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{x}}_{h\left(r_h\right)}+{\beta }_2\left(x_h\right))}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill T_{m{m}_3}^c& ={\overline{y}}_{\left[\mathrm{srss}\right]}\frac{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{X}}_h{\beta }_2\left(x_h\right)+C_{x_h})}{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{x}}_{h\left(r_h\right)}{\beta }_2\left(x_h\right)+C_{x_h})}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill T_{m{m}_4}^c& ={\overline{y}}_{\left[\mathrm{srss}\right]}\frac{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{X}}_h{C}_{x_h}+{\beta }_2\left(x_h\right))}{{\displaystyle \sum _{h=1}^L}{W}_h({\overline{x}}_{h\left(r_h\right)}{C}_{x_h}+{\beta }_2\left(x_h\right))}\hfill \end{array}$$

(20)

Ref. [38] examined an advanced ratio estimator of [39] under SRSS as

$$T_{mm}^c=k{\overline{y}}_{\left[\mathrm{srss}\right]}\left(\frac{\overline{X}}{{\overline{x}}_{\left(\mathrm{srss}\right)}}\right)$$

(21)

where k is a characterizing scalar.

Following [4], one may define the following combined class of estimators under SRSS as

$$\begin{array}{c}\hfill T_{s{s}_3}^c={\kappa }_3{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{{\overline{X}}^{*}}{{\overline{x}}_{\left(\mathrm{srss}\right)}^{*}}\right)}^{\alpha }exp\left(\frac{\beta ({\overline{X}}^{*}-{\overline{x}}_{\left(\mathrm{srss}\right)})}{({\overline{X}}^{*}+{\overline{x}}_{\left(\mathrm{srss}\right)})}\right)+{\psi }_3{\left(\frac{{\overline{x}}_{\left(\mathrm{srss}\right)}^{*}}{{\overline{X}}^{*}}\right)}^{\eta }{\overline{y}}_{\left[\mathrm{srss}\right]}exp\left(\frac{\delta ({\overline{x}}_{\left(\mathrm{srss}\right)}-{\overline{X}}^{*})}{({\overline{x}}_{\left(\mathrm{srss}\right)}+{\overline{X}}^{*})}\right)\end{array}$$

(22)

where ${\kappa }_3$ and ${\psi }_3$ are characterizing scalars and $\alpha$, $\beta$, $\eta$, and $\delta$ are real constants.

Ref. [14] suggested ratio estimators as

$$\begin{array}{c}\hfill T_{s{k}_t}^c={\overline{y}}_{\left[\mathrm{srss}\right]}\left(\frac{\overline{X}-{\overline{x}}_{\left(\mathrm{srss}\right)}+q_t}{\overline{X}+{\overline{x}}_{\left(\mathrm{srss}\right)}+q_t}\right);t=1,3\end{array}$$

(23)

where $q_t,t=1,3$ is the $t^{th}$ quartile.

The readers are referred to Appendix A for the mean square error (MSE) expressions of these estimators.

2.2. Separate Estimators

The separate conventional mean estimator may be defined as

$$T_m^s=\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}$$

(24)

The superscript “s” means “separate” and the subscript “rss” means “ranked set sampling.”

Ref. [8] developed the classical separate ratio estimator as

$$T_r^s=\sum _{h=1}^L{W}_h\frac{{\overline{y}}_{h\left[\mathrm{rss}\right]}}{{\overline{x}}_{h\left(\mathrm{rss}\right)}}{\overline{X}}_h$$

(25)

Ref. [10] considered the separate regression estimator as

$$T_{\beta }^s=\sum _{h=1}^L{W}_h\left({\overline{y}}_{h\left[\mathrm{rss}\right]}+{\beta }_h({\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)})\right)$$

(26)

where ${\beta }_h$ is the regression coefficient of Y on X in stratum h.

Following [31], one may define a separate regression cum ratio estimator under SRSS as

$$T_{s{g}_1}^s=\sum _{h=1}^L{W}_h{\lambda }_{s_h}\left({\overline{y}}_{h\left[\mathrm{rss}\right]}+{\beta }_h({\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)})\right)\left(\frac{{\overline{z}}_{h\left(\mathrm{rss}\right)}}{{\overline{Z}}_h}\right)$$

(27)

where ${\lambda }_{s_h}$ is a suitably chosen scalar, $z_{h\left(\mathrm{rss}\right)}={\overline{x}}_{h\left(\mathrm{rss}\right)}+X_h$, ${\overline{Z}}_h={\overline{X}}_h+X_h$ and $X_h$ is the population total.

Motivated by [32], one may consider a separate family of estimators under SRSS as

$$\begin{array}{c}\hfill T_{s{v}_1}^s=\sum _{h=1}^L{W}_h\left({\lambda }_{1_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}+{\lambda }_{2_h}{b}_h({\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)})\right){\left(\frac{{\overline{z}}_{h\left(\mathrm{rss}\right)}^{*}}{{\overline{Z}}_h^{*}}\right)}^{\alpha }\end{array}$$

(28)

where ${\lambda }_{1_h},{\lambda }_{2_h}$, and $\alpha$ are suitably chosen scalars.

Motivated by [33], one may consider a separate family of estimators under SRSS as

$$T_{k{k}_1}^s=\sum _{h=1}^L{W}_h{\lambda }_{k_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}{\left(\frac{a_h{\overline{X}}_h+b_h}{{\alpha }_h(a_h{\overline{x}}_{h\left(\mathrm{rss}\right)}+b_h)+(1-{\alpha }_h)(a_h{\overline{X}}_h+b_h)}\right)}^g$$

(29)

where ${\lambda }_{k_h}$ is a suitably chosen scalar, ${\alpha }_h$ is a fixed constant, and g considers real values to produce the product and ratio type estimators. Furthermore, $(a_h\ne 0)$ and $b_h$ are real values generally taken to be a function of the available parameters of the auxiliary variable X namely, mean ${\overline{X}}_h$ standard deviation $S_{x_h}$, coefficient of kurtosis ${\beta }_2\left(x_h\right)$, coefficient of variation $C_{x_h}$, and coefficient of correlation ${\rho }_{x{y}_h}$ in the stratum h.

Following [34], one may consider the following separate families of estimators as

$$\begin{array}{c}\hfill T_{k{k}_2}^s=\sum _{h=1}^L{W}_h\left(k_{1_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}+k_{2_h}({\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)})\right)\left(\frac{a_h{\overline{X}}_h+b_h}{a_h{\overline{x}}_{h\left(\mathrm{rss}\right)}+b_h}\right)\end{array}$$

(30)

where $k_{1_h}$ and $k_{2_h}$ are suitably chosen scalars.

Motivated by [35], one may propose a separate regression cum exponential type estimator utilizing SRSS as

$$T_{s{g}_2}^s=\sum _{h=1}^L{W}_h\{d_{1_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}+d_{2_h}({\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)})\}exp\left(\frac{{\overline{Z}}_h-{\overline{z}}_{h\left(\mathrm{rss}\right)}}{{\overline{Z}}_h+{\overline{z}}_{h\left(\mathrm{rss}\right)}}\right)$$

(31)

where $d_{1_h}$ and $d_{2_h}$ are optimizing scalars.

Following [36], one may suggest a separate general estimation procedure for SRSS as

$$\begin{array}{c}\hfill T_{s{v}_2}^s=\sum _{h=1}^L{W}_h\left({\mathrm{\Lambda }}_{1_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}+{\mathrm{\Lambda }}_{2_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}{\left(\frac{{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}}{{\overline{X}}_h^{*}}\right)}^{\beta }\right)\end{array}$$

(32)

where ${\mathrm{\Lambda }}_{1_h}$ and ${\mathrm{\Lambda }}_{2_h}$ are optimizing scalars, ${\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}=(a_h{\overline{x}}_h+b_h)$, and ${\overline{X}}_h^{*}=(a_h{\overline{X}}_h+b_h)$.

On the lines of [1], one may consider a separate estimator for $\overline{Y}$ utilizing SRSS as

$$\begin{array}{c}\hfill T_{s{s}_1}^s=\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\begin{array}{c}{\kappa }_{1_h}{\left(\frac{{\alpha }_h{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}+(1-{\alpha }_h){\overline{X}}_h^{*}}{{\overline{X}}_h^{*}}\right)}^{\delta }+{\psi }_{1_h}{\left(\frac{{\overline{X}}_h^{*}}{{\alpha }_h{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}+(1-{\alpha }_h){\overline{X}}_h^{*}}\right)}^g\hfill \end{array}\right)\end{array}$$

(33)

where ${\mathrm{\kappa }}_{1_h}$ and ${\psi }_{1_h}$ are suitably chosen scalars and $\delta$, g, and $\alpha$ are constants.

Following [3], one may suggest a separate version of the ratio exponential kind of estimators of $\overline{Y}$ utilizing SRSS as

$$\begin{array}{c}\hfill T_u^s=\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}exp\left(\frac{{\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)}}{{\overline{X}}_h+(a_h-1){\overline{x}}_{h\left(\mathrm{rss}\right)}}\right)\end{array}$$

(34)

where $a_h$ is the positive real constant.

Motivated by [2], one may envisage a separate class of estimators under SRSS as

$$\begin{array}{c}\hfill T_{s{s}_2}^s=\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left({\kappa }_{2_h}\left(\frac{{\overline{X}}_h^{*}}{{\alpha }_h{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}+(1-{\alpha }_h){\overline{X}}_h^{*}}\right)+{\psi }_{2_h}exp\left(\frac{\delta ({\overline{X}}_h^{*}-{\overline{x}}_{h\left(\mathrm{rss}\right)})}{({\overline{X}}_h^{*}+{\overline{x}}_{h\left(\mathrm{rss}\right)})}\right)\right)\end{array}$$

(35)

where $\alpha$, $\delta$ are real constants and ${\kappa }_{2_h},{\psi }_{2_h}$ are suitably chosen scalars.

The separate version of [9] estimator under SRSS is given by

$$\begin{array}{cc}\hfill T_{m{m}_1}^s& =\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h+C_{x_h}}{{\overline{x}}_{h\left(\mathrm{rss}\right)}+C_{x_h}}\right)\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill T_{m{m}_2}^s& =\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h+{\beta }_2\left(x_h\right)}{{\overline{x}}_{h\left(\mathrm{rss}\right)}+{\beta }_2\left(x_h\right)}\right)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill T_{m{m}_3}^s& =\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h{\beta }_2\left(x_h\right)+C_{x_h}}{{\overline{x}}_{h\left(\mathrm{rss}\right)}{\beta }_2\left(x_h\right)+C_{x_h}}\right)\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill T_{m{m}_4}^s& =\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h{C}_{x_h}+{\beta }_2\left(x_h\right)}{{\overline{x}}_{h\left(\mathrm{rss}\right)}{C}_{x_h}+{\beta }_2\left(x_h\right)}\right)\hfill \end{array}$$

(39)

The separate version of [38] estimator under SRSS is given by

$$T_{mm}^s=\sum _{h=1}^L{W}_h{k}_h{\overline{y}}_{\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h}{{\overline{x}}_{h\left(\mathrm{rss}\right)}}\right)$$

(40)

where $k_h$ is the characterizing scalar.

Following [4], one may consider the following separate class of estimators under SRSS as

$$\begin{array}{c}\hfill T_{s{s}_3}^s=\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\begin{array}{c}{\kappa }_{3_h}{\left(\frac{{\overline{X}}_h^{*}}{{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}}\right)}^{\alpha }exp\left(\frac{\beta ({\overline{X}}_h^{*}-{\overline{x}}_{h\left(\mathrm{rss}\right)})}{({\overline{X}}_h^{*}+{\overline{x}}_{h\left(\mathrm{rss}\right)})}\right)+{\psi }_{3_h}{\left(\frac{{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}}{{\overline{X}}_h^{*}}\right)}^{\eta }exp\left(\frac{\delta ({\overline{x}}_{h\left(\mathrm{rss}\right)}-{\overline{X}}_h^{*})}{({\overline{x}}_{h\left(\mathrm{rss}\right)}+{\overline{X}}_h^{*})}\right)\hfill \end{array}\right)\end{array}$$

(41)

where ${\kappa }_{3_h}$ and ${\psi }_{3_h}$ are characterizing scalars and $\alpha$, $\beta$, $\eta$, and $\delta$ are real constants.

The separate form of [14] estimator is given by

$$\begin{array}{c}\hfill T_{s{k}_t}^s=\sum _{h=1}^L{W}_h{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h-{\overline{x}}_{h\left(\mathrm{rss}\right)}+q_{t_h}}{{\overline{X}}_h+{\overline{x}}_{h\left(\mathrm{rss}\right)}+q_{t_h}}\right);t=1,3\end{array}$$

(42)

The readers are referred to Appendix B for the MSE expressions of these estimators.

3. Proposed Estimators

The motivation of the present manuscript is to examine some efficient combined and separate estimators for the computation of population mean under SRSS. These estimators furnish a better choice to the conventional estimators brushed up in Section 2. In our proposal, we have suggested some improved class of estimators utilizing auxiliary information under $SRSS$.

3.1. Combined Estimators

We propose some improved combined class of difference and ratio kind of estimators as

$$\begin{array}{cc}\hfill T_{y_1}^c& ={\alpha }_1{\overline{y}}_{\left[\mathrm{srss}\right]}+{\beta }_1({\overline{x}}_{\left(\mathrm{srss}\right)}-\overline{X})\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill T_{y_2}^c& ={\alpha }_2{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{\overline{X}}{{\overline{x}}_{\left(\mathrm{srss}\right)}}\right)}^{{\beta }_2}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill T_{y_3}^c& ={\alpha }_3{\overline{y}}_{\left[\mathrm{srss}\right]}\left(\frac{\overline{X}}{\overline{X}+{\beta }_3({\overline{x}}_{\left(\mathrm{srss}\right)}-\overline{X})}\right)\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill T_{y_4}^c& ={\alpha }_4{\overline{y}}_{\left[\mathrm{srss}\right]}+{\beta }_4({\overline{x}}_{\left(\mathrm{srss}\right)}^{*}-{\overline{X}}^{*})\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill T_{y_5}^c& ={\alpha }_5{\overline{y}}_{\left[\mathrm{srss}\right]}{\left(\frac{{\overline{X}}^{*}}{{\overline{x}}_{\left(\mathrm{srss}\right)}^{*}}\right)}^{{\beta }_5}\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill T_{y_6}^c& ={\alpha }_6{\overline{y}}_{\left[\mathrm{srss}\right]}\left(\frac{{\overline{X}}^{*}}{{\overline{X}}^{*}+{\beta }_6({\overline{x}}_{\left(\mathrm{srss}\right)}^{*}-{\overline{X}}^{*})}\right)\hfill \end{array}$$

(48)

where ${\alpha }_i$ and ${\beta }_i$, $i=1,2,\dots ,6$ are characterizing scalars. Further, we noticed that the proposed estimators $T_{y_i}^c$, $i=1,2,\dots ,6$ reduce to some known estimators, such as:

• Classical mean estimator $T_m^c$, for $({\alpha }_i,{\beta }_i)=(1,0)$;
• Classical regression estimator $T_{\beta }^c$, for $({\alpha }_1,{\beta }_1)=(1,{\beta }_1)$;
• Classical ratio estimator $T_r^c$, for $({\alpha }_2,{\beta }_2)=(1,1)$;
• [9] estimator $T_{m{m}_1}^c$, for $({\alpha }_5,{\beta }_5,a,b)=(1,1,1,C_x)$;
• [9] estimator $T_{m{m}_2}^c$, for $({\alpha }_5,{\beta }_5,a,b)=(1,1,1,{\beta }_2\left(x\right))$;
• [9] estimator $T_{m{m}_3}^c$, for $({\alpha }_5,{\beta }_5,a,b)=(1,1,{\beta }_2\left(x\right),C_x)$;
• [9] estimator $T_{m{m}_4}^c$, for $({\alpha }_5,{\beta }_5,a,b)=(1,1,C_x,{\beta }_2\left(x\right))$, and
• [38] estimator $T_{mm}^c$, for $({\alpha }_2,{\beta }_2)=({\alpha }_2,1)$.

Similarly, putting different values of the scalars, one may develop several other class of estimators.

Theorem 1.

The MSE of the proposed classes of estimators $T_{y_i}^c,i=1,2,\dots ,6$ is given by

$$\begin{array}{cc}\hfill MSE\left(T_{y_1}^c\right)& ={\overline{Y}}^2\left(\begin{array}{c}{({\alpha }_1-1)}^2+{\alpha }_1^2{V}_{0,2}+\frac{{\overline{X}}^2}{{\overline{Y}}^2}{\beta }_1^2{V}_{2,0}+2\frac{\overline{X}}{\overline{Y}}{\alpha }_1{\beta }_1{V}_{1,1}\hfill \end{array}\right)\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill MSE\left(T_{y_2}^c\right)& ={\overline{Y}}^2\left(\begin{array}{c}1+{\alpha }_2^2\left(\begin{array}{c}1+V_{0,2}+{\beta }_2(2{\beta }_2+1)V_{2,0}-4{\beta }_2{V}_{1,1}\hfill \end{array}\right)\hfill \\ -2{\alpha }_2\left(\begin{array}{c}1+\frac{{\beta }_2({\beta }_2+1)}{2}{V}_{2,0}-{\beta }_2{V}_{1,1}\hfill \end{array}\right)\hfill \end{array}\right)\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill MSE\left(T_{y_3}^c\right)& ={\overline{Y}}^2(1+{\alpha }_3^2(1+V_{0,2}+3{\beta }_3^2{V}_{2,0}-4{\beta }_3{V}_{1,1})-2{\alpha }_3(1+{\beta }_3^2{V}_{2,0}-{\beta }_3{V}_{1,1}\left)\right)\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill MSE\left(T_{y_4}^c\right)& ={\overline{Y}}^2\left({({\alpha }_4-1)}^2+{\alpha }_4^2{V}_{0,2}+\frac{{\overline{X}}^2}{{\overline{Y}}^2}{\beta }_4^2{\upsilon }^2{V}_{2,0}+2\frac{\overline{X}}{\overline{Y}}{\alpha }_4{\beta }_4\upsilon V_{1,1}\right)\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill MSE\left(T_{y_5}^c\right)& ={\overline{Y}}^2\left(\begin{array}{c}1+{\alpha }_5^2\left(1+V_{0,2}-4{\beta }_5\upsilon V_{1,1}+{\beta }_5(2{\beta }_5+1){\upsilon }^2{V}_{2,0}\right)\hfill \\ -2{\alpha }_5\left(1+\frac{{\beta }_5({\beta }_5+1)}{2}{\upsilon }^2{V}_{2,0}-{\beta }_5\upsilon V_{1,1}\right)\hfill \end{array}\right)\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill MSE\left(T_{y_6}^c\right)& ={\overline{Y}}^2\left(\begin{array}{c}1+{\alpha }_6^2\left(1+V_{0,2}+3{\beta }_6^2{\upsilon }^2{V}_{2,0}-4{\beta }_6\upsilon V_{1,1}\right)\hfill \\ -2{\alpha }_6\left(1+{\beta }_6^2{\upsilon }^2{V}_{2,0}-{\beta }_6\upsilon V_{1,1}\right)\hfill \end{array}\right)\hfill \end{array}$$

(54)

Proof. 

The outline of the derivations are reported in Appendix C. □

Corollary 1.

The minimum MSE of the proposed combined estimators at the optimum values of ${\alpha }_i$ and ${\beta }_i$ is given by

$$\begin{array}{cc}\hfill minMSE\left(T_{y_i}^c\right)& ={\overline{Y}}^2(1-{\alpha }_{i\left(\mathrm{opt}\right)})={\overline{Y}}^2\left(1-\frac{P_i^2}{Q_i}\right);i=1,3,4,6\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill minMSE\left(T_{y_i}^c\right)& ={\overline{Y}}^2\left(1-\frac{P_i^2}{Q_i}\right);i=2,5\hfill \end{array}$$

(56)

Proof. 

The outline of the derivation and definitions of the parametric functions $P_i$ and $Q_i,i=1,2,\dots ,6$ are given in Appendix C. □

3.2. Separate Estimators

We also propose some separate class of difference and ratio kind of estimators of as

$$\begin{array}{cc}\hfill T_{y_1}^s& =\sum _{h=1}^L{W}_h\left({\alpha }_{1_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}+{\beta }_{1_h}({\overline{x}}_{h\left(\mathrm{rss}\right)}-{\overline{X}}_h)\right)\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill T_{y_2}^s& =\sum _{h=1}^L{W}_h{\alpha }_{2_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}{\left(\frac{{\overline{X}}_h}{{\overline{x}}_{h\left(\mathrm{rss}\right)}}\right)}^{{\beta }_{2_h}}\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill T_{y_3}^s& =\sum _{h=1}^L{W}_h{\alpha }_{3_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h}{{\overline{X}}_h+{\beta }_{3_h}({\overline{x}}_{h\left(\mathrm{rss}\right)}-{\overline{X}}_h)}\right)\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill T_{y_4}^s& =\sum _{h=1}^L{W}_h\left({\alpha }_{4_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}+{\beta }_{4_h}({\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}-{\overline{X}}_h^{*})\right)\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill T_{y_5}^s& =\sum _{h=1}^L{W}_h{\alpha }_{5_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}{\left(\frac{{\overline{X}}_h^{*}}{{\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}}\right)}^{{\beta }_{5_h}}\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill T_{y_6}^s& =\sum _{h=1}^L{W}_h{\alpha }_{6_h}{\overline{y}}_{h\left[\mathrm{rss}\right]}\left(\frac{{\overline{X}}_h^{*}}{{\overline{X}}_h^{*}+{\beta }_{6_h}({\overline{x}}_{h\left(\mathrm{rss}\right)}^{*}-{\overline{X}}_h^{*})}\right)\hfill \end{array}$$

(62)

where ${\alpha }_{i_h}$ and ${\beta }_{i_h}$, $i=1,2,\dots ,6$ are suitably chosen scalars. It is worth mentioning that the proposed estimators $T_{y_i}^s$, ($i=1,2,\dots ,6$) reduce to some known estimators such as:

• Classical regression estimator $T_{\beta }^s$ for $({\alpha }_{1_h},{\beta }_{1_h})=(1,{\beta }_{1_h})$;
• Conventional mean estimator $T_m^s$ for $({\alpha }_{2_h},{\beta }_{2_h})=(1,0)$;
• Classical ratio estimator $T_r^s$ for $({\alpha }_{2_h},{\beta }_{2_h})=(1,1)$;
• [9] estimator $T_{m{m}_1}^s$ for $({\alpha }_{5_h},{\beta }_{5_h},a_h,b_h)=(1,1,1,C_{x_h})$;
• [9] estimator $T_{m{m}_2}^s$ for $({\alpha }_{5_h},{\beta }_{5_h},a_h,b_h)=(1,1,1,{\beta }_2\left(x_h\right))$;
• [9] estimator $T_{m{m}_3}^s$ for $({\alpha }_{5_h},{\beta }_{5_h},a_h,b_h)=(1,1,{\beta }_2\left(x_h\right),C_{x_h})$;
• [9] estimator $T_{m{m}_4}^s$ for $({\alpha }_{5_h},{\beta }_{5_h},a_h,b_h)=(1,1,C_{x_h},{\beta }_2\left(x_h\right))$, and
• [38] estimator $T_{mm}^s$ for $({\alpha }_{2_h},{\beta }_{2_h})=({\alpha }_{2_h},1)$.

Likewise, one may develop various other classes of estimators for different values of scalars.

Theorem 2.

The MSE of the proposed estimators $T_{y_i}^s,i=1,2,\dots ,6$ is given by:

$$\begin{array}{cc}\hfill MSE\left(T_{y_1}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(\begin{array}{c}{({\alpha }_{1_h}-1)}^2+{\alpha }_{1_h}^2{U}_0+\frac{{\overline{X}}_h^2}{{\overline{Y}}_h^2}{\beta }_{1_h}^2{U}_1+2\frac{{\overline{X}}_h}{{\overline{Y}}_h}{\alpha }_{1_h}{\beta }_{1_h}{U}_{10}\hfill \end{array}\right)\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill MSE\left(T_{y_2}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(\begin{array}{c}1+{\alpha }_{2_h}^2\left(\begin{array}{c}1+U_0+{\beta }_{2_h}(2{\beta }_{2_h}+1)U_1-4{\beta }_{2_h}{U}_{10}\hfill \end{array}\right)\hfill \\ -2{\alpha }_{2_h}\left(\begin{array}{c}1+\frac{{\beta }_{2_h}({\beta }_{2_h}+1)}{2}{U}_1-{\beta }_{2_h}{U}_{10}\hfill \end{array}\right)\hfill \end{array}\right)\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill MSE\left(T_{y_3}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(\begin{array}{c}1+{\alpha }_{3_h}^2\left(1+U_0+3{\beta }_{3_h}^2{U}_1-4{\beta }_{3_h}{U}_{10}\right)\hfill \\ -2{\alpha }_{3_h}\left(1+{\beta }_{3_h}^2{U}_1-{\beta }_{3_h}{U}_{10}\right)\hfill \end{array}\right)\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill MSE\left(T_{y_4}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left({({\alpha }_4-1)}^2+{\alpha }_{4_h}^2{U}_0+\frac{{\overline{X}}_h^2}{{\overline{Y}}_h^2}{\beta }_{4_h}^2{\upsilon }_h^2{U}_1+2\frac{{\overline{X}}_h}{{\overline{Y}}_h}{\alpha }_{4_h}{\beta }_{4_h}{\upsilon }_h{U}_{10}\right)\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill MSE\left(T_{y_5}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(\begin{array}{c}1+{\alpha }_{5_h}^2\left(1+U_0-4{\beta }_{5_h}{\upsilon }_h{U}_{10}+{\beta }_{5_h}(2{\beta }_{5_h}+1){\upsilon }_h^2{U}_1\right)\hfill \\ -2{\alpha }_{5_h}\left(1+\frac{{\beta }_{5_h}({\beta }_{5_h}+1)}{2}{\upsilon }_h^2{U}_1-{\beta }_{5_h}\upsilon U_{10}\right)\hfill \end{array}\right)\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill MSE\left(T_{y_6}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(\begin{array}{c}1+{\alpha }_{6_h}^2\left(1+U_0+3{\beta }_{6_h}^2{\upsilon }_h^2{U}_1-4{\beta }_{6_h}{\upsilon }_h{U}_{10}\right)\hfill \\ -2{\alpha }_{6_h}\left(1+{\beta }_{6_h}^2{\upsilon }_h^2{U}_1-{\beta }_{6_h}{\upsilon }_h{U}_{10}\right)\hfill \end{array}\right)\hfill \end{array}$$

(68)

Proof. 

The outline of the derivations are reported in Appendix D. □

Corollary 2.

The minimum MSE of the proposed estimators $T_{y_i}^s,i=1,2,\dots ,6$ is given by

$$\begin{array}{cc}\hfill minMSE\left(T_{y_i}^s\right)& =\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2(1-{\alpha }_{i_h\left(\mathrm{opt}\right)})=\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right);i=1,3,4,6\hfill \end{array}$$

(69)

$$minMSE\left(T_{y_i}^s\right)=\sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right);i=2,5$$

(70)

Proof. 

The outline of the derivation and definitions of the parametric functions $P_{i_h}$ and $Q_{i_h},i=1,2,\dots ,6$ are given in Appendix D. □

We would like to note that the MSE expressions of Theorem 1, Theorem 2, Corollary 1 and Corollary 2 are important to determine the conditions of the succeeding section.

4. Efficiency Comparison

In the present section, we perform an efficiency comparison of the proposed combined and separate estimators with the existing combined and separate estimators.

4.1. Combined Estimators

By likening the minimum MSE of the proposed combined estimators $T_{y_i}^c,i=1,2,\dots ,6$ from (55) and (56) with the minimum MSE of the brushed up estimators from (A1), (A2), (A5), (A6), (A8), (A10), (A12), (A14), (A16), (A18), (A20), (A21), (A22) and (A23), we obtain the efficiency conditions given hereunder.

$$\begin{array}{cc}\hfill MSE\left(T_m^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-V_{0,2}\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill MSE\left(T_r^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-V_{0,2}-V_{2,0}+2V_{1,1}\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill MSE(T_{\beta }^c)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-V_{0,2}+\frac{V_{1,1}^2}{V_{2,0}}\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill MSE\left(T_{s{g}_1}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>{\lambda }_{s\left(\mathrm{opt}\right)}\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill MSE\left(T_{s{v}_1}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>\frac{B_1{D}_1^2+A_1^2{E}_1-2C_1{D}_1{E}_1}{A_1{B}_1-C_1^2}\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill MSE(T_{k{k}_1}^c)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>\frac{A^2}{4B}\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill MSE(T_{k{k}_2}^c)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-\frac{({\upsilon }^2{V}_{2,0}-1)(V_{2,0}{V}_{0,2}-V_{1,1}^2)}{{\upsilon }^2{V}_{2,0}^2+V_{1,1}^2-V_{2,0}(1+V_{0,2})}\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill MSE\left(T_{s{g}_2}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>\left(\frac{V_{2,0}}{4{(N+1)}^2}+\frac{{\left(1-\frac{V_{2,0}}{8{(N+1)}^2}\right)}^2}{\left(1+V_{0,2}-\frac{V_{1,1}^2}{V_{2,0}}\right)}\right)\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill MSE\left(T_{s{v}_2}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>\frac{(A_2{D}_2^2+B_2-2C_2{D}_2)}{(A_2{B}_2-C_2^2)}\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill MSE\left(T_{s{s}_i}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>\frac{(F_i{J}_i^2+G_i{I}_i^2-2H_i{I}_i{J}_i)}{(F_i{G}_i-H_i^2)}\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill MSE\left(T_u^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-V_{0,2}+\frac{V_{1,1}^2}{V_{2,0}}\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill MSE\left(T_{m{m}_i}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-V_{0,2}-{\lambda }_i^2{V}_{2,0}+2{\lambda }_i{V}_{1,1}\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill MSE\left(T_{mm}^c\right)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-{(k^{*}-1)}^2-V_{0,2}-k^{{*}^2}{V}_{2,0}+2k^{*}{V}_{1,1}\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill MSE(T_{s{k}_t}^c)& >MSE\left(T_{y_i}^c\right)\Rightarrow \frac{P_i^2}{Q_i}>1-\left(V_{0,2}+P_t{V}_{2,0}^2-2P_t^2{V}_{1,1}\right)\hfill \end{array}$$

(84)

If the conditions (71) to (84) satisfy, then the proposed combined estimators dominate the existing combined estimators brushed up in Section 2.1.

4.2. Separate Estimators

On likening the minimum MSE of the proposed separate estimators $T_{y_i}^s,i=1,2,\dots ,6$ from (69) and (70) with the minimum MSE of the brushed up estimators from (A24), (A25), (A27), (A29), (A31), (A33), (A35), (A37), (A39), (A41), (A43), (A44), (A45) and (A46), we obtain the efficiency conditions given hereunder.

$$\begin{array}{ccc}\hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_m^s\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2{U}_0\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_r^s\right)\hfill \end{array}$$

(85)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2(U_0+U_1-2U_{10})\hfill \\ \hfill MSE(T_{y_i}^s)& <& MSE\left(T_{\beta }^s\right)\hfill \end{array}$$

(86)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(U_0-\frac{U_{10}^2}{U_1}\right)\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_{s{g}_1}^s\right)\hfill \end{array}$$

(87)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2(1-\frac{P_{i_h}^2}{Q_{i_h}})& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-{\lambda }_{s_h\left(\mathrm{opt}\right)}\right)\hfill \\ \hfill MSE(T_{y_i}^s)& <& MSE(T_{s{v}_1}^s)\hfill \end{array}$$

(88)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{(B_{1_h}{D}_{1_h}^2+A_{1_h}^2{E}_{1_h}-2C_{1_h}{D}_{1_h}{E}_{1_h})}{(A_{1_h}{B}_{1_h}-C_{1_h}^2)}\right)\hfill \\ \hfill MSE(T_{y_i}^s)& <& MSE(T_{k{k}_1}^s)\hfill \end{array}$$

(89)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{A_h^2}{4B_h}\right)\hfill \\ \hfill MSE(T_{y_i}^s)& <& MSE\left(T_{k{k}_2}^s\right)\hfill \end{array}$$

(90)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2\left(\frac{{\overline{Y}}_h^2({\upsilon }_h^2{U}_1-1)(U_1{U}_0-U_{10}^2)}{{\upsilon }_h^2{U}_1^2+U_{10}^2-U_1(1+U_0)}\right)\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_{s{g}_2}^s\right)\hfill \end{array}$$

(91)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{U_1}{4{(N+1)}^2}-\frac{{\left(1-\frac{U_1}{8{(N+1)}^2}\right)}^2}{\left(1+U_0-\frac{U_{10}^2}{U_1}\right)}\right)\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_{s{v}_2}^s\right)\hfill \end{array}$$

(92)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{(A_{2_h}{D}_{2_h}^2+B_{2_h}-2C_{2_h}{D}_{2_h})}{(A_{2_h}{B}_{2_h}-C_{2_h}^2)}\right)\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_{s{s}_i}^s\right)\hfill \end{array}$$

(93)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{(F_{i_h}{J}_{i_h}^2+G_{i_h}{I}_{i_h}^2-2H_{i_h}{I}_{i_h}{J}_{i_h})}{(F_{i_h}{G}_{i_h}-H_{i_h}^2)}\right)\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_u^s\right)\hfill \end{array}$$

(94)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(U_0-\frac{U_{10}^2}{U_1}\right)\hfill \end{array}$$

(95)

$$\begin{array}{ccc}\hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_{m{m}_i}^s\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2(U_0+{\lambda }_{i_h}^2{U}_1-2{\lambda }_{i_h}{U}_{10})\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE\left(T_{mm}^s\right)\hfill \end{array}$$

(96)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left({(k_h^{*}-1)}^2+\left(\begin{array}{c}{U}_0+k_h^{{*}^2}{U}_1-2k_h^{*}{U}_{10}\hfill \end{array}\right)\right)\hfill \\ \hfill MSE\left(T_{y_i}^s\right)& <& MSE(T_{s{k}_t}^s)\hfill \end{array}$$

(97)

$$\begin{array}{ccc}\hfill \Rightarrow \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2\left(1-\frac{P_{i_h}^2}{Q_{i_h}}\right)& <& \sum _{h=1}^L{W}_h^2{\overline{Y}}_h^2(U_0+P_t{U}_1^2-2P_t{U}_{10})\hfill \end{array}$$

(98)

If conditions (85) to (98) satisfy, then the proposed separate estimators dominate the existing separate estimators brushed up in Section 2.2.

4.3. Comparison of Proposed Combined and Separate Estimators

We compare the minimum MSE of the proposed combined and separate classes of estimators $T_{y_i}^c$ and $T_{y_i}^s$, $i=1,2,\dots ,6$ as

$$\begin{array}{c}\hfill minMSE\left(T_{y_i}^c\right)-minMSE\left(T_{y_i}^s\right)=\sum _{h=1}^L\left(({\overline{Y}}^2-W_h^2{\overline{Y}}_h^2)-\left({\overline{Y}}^2\frac{P_i^2}{Q_i}-W_h^2{\overline{Y}}_h^2\frac{P_{i_h}^2}{Q_{i_h}}\right)\right)\end{array}$$

(99)

In this case, if the proposed combined and separate estimators are valid and the relationship between the variables X and Y within each stratum is a straight line that passes through origin, then the last term of (99) is usually small and it vanishes.

Additionally, unless $R_h$ is invariant from stratum to stratum, separate estimator probably become more efficient in each stratum whether the sample in every stratum is adequately substantial such that the estimated expression of $MSE\left(T_{y_i}^s\right),i=1,2,\dots ,6$ is reasonable and the cumulative bias which can alter the proposed estimators is negligible, whereas the proposed combined estimators are to be preferably recommended with only a few samples in every stratum (see, [40]).

5. Empirical Study

An empirical study is performed in two subsections, the numerical study utilizing natural populations and the simulation study utilizing hypothetically drawn populations.

5.1. Numerical Study

We consider two real populations to exhibit the execution of the proposed estimators against the estimators brushed up in Section 2. The details of the population are discussed hereunder.

Population 1 [Source: [41], p. 208]: The number of refrigerators sold in the last summer is considered as the auxiliary variable X and the expected sale for the current summer is considered as study variable Y. The descriptive statistics are given in

Table 1. Population 1 summaries.

	Total	Stratum h	1	2	3	4
Population mean	$\overline{X}$ = 85,838	${\overline{X}}_h$	72.08	55.8	73.0	66.1
Population mean	$\overline{Y}$ = 70.02	${\overline{Y}}_h$	79.4	59.4	76.7	64.6
Kurtosis coefficient	${\beta }_2\left(x\right)$ = 0.8678	${\beta }_2\left(x_h\right)$	0.6940	0.8361	0.9756	0.9655
Correlation coefficient	${\rho }_{xy}$ = 0.8673	${\rho }_{x{y}_h}$	0.7810	0.8891	0.9004	0.8990
Variation coefficient	$C_x$ = 0.206	$C_{x_h}$	0.1906	0.2416	0.201	0.1908
Standard deviation	$S_x$ = 13.684	$S_{x_h}$	14.529	9.853	13.892	12.161
Standard deviation	$S_y$ = 13.584	$S_{y_h}$	12.911	13.202	15.053	13.061
Population size	N = 1254	$N_1$	400	216	364	274

for ready reference.

Population 2 [Source: [39]]: The number of production of apple is considered as study variable Y and the quantity of apple trees are considered as auxiliary variable X in 854 villages of Turkey in 1999. The data are collected from the regions of Turkey considered as stratum. The required statistics are given in

Table 2. Population 2 summaries.

	Total	Stratum h	1	2	3	4	5	6
Population mean	$\overline{X}$ = 37,600	${\overline{X}}_h$	24,375	27,421	72,409	74,365	26,441	9844
Population mean	$\overline{Y}$ = 2930	${\overline{Y}}_h$	1536	2212	9384	5588	967	404
Kurtosis coefficient	${\beta }_2\left(x\right)$ = 312.07	${\beta }_2\left(x_h\right)$	25.71	34.57	26.14	97.60	27.47	28.10
Correlation coefficient	${\rho }_{xy}$ = 0.92	${\rho }_{x{y}_h}$	0.82	0.86	0.9	0.99	0.71	0.89
Variation coefficient	$C_x=3.8509$	$C_{x_h}$	2.0180	2.0955	2.2201	3.8404	1.7171	1.9091
Standard deviation	$S_x$ = 144,794	$S_{x_h}$	49,189	57,461	160,757	285,603	45,403	18,794
Standard deviation	$S_y$ = 17,106	$S_{y_h}$	6425	11,552	29,907	28,643	2390	946
Population size	N = 854	$N_h$	106	106	94	171	204	173

for ready reference.

By using the SRSS methodology discussed in the earlier section, we have quantified a stratified ranked set sample of size 9 units with set size 3 and number of cycles 3 from each stratum of both populations. Utilizing the above populations, the MSE and percent relative efficiency (PRE) of various combined and separate estimators $T_i$ regarding the sample mean estimator $T_m$ have been computed. The PRE is tabulated using the expression given hereunder.

$$PRE=\frac{MSE\left(T_m\right)}{MSE\left(T_i\right)}\times 100$$

(100)

The outcomes of the numerical study are exploited in

Table 3. MSE and PRE of combined and separate estimators utilizing real populations.

Combined	Population 1		Population 2		Separate	Population 1		Population 2
Estimators	MSE	PRE	MSE	PRE	Estimators	MSE	PRE	MSE	PRE
$T_m^c$	1219.903	100	43,839.760	100	$T_m^s$	1211.201	100	43.829.561	100
$T_r^c$	416.781	292	8491.407	516	$T_r^s$	411.618	294	8476.984	517
$T_{\beta }^c$/$T_u^c$	27.995	4357	8410.157	521	$T_{\beta }^s$/$T_u^s$	27.781	4359	8396.054	522
$T_{s{g}_1}^c$	36.760	3318	3110.717	1409	$T_{s{g}_1}^s$	36.480	3320	3106.152	1411
$T_{s{v}_1}^c$	27.846	4380	3187.773	1375	$T_{s{v}_1}^s$	27.636	4382	3181.980	1377
$T_{k{k}_1}^c$	416.524	292	4901.103	894	$T_{k{k}_1}^s$	414.648	292	4891.648	896
$T_{k{k}_2}^c$	27.995	4357	8414.176	521	$T_{k{k}_2}^s$	27.789	4358	8401.425	521
$T_{s{g}_2}^c$	27.896	4372	3543.575	1237	$T_{s{g}_2}^s$	27.690	4374	3536.586	1239
$T_{s{v}_2}^c$	27.846	4382	3187.773	1375	$T_{s{v}_2}^s$	27.621	4385	3183.124	1376
$T_{s{s}_1}^c$	27.995	4357	4299.486	1019	$T_{s{s}_1}^s$	27.784	4359	4294.254	1020
$T_{m{m}_1}^c$	416.673	292	8492.241	516	$T_{m{m}_1}^s$	412.587	293	8481.920	516
$T_{m{m}_2}^c$	410.253	297	8456.571	518	$T_{m{m}_2}^s$	405.982	298	8447.043	518
$T_{m{m}_3}^c$	416.778	292	8491.046	516	$T_{m{m}_3}^s$	411.971	294	8481.816	516
$T_{m{m}_4}^c$	403.815	302	8436.252	519	$T_{m{m}_4}^s$	398.127	304	8421.512	520
$T_{mm}^c$	416.993	292	8815.222	497	$T_{mm}^s$	413.002	293	8800.100	498
$T_{s{s}_2}^c$	27.837	4382	8254.505	531	$T_{s{s}_2}^s$	27.620	4385	8238.837	531
$T_{s{s}_3}^c$	27.341	4461	8459.135	518	$T_{s{s}_3}^s$	27.760	4362	8441.812	519
$T_{s{k}_1}^c$	83.812	1455	16,419.040	267	$T_{s{k}_1}^s$	83.128	1457	16,392.575	267
$T_{s{k}_2}^c$	120.263	1014	16,980.127	258	$T_{s{k}_2}^s$	119.201	1016	16,967.403	258
$T_{y_i}^c,i=1,3,4,6$	25.092	4861	3097.777	1415	$T_{y_i}^s,i=1,3,4,6$	24.898	4864	3094.315	1416
$T_{y_i}^c,i=2,5$	23.850	5114	2684.362	1633	$T_{y_i}^s,i=2,5$	23.672	5116	2681.005	1634

that demonstrate the superiority of the proposed classes of estimators over the reviewed estimators. Furthermore, to generalize these results, a simulation study is accomplished in the next section.

5.2. Simulation Study

Following [42], we have conducted a simulation study by utilizing some hypothetically drawn populations, such as Normal, Weibull, and Log-normal each of size N = 1200 with an auxiliary variable X and study variable Y whose values are produced by utilizing the model given hereunder.

$$\begin{array}{cc}\hfill Y_i& =2.8+\sqrt{(1-{\rho }_{xy}^2)}{Y}_i^{*}+{\rho }_{xy}\left(\frac{S_y}{S_x}\right)X_i^{*}\mathrm{and}{X}_i=2.4+X_i^{*}\hfill \end{array}$$

where $X_i^{*}$ and $Y_i^{*}$ are the independent variates of certain distributions to design different populations with various amounts of coefficient of correlation ${\rho }_{xy}$ = 0.6, 0.7, 0.8, and 0.9 between the variables X and Y in each population. We have chosen different valuations of ${\rho }_{xy}$ to notice the deportment of the proposed classes of estimators. Now, each population is stratified into three equally disjoint strata and a ranked set sample of size 9 units with set size 3 and number of cycles 3 is drawn from each stratum by utilizing the sampling methodology described in Section 2. With 10,000 iterations, the MSE and PRE of the combined and separate estimators are tabulated regarding the sample mean estimator as

$$\begin{array}{cc}\hfill PRE& =\frac{{\sum }_{i=1}^{\mathrm{10,000}}{(T_m-\overline{Y})}^2}{{\sum }_{i=1}^{\mathrm{10,000}}{(T_i-\overline{Y})}^2}\times 100\hfill \end{array}$$

The simulation findings are reported from

Table 4. MSE and PRE of combined estimators utilizing hypothetically drawn Normal population.

${\mathit{\rho }}_{\mathit{xy}}$	0.6		0.7		0.8		0.9
Estimators	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$X^{*}\sim N(3,4)$
$Y^{*}\sim N(2,3)$
$T_m^c$	0.123	100	0.123	100	0.133	100	0.138	100
$T_r^c$	0.290	42.373	0.293	42.005	0.303	44.097	0.293	47.168
$T_{\beta }^c$/$T_u^c$	0.096	128.039	0.096	128.399	0.103	128.845	0.106	130.174
$T_{s{g}_1}^c$	0.094	130.587	0.094	130.900	0.101	131.497	0.103	133.015
$T_{s{v}_1}^c$	0.094	130.459	0.094	130.658	0.102	131.132	0.104	132.477
$T_{k{k}_1}^c$	0.227	54.059	0.230	53.598	0.238	56.117	0.231	59.766
$T_{k{k}_2}^c$	0.093	131.091	0.093	131.410	0.101	132.006	0.103	133.531
$T_{s{g}_2}^c$	0.145	84.849	0.145	84.664	0.158	84.598	0.164	84.209
$T_{s{v}_2}^c$	0.094	129.599	0.094	129.923	0.102	130.426	0.104	131.792
$T_{s{s}_1}^c$	0.093	131.361	0.093	131.682	0.101	132.258	0.103	133.727
$T_{m{m}_1}^c$	0.133	92.197	0.134	91.969	0.141	94.788	0.139	99.172
$T_{m{m}_2}^c$	0.144	85.181	0.145	84.892	0.152	87.778	0.149	92.213
$T_{m{m}_3}^c$	0.189	64.890	0.191	64.504	0.199	67.239	0.193	71.345
$T_{m{m}_4}^c$	0.212	57.987	0.214	57.592	0.222	60.178	0.215	64.033
$T_{mm}^c$	0.287	42.793	0.290	42.398	0.300	44.556	0.289	47.705
$T_{s{s}_2}^c$	0.095	129.599	0.094	129.923	0.102	130.426	0.104	131.792
$T_{s{s}_3}^c$	0.095	128.818	0.095	129.240	0.103	129.591	0.105	130.862
$T_{s{k}_1}^c$	0.100	122.501	0.100	122.715	0.107	124.152	0.109	126.759
$T_{s{k}_2}^c$	0.096	126.961	0.096	127.320	0.105	127.414	0.107	128.047
$T_{y_i}^c,i=1,3,4,6$	0.093	131.026	0.093	131.347	0.101	131.939	0.103	133.460
$T_{y_i}^c,i=2,5$	0.092	133.718	0.091	134.049	0.099	134.691	0.101	136.312

,

Table 5. MSE and PRE of combined estimators utilizing hypothetically drawn Weibull population.

${\mathit{\rho }}_{\mathit{xy}}$	0.6		0.7		0.8		0.9
Estimators	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$X^{*}\sim Weibull(2,3)$
$Y^{*}\sim Weibull(2,5)$
$T_m^c$	0.074	100	0.071	100	0.067	100	0.067	100
$T_r^c$	0.070	105.900	0.068	104.239	0.063	105.726	0.062	106.929
$T_{\beta }^c$/$T_u^c$	0.059	125.106	0.056	125.607	0.052	128.128	0.052	128.355
$T_{s{g}_1}^c$	0.058	125.751	0.056	126.191	0.052	128.655	0.052	128.842
$T_{s{v}_1}^c$	0.058	125.809	0.056	126.262	0.052	128.746	0.052	129.087
$T_{k{k}_1}^c$	0.069	107.394	0.067	105.696	0.062	107.150	0.062	109.339
$T_{k{k}_2}^c$	0.058	125.946	0.056	126.395	0.052	128.881	0.052	129.132
$T_{s{g}_2}^c$	0.088	84.128	0.085	83.862	0.081	82.757	0.081	83.312
$T_{s{v}_2}^c$	0.058	125.577	0.056	126.043	0.052	128.517	0.052	129.101
$T_{s{s}_1}^c$	0.058	126.022	0.056	126.484	0.052	128.972	0.052	129.333
$T_{m{m}_1}^c$	0.064	114.809	0.062	113.805	0.057	115.871	0.057	117.474
$T_{m{m}_2}^c$	0.060	122.306	0.058	122.129	0.053	124.656	0.054	125.508
$T_{m{m}_3}^c$	0.066	110.954	0.065	109.635	0.060	111.447	0.059	113.355
$T_{m{m}_4}^c$	0.059	124.108	0.057	124.236	0.052	126.833	0.053	127.392
$T_{mm}^c$	0.070	105.903	0.068	104.280	0.063	105.786	0.062	107.949
$T_{s{s}_2}^c$	0.059	125.577	0.056	126.043	0.052	128.517	0.052	128.842
$T_{s{s}_3}^c$	0.059	125.253	0.056	125.729	0.052	128.216	0.052	128.355
$T_{s{k}_1}^c$	0.061	120.840	0.058	121.575	0.054	123.404	0.055	123.230
$T_{s{k}_2}^c$	0.062	118.397	0.059	119.087	0.055	120.633	0.056	120.603
$T_{y_i}^c,i=1,3,4,6$	0.058	125.944	0.056	126.394	0.052	128.879	0.052	129.118
$T_{y_i}^c,i=2,5$	0.058	126.133	0.056	126.584	0.052	129.074	0.052	129.870

,

Table 6. MSE and PRE of combined estimators utilizing hypothetically drawn Log-normal population.

${\mathit{\rho }}_{\mathit{xy}}$	0.6		0.7		0.8		0.9
Estimators	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$X^{*}\sim Log-normal(0.2,1.5)$
$Y^{*}\sim Log-normal(0.4,2.5)$
$T_m^c$	159.784	100	197.205	100	246.578	100	299.852	100
$T_r^c$	224.332	71	278.205	70	347.856	70	424.957	70
$T_{\beta }^c$/$T_u^c$	159.723	100	197.195	100	246.565	100	299.674	100
$T_{s{g}_1}^c$	113.604	140	139.291	141	171.861	143	206.584	145
$T_{s{v}_1}^c$	113.603	140	139.292	141	171.867	143	206.596	145
$T_{k{k}_1}^c$	111.622	143	136.703	144	167.930	146	201.401	148
$T_{k{k}_2}^c$	99.734	160	122.035	161	149.984	164	179.713	166
$T_{s{g}_2}^c$	113.653	140	139.290	141	171.868	143	206.754	145
$T_{s{v}_2}^c$	113.603	140	139.292	141	171.867	143	206.596	145
$T_{s{s}_1}^c$	113.597	140	139.216	141	171.709	143	206.371	145
$T_{m{m}_1}^c$	193.147	82	235.740	83	293.655	83	354.142	84
$T_{m{m}_2}^c$	188.905	84	232.480	84	289.162	85	348.091	86
$T_{m{m}_3}^c$	217.253	73	267.497	73	333.200	74	398.116	75
$T_{m{m}_4}^c$	213.813	74	263.261	74	328.011	75	394.085	76
$T_{mm}^c$	223.925	71	274.947	71	341.636	72	408.107	73
$T_{s{s}_2}^c$	113.621	140	139.292	141	171.868	143	206.650	145
$T_{s{s}_3}^c$	120.366	132	147.403	133	181.597	135	218.016	137
$T_{s{k}_1}^c$	173.153	92	212.822	92	264.191	93	315.957	94
$T_{s{k}_2}^c$	170.588	93	209.588	94	259.154	95	314.161	95
$T_{y_i}^c,i=1,3,4,6$	77.587	205	95.137	207	117.838	209	140.501	213
$T_{y_i}^c,i=2,5$	71.162	224	86.985	226	108.010	228	128.979	232

,

Table 7. MSE and PRE of separate estimators utilizing hypothetically drawn Normal population.

${\mathit{\rho }}_{\mathit{xy}}$	0.6		0.7		0.8		0.9
Estimators	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$X^{*}\sim N(3,4)$
$Y^{*}\sim N(2,3)$
$T_m^s$	0.134	100	0.133	100	0.149	100	0.154	100
$T_r^s$	0.185	72.745	0.185	71.892	0.196	76.250	0.191	80.579
$T_{\beta }^s$/$T_u^s$	0.105	128.039	0.104	128.399	0.116	128.845	0.118	130.174
$T_{s{g}_1}^s$	0.103	130.365	0.102	130.629	0.113	131.359	0.116	132.895
$T_{s{v}_1}^s$	0.102	131.501	0.101	131.793	0.112	132.510	0.115	134.071
$T_{k{k}_1}^s$	0.163	82.331	0.164	81.382	0.173	86.105	0.170	90.728
$T_{k{k}_2}^s$	0.102	131.479	0.101	131.761	0.112	132.480	0.115	134.034
$T_{s{g}_2}^s$	0.102	131.712	0.101	132.007	0.112	132.702	0.114	134.245
$T_{s{v}_2}^s$	0.103	129.747	0.102	130.051	0.114	130.611	0.116	131.982
$T_{s{s}_1}^s$	0.102	131.051	0.101	131.347	0.113	131.977	0.115	133.423
$T_{m{m}_1}^s$	0.151	88.651	0.151	87.911	0.162	92.220	0.159	96.633
$T_{m{m}_2}^s$	0.118	113.339	0.118	113.066	0.128	116.112	0.129	119.590
$T_{m{m}_3}^s$	0.168	79.706	0.169	78.887	0.179	83.283	0.175	87.702
$T_{m{m}_4}^s$	0.111	120.551	0.110	120.507	0.121	122.746	0.122	125.549
$T_{mm}^s$	0.181	74.099	0.182	73.190	0.192	77.732	0.187	82.204
$T_{s{s}_2}^s$	0.123	109.412	0.122	109.431	0.136	109.541	0.140	109.729
$T_{s{s}_3}^s$	0.105	128.046	0.104	128.400	0.116	128.859	0.118	130.193
$T_{s{k}_1}^s$	0.106	125.888	0.105	126.071	0.117	127.379	0.119	129.364
$T_{s{k}_2}^s$	0.105	128.037	0.104	128.391	0.116	128.812	0.118	129.953
$T_{y_i}^s,i=1,3,4,6$	0.102	131.307	0.101	131.592	0.113	132.298	0.115	133.841
$T_{y_i}^s,i=2,5$	0.101	132.996	0.100	133.286	0.111	134.017	0.113	135.599

,

Table 8. MSE and PRE of separate estimators utilizing hypothetically drawn Weibull population.

${\mathit{\rho }}_{\mathit{xy}}$	0.6		0.7		0.8		0.9
Estimators	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$X^{*}\sim Weibull(2,3)$
$Y^{*}\sim Weibull(2,5)$
$T_m^s$	0.075	100	0.072	100	0.068	100	0.069	100
$T_r^s$	0.067	112.388	0.065	111.345	0.060	112.996	0.060	115.448
$T_{\beta }^s$/$T_u^s$	0.060	125.106	0.058	125.607	0.053	128.128	0.053	128.942
$T_{s{g}_1}^s$	0.060	125.437	0.057	125.865	0.053	128.285	0.053	129.182
$T_{s{v}_1}^s$	0.059	126.023	0.057	126.478	0.052	128.957	0.053	129.906
$T_{k{k}_1}^s$	0.066	113.728	0.064	112.651	0.059	114.264	0.059	116.806
$T_{k{k}_2}^s$	0.059	125.963	0.057	126.415	0.052	128.894	0.053	129.664
$T_{s{g}_2}^s$	0.059	126.189	0.057	126.654	0.052	129.158	0.053	129.906
$T_{s{v}_2}^s$	0.060	125.585	0.057	126.053	0.053	128.522	0.053	129.422
$T_{s{s}_1}^s$	0.059	125.988	0.057	126.452	0.052	128.932	0.053	129.905
$T_{m{m}_1}^s$	0.065	114.875	0.064	114.023	0.058	115.865	0.058	118.197
$T_{m{m}_2}^s$	0.060	124.137	0.058	124.314	0.053	126.848	0.054	127.992
$T_{m{m}_3}^s$	0.066	113.702	0.064	112.757	0.059	114.509	0.059	116.806
$T_{m{m}_4}^s$	0.062	119.748	0.060	120.430	0.055	122.188	0.056	122.574
$T_{mm}^s$	0.066	112.707	0.065	111.661	0.060	113.306	0.060	115.833
$T_{s{s}_2}^s$	0.072	104.460	0.069	104.540	0.065	104.759	0.065	105.623
$T_{s{s}_3}^s$	0.060	125.294	0.058	125.769	0.053	128.250	0.053	129.182
$T_{s{k}_1}^s$	0.061	123.301	0.058	123.997	0.053	126.200	0.054	126.824
$T_{s{k}_2}^s$	0.061	122.517	0.059	123.223	0.054	125.320	0.055	125.905
$T_{y_i}^s,i=1,3,4,6$	0.059	125.959	0.057	126.411	0.052	128.890	0.053	129.785
$T_{y_i}^s,i=2,5$	0.059	126.298	0.057	126.782	0.052	129.222	0.053	130.149

and

Table 9. MSE and PRE of separate estimators utilizing hypothetically drawn Log-normal population.

${\mathit{\rho }}_{\mathit{xy}}$	0.6		0.7		0.8		0.9
Estimators	MSE	PRE	MSE	PRE	MSE	PRE	MSE	PRE
$X^{*}\sim Log-normal(0.2,1.5)$
$Y^{*}\sim Log-normal(0.4,2.5)$
$T_m^s$	159.785	100	197.206	100	246.579	100	299.853	100
$T_r^s$	224.322	71	277.980	70	341.128	72	398.128	75
$T_{\beta }^s$/$T_u^s$	159.051	100	197.195	100	246.565	100	299.674	100
$T_{s{g}_1}^s$	113.592	140	138.924	141	171.018	144	205.148	146
$T_{s{v}_1}^s$	113.591	140	138.998	141	170.985	144	205.051	146
$T_{k{k}_1}^s$	111.013	143	135.784	145	167.018	147	200.744	149
$T_{k{k}_2}^s$	98.998	161	121.089	162	149.001	165	178.673	167
$T_{s{g}_2}^s$	113.085	141	138.214	142	170.986	144	205.018	146
$T_{s{v}_2}^s$	113.053	141	138.114	142	170.889	144	205.324	146
$T_{s{s}_1}^s$	113.127	141	139.010	141	171.001	144	205.231	146
$T_{m{m}_1}^s$	193.001	82	235.742	83	293.100	84	352.050	85
$T_{m{m}_2}^s$	188.142	84	232.481	84	288.903	85	346.267	86
$T_{m{m}_3}^s$	217.007	73	267.498	73	332.000	74	394.512	76
$T_{m{m}_4}^s$	213.090	74	263.262	74	327.100	75	389.248	77
$T_{mm}^s$	223.102	71	274.901	71	340.048	72	405.154	74
$T_{s{s}_2}^s$	113.053	141	138.154	142	170.768	144	205.115	146
$T_{s{s}_3}^s$	119.987	133	146.781	134	180.498	136	217.105	138
$T_{s{k}_1}^s$	173.889	92	212.819	92	264.188	93	314.421	95
$T_{s{k}_2}^s$	170.000	93	209.558	94	259.151	95	312.121	96
$T_{y_i}^s,i=1,3,4,6$	77.412	206	94.589	208	117.014	210	139.652	214
$T_{y_i}^s,i=2,5$	70.912	225	86.584	227	107.290	229	127.889	234

that demonstrate the superiority of the proposed estimators against the reviewed estimators for reasonably chosen values of ${\rho }_{xy}$.

6. Interpretation of Empirical Results

The clarification of the empirical results is given in the following points.

(i).

The results of the real populations displayed in Table 3 demonstrate the ascendancy of the proposed combined estimators $T_{y_i}^{\mathrm{c}},i=1,2,\dots ,6$ over the reviewed combined estimators, namely, conventional mean estimator $T_m^{\mathrm{c}}$, classical ratio and regression estimators $T_r^{\mathrm{c}}$ and $T_{\beta }^{\mathrm{c}}$, [31,35] type estimators $T_{s{g}_1}^{\mathrm{c}}$ and $T_{s{g}_2}^{\mathrm{c}}$, [33,34] type estimators $T_{k{k}_1}^{\mathrm{c}}$ and $T_{k{k}_2}^{\mathrm{c}}$, [32,36] type estimators $T_{s{v}_1}^{\mathrm{c}}$ and $T_{s{v}_2}^{\mathrm{c}}$, [1] type estimator $T_{s{s}_1}^{\mathrm{c}}$, [2,4] type estimators $T_{s{s}_2}^{\mathrm{c}}$ and $T_{s{s}_3}^{\mathrm{c}}$, [3] type estimator $T_u^{\mathrm{c}}$, [9] estimators $T_{m{m}_i}^{\mathrm{c}},i=1,2,3,4$, [38] estimator $T_{mm}^{\mathrm{c}}$, and [14] estimators $T_{s{k}_t}^{\mathrm{c}},t=1,3$.

(ii).

The results of normal population disclosed in Table 4 show the superiority of the proposed combined estimators $T_{y_i}^{\mathrm{c}},i=2,5$ over the existing combined estimators. Additionally, the proposed combined estimators $T_{y_i}^{\mathrm{c}},i=1,3,4,6$ are found to be fare efficient than most of the combined estimators existing till date but found to be less efficient than [34] type estimator $T_{k{k}_2}^{\mathrm{c}}$ and [1] type estimator $T_{s{s}_1}^{\mathrm{c}}$. The similar interpretation can also be drawn from the results of Weibull population disclosed in Table 5.

(iii).

From the results of Log-normal population reported in Table 6, the ascendancy of the proposed combined estimators $T_{y_i}^{\mathrm{c}},i=1,2,\dots ,6$ can be observed over the existing combined estimators.

(iv).

From Table 3, Table 4, Table 5 and Table 6, the proposed combined class of estimators $T_{y_i}^{\mathrm{c}},i=2,5$ perform fare better among the proposed combined classes of estimators.

(v).

The similar conclusions, such as points (i) to (iv), can also be drawn regarding the proposed separate classes of estimators $T_{y_i}^{\mathrm{s}},i=1,2,\dots ,6$ from the results summarized in Table 3, Table 7, Table 8 and Table 9.

(vi).

From the results of Table 4 and Table 7 based on Normal population, the proposed separate estimators $T_{y_i}^{\mathrm{s}},i=1,3,4,6$ dominate the proposed combined estimators $T_{y_i}^{\mathrm{c}},$$i=1,3,4,6$ whereas, the proposed combined estimators $T_{y_i}^{\mathrm{c}},i=2,5$ surpass the proposed separate estimators $T_{y_i}^{\mathrm{s}},i=2,5$.

(vii).

From the results of Table 3 consisting of the real populations and Table 5, Table 6, Table 8 and Table 9 consisting of the simulated Weibull and Log-normal populations, the proposed separate estimators $T_{y_i}^{\mathrm{s}},i=1,2,\dots ,6$ dominate the proposed combined estimators $T_{y_i}^{\mathrm{c}},i=1,2,\dots ,6$.

(viii).

Moreover, from the results of Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, the PRE of the proposed combined and separate classes of estimators increases as the value of ${\rho }_{xy}$ from 0.6 to 0.9 over the successive increase of 0.1.

(ix).

The results of the numerical study using real populations, which are reported in Table 3 are also presented through the line diagrams given in Figure 1. The dominance of the proposed combined and separate estimators can easily be observed from Figure 1. As the PRE of the simulation results of Table 4, Table 5, Table 6, Table 7 and Table 8 also exhibit the same pattern and can be easily presented through line diagrams, if required.

7. Conclusions

This manuscript considers some improved classes of combined and separate difference and ratio type estimators for the population mean utilizing SRSS. Some prominent existing estimators, namely, the usual mean estimator, classical ratio estimator envisaged by [8], classical regression estimator suggested by [9,10] estimators, and [38] estimators are identified as the members of the proposed classes of estimators for particularly chosen values of optimizing scalars. The MSE expressions of the proposed combined and separate estimators have been reported to the first order of approximation. The efficiency conditions are determined under which the proposed classes of estimators surpass the reviewed estimators. Furthermore, an empirical study is carried out extensively by using some hypothetically drawn and real populations. The empirical findings are established to be highly rewarding by means of lesser MSE and greater PRE providing improvement over the contemporary estimators discussed till date under SRSS. Thus, the motivation of the study is justified.

On the lines of [8], the entire study can be reprized when the ranking is accomplished on the variable Y and the ranking of the variable X is imperfect.