Almost unbiased optimum estimators for population mean using dual auxiliary information

1 Introduction

Utilizing the auxiliary information to boost the efficiency of estimators is a common practice in the theory of survey sampling. The auxiliary information such as standard deviation $S_x$, coefficient of variation$C_x$, coefficient of skewness ${\beta }_{1\left(x\right)}$, coefficient of kurtosis ${\beta }_{2\left(x\right)}$, coefficient of correlation ${\rho }_{\mathit{yx}}$ etc. may play positive role in the selection of sample, strata, type of estimators or in estimation. If this auxiliary information is positively (high) correlated with study variable, ratio estimators are preferred and in case it is negatively (high) correlated, product estimators are used. In this context, some notable contributions were made by Upadhyaya and Singh (1999), Singh and Tailor (2003), Kadilar and Cingi (2006a, 2006b), Gupta and Shabbir (2008), Shabbir and Gupta (2011), Haq and Shabbir (2013), Singh and Solanki (2013), Irfan et al. (2019a, 2019b), Raza et al. (2020) and many others.

Consider a sample of $n$ pair of observations ($y_i,x_i),i=1,2,3,\cdots ,n$ for the study and auxiliary variables, respectively are selected from a finite population $\mathrm{\Theta }=\left\{{\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2,{\mathrm{\Theta }}_3,\cdots ,{\mathrm{\Theta }}_N\right\}$of size $"N"$ under simple random sampling without replacement (SRSWOR) subject to the constraint$n<N$. Let $r$ denotes the ranks of auxiliary variable and $r_i$ denotes the $\mathit{ith}$value of $r$ in the population. Important measures related to study variable $y$, auxiliary variable $x$ and the ranks of auxiliary variable $r$ are described in Table 1.

• $s_y^2,s_x^2\mathrm{a}\mathrm{n}\mathrm{d}{s}_r^2$ are also unbiased estimators of $S_y^2,S_x^2\mathrm{a}\mathrm{n}\mathrm{d}{S}_r^2,$ respectively.

• Similarly,$s_{\mathit{yx}},s_{\mathit{yr}}\mathrm{a}\mathrm{n}\mathrm{d}{s}_{\mathit{xr}}$are the unbiased estimators of their population parameters $S_{\mathit{yx}},S_{\mathit{yr}}\mathrm{a}\mathrm{n}\mathrm{d}{S}_{\mathit{xr}}$ respectively.

2 Unbiased/Almost unbiased estimators from literature

Usually, ratio and product type of estimators of population mean are biased and inconsistent and thus can lead to erroneous inferences. Several researchers have attempted to reduce the bias from these estimators as unbiasedness is one of the important properties of estimators. Unbiased ratio and product type estimators have also been discussed by Hartely and Ross (1954), Robson (1957), Murthy and Nanjamma (1959), Biradar and Singh (1992a, 1992b, 1995), Sahoo et al. (1994) and Javed et al. (2019).

This section presents a comprehensive detail of unbiased/almost unbiased estimators of population mean under simple random sampling scheme from literature.

3 Methodology

All contributions for efficient estimation of population mean under simple random sampling scheme and alike published work are based on only the utilization of original auxiliary information. None of them tried the dual use of auxiliary information to explore the unbiased estimators for population mean under simple random sampling.

Recently, Irfan et al. (2020) and Javed and Irfan (2020) used an additional information of the auxiliary variable called ranked auxiliary variable to develop efficient estimators under simple and stratified random sampling.

First time, we initiated a blend of three concepts to explore an optimum class of almost unbiased estimators for estimating the population mean:

1. information on auxiliary variable

2. the ranks of auxiliary variable

3. Hartley-Ross type unbiased estimation

A class of biased estimators proposed by Haq et al. (2017) is as follows:

Bias of the class given in Eq. (14) is derived, up to first order of approximation as

Subtracting Eq. (15) from Eq. (14), we obtained the expression given below

After some simplification and replacing the parameters $\stackrel{-}{Y}$ and $S_{\mathit{yx}}$ by their unbiased estimators $\stackrel{-}{y}$ and $s_{\mathit{yx}}$ in Eq. (16), we have$${\stackrel{-}{y}}_H+\stackrel{-}{y}-\frac{1}{2}{k}_3\phi \lambda \stackrel{-}{X}{C}_x^2-\frac{1}{2}{k}_4\phi \lambda C_{\mathit{xr}}\stackrel{-}{R}-k_2\stackrel{-}{y}-\frac{3}{8}{k}_2\phi {\lambda }^2\stackrel{-}{y}{C}_x^2+\frac{1}{2}{k}_2\phi \lambda \frac{s_{\mathit{yx}}}{\stackrel{-}{X}}$$

So, the proposed class of almost unbiased estimators is as follows:

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\lambda =\frac{\alpha \stackrel{-}{X}}{\alpha \stackrel{-}{X}+\beta },\left(k_2,k_3\mathrm{a}\mathrm{n}\mathrm{d}{k}_4\right)$ are the suitable weights to be chosen. $\alpha \left(\ne 0\right)$ and $\beta$ are either known constants or functions of any known population parameters of auxiliary variable including coefficient of skewness ${\beta }_{1\left(x\right)}$, coefficient of kurtosis ${\beta }_{2\left(x\right)}$, coefficient of variation $C_x$ and coefficient of correlation ${\rho }_{\mathit{yx}}$etc.

Following are the relative error terms along with their expectations, used to derive the expressions for the bias, variance and minimum variance of the proposed estimators.$${\omega }_0=\frac{\stackrel{-}{y}-\stackrel{-}{Y}}{\stackrel{-}{Y}},{\omega }_1=\frac{\stackrel{-}{x}-\stackrel{-}{X}}{\stackrel{-}{X}},{\omega }_2=\frac{\stackrel{-}{r}-\stackrel{-}{R}}{\stackrel{-}{R}},{\omega }_3=\frac{s_{\mathit{yx}}-S_{\mathit{yx}}}{S_{\mathit{yx}}}$$

such that$$E\left({\omega }_i\right)=0\mathrm{f}\mathrm{or}i=0,1,2,3$$$$E\left({\omega }_0^2\right)=\phi C_y^2,E\left({\omega }_1^2\right)=\phi C_x^2,E\left({\omega }_2^2\right)=\phi C_r^2,E\left({\omega }_3^2\right)=\phi \left(\frac{{\theta }_{22x}}{{\rho }_{\mathit{yx}}}-1\right),$$$$E\left({\omega }_0{\omega }_1\right)=\phi {\rho }_{\mathit{yx}}{C}_y{C}_x=\phi C_{\mathit{yx}},E\left({\omega }_0{\omega }_2\right)=\phi {\rho }_{\mathit{yr}}{C}_y{C}_r=\phi C_{\mathit{yr}},$$$$E\left({\omega }_0{\omega }_3\right)=\phi \left(\frac{C_y{\theta }_{21x}}{{\rho }_{\mathit{yx}}}\right),E\left({\omega }_1{\omega }_2\right)=\phi {\rho }_{\mathit{xr}}{C}_x{C}_r=\phi C_{\mathit{xr}},$$$$E\left({\omega }_1{\omega }_3\right)=\phi \left(\frac{C_x{\theta }_{12x}}{{\rho }_{\mathit{yx}}}\right),E\left({\omega }_2{\omega }_3\right)=\phi \left(\frac{C_r{\theta }_{12r}}{{\rho }_{\mathit{yr}}}\right).$$

In order to obtain the values of ${\theta }_{21x},{\theta }_{12x},{\theta }_{22x}\mathrm{a}\mathrm{n}\mathrm{d}{\theta }_{12r}$, following expressions are helpful.$$\left.\begin{array}{c}{\theta }_{\mathit{abx}}=\frac{{\mu }_{\mathit{abx}}}{\left({\mu }_{20x}^{\frac{a}{2}}\right)\left({\mu }_{02x}^{\frac{b}{2}}\right)},{\mu }_{\mathit{abx}}=\frac{{\sum }_{i=1}^N{\left(y_i-\stackrel{-}{Y}\right)}^a{\left(x_i-\stackrel{-}{X}\right)}^b}{N}\\ {\theta }_{\mathit{abr}}=\frac{{\mu }_{\mathit{abr}}}{\left({\mu }_{20r}^{\frac{a}{2}}\right)\left({\mu }_{02r}^{\frac{b}{2}}\right)},{\mu }_{\mathit{abr}}=\frac{{\sum }_{i=1}^N{\left(y_i-\stackrel{-}{Y}\right)}^a{\left(r_i-\stackrel{-}{R}\right)}^b}{N}\end{array}\right\}\mathrm{f}\mathrm{or}a=b=0,1,2$$

After rewriting ${\stackrel{-}{y}}_P^{\left(u\right)}$ in terms of relative errors and expanding up to first order of approximation, we get

Subtracting $\stackrel{-}{Y}$ from both sides of Eq. (18), we have

Taking expectation on both sides of Eq. (19) to get the $Bias\left({\stackrel{-}{y}}_P^{\left(u\right)}\right)$$$Bias\left({\stackrel{-}{y}}_P^{\left(u\right)}\right)=E\left({\stackrel{-}{y}}_P^{\left(u\right)}-\stackrel{-}{Y}\right)=-\frac{1}{2}{k}_2\stackrel{-}{Y}\lambda \phi C_{\mathit{yx}}+\frac{1}{2}{k}_4\stackrel{-}{R}\lambda \phi C_{\mathit{xr}}+\left(\frac{3}{8}{k}_2\stackrel{-}{Y}{\lambda }^2+\frac{1}{2}{k}_3\stackrel{-}{X}\lambda \right)\phi C_x^2-\frac{1}{2}\phi \lambda \left(k_3\stackrel{-}{X}{C}_x^2+k_4{C}_{\mathit{xr}}\stackrel{-}{R}+\frac{3}{4}{k}_2\stackrel{-}{Y}\lambda C_x^2-k_2\stackrel{-}{Y}{C}_{\mathit{yx}}\right)$$$$E\left({\stackrel{-}{y}}_P^{\left(u\right)}-\stackrel{-}{Y}\right)=-\frac{1}{2}{k}_2\stackrel{-}{Y}\lambda \phi C_{\mathit{yx}}+\frac{1}{2}{k}_4\stackrel{-}{R}\lambda \phi C_{\mathit{xr}}+\frac{3}{8}{k}_2\stackrel{-}{Y}{\lambda }^2\phi C_x^2+\frac{1}{2}{k}_3\stackrel{-}{X}\lambda \phi C_x^2+\frac{1}{2}{k}_2\stackrel{-}{Y}\lambda \phi C_{\mathit{yx}}-\frac{1}{2}{k}_4\stackrel{-}{R}\lambda \phi C_{\mathit{xr}}-\frac{3}{8}{k}_2\stackrel{-}{Y}{\lambda }^2\phi C_x^2-\frac{1}{2}{k}_3\stackrel{-}{X}\lambda \phi C_x^2$$

During simplification, all the terms cancel out and we get zero bias which shows that the proposed class generates almost unbiased estimators. As the first order approximation is used in deriving the expression therefore the term “almost” is added here. So,$$Bias\left({\stackrel{-}{y}}_P^{\left(u\right)}\right)=E\left({\stackrel{-}{y}}_P^{\left(u\right)}-\stackrel{-}{Y}\right)\cong 0$$

Squaring both sides of Eq. (19) and taking the expectation, we get the variance of proposed estimators up to first order of approximation as:

where$$A_1=\frac{3}{4}\phi \lambda C_y^2{C}_x^2+C_{\mathit{yx}}-\frac{\phi C_{\mathit{yx}}{C}_y{\theta }_{21x}}{{\rho }_{\mathit{yx}}}$$$$A_2=\frac{1}{4}{C}_x^2-\frac{9}{64}\phi {\lambda }^2{C}_x^4-\frac{1}{4}{\phi C}_{\mathit{yx}}^2-\frac{\phi C_{\mathit{yx}}{C}_x{\theta }_{12x}}{2{\rho }_{\mathit{yx}}}+\frac{3}{4}\phi \lambda C_x^2{C}_{\mathit{yx}}$$$$A_3=1-\frac{1}{4}\phi {\lambda }^2{C}_x^2$$$$A_4=1-\frac{1}{4}\phi {\lambda }^2{C}_x^2{\rho }_{\mathit{xr}}^2$$$$A_5=C_x+\frac{5}{4}\phi \lambda C_x{C}_{\mathit{yx}}-\frac{\phi C_{\mathit{yx}}{\theta }_{12x}}{{\rho }_{\mathit{yx}}}-\frac{3}{8}\phi {\lambda }^2{C}_x^3$$$$A_6=\frac{3}{4}\phi \lambda C_x^2{C}_{\mathit{yr}}+C_{\mathit{xr}}-\frac{\phi C_{\mathit{yx}}{C}_r{\theta }_{12r}}{{\rho }_{\mathit{yr}}}+\frac{1}{2}\phi \lambda C_{\mathit{xr}}{C}_{\mathit{yx}}-\frac{3}{8}\phi {\lambda }^2{C}_x^2{C}_{\mathit{xr}}$$

Partially differentiating Eq. (20) with respect to $k_2,k_3$ and $k_4$ and equating them to zero, we get the optimal values of $k_2,k_3$ and $k_4$ as follows.$$k_{2\left(opt\right)}=\frac{2C_1}{\lambda C_2}$$$$k_{3\left(opt\right)}=\frac{\stackrel{-}{Y}\left[A_1{B}_2{C}_2-4A_2{B}_2{C}_1-A_6\left(B_3{C}_2-B_6{C}_1\right)\right]}{\stackrel{-}{X}{C}_x{A}_5{B}_2{C}_2}$$$$k_{4\left(opt\right)}=\frac{\stackrel{-}{Y}\left[B_3{C}_2-B_6{C}_1\right]}{{\stackrel{-}{R}B}_2{C}_2}$$

Placing these optimal values in Eq. (20), we obtained the minimum variance as given by$$V_{\mathit{min}}\left({\stackrel{-}{\mathrm{y}}}_{\mathrm{P}}^{\left(\mathrm{u}\right)}\right)\cong \frac{1}{C_x{A}_5^2{B}_2^2{C}_2^2}{\stackrel{-}{Y}}^2\phi \left[C_x{A}_5^2{B}_2^2\left\{C_y^2{C}_2^2-2A_1{C}_1{C}_2+4A_2{C}_1^2\right\}+D_1\left\{C_x{A}_3{D}_1-2C_{\mathit{yx}}{A}_5{B}_2{C}_2+2C_x{A}_5^2{B}_2{C}_1\right\}+C_x{A}_5^2{D}_2\left\{C_r^2{A}_4{D}_2-2C_{\mathit{yr}}{B}_2{C}_2+2A_6{B}_2{C}_1\right\}+2C_{\mathit{xr}}{A}_3{A}_5{D}_1{D}_2\right](21)$$

where$$B_1=C_x{A}_1{A}_3-C_{\mathit{yx}}{A}_5,B_2=C_{\mathit{xr}}{A}_3{A}_6-C_r^2{C}_x{A}_4{A}_5,B_3=C_{\mathit{xr}}{A}_1{A}_3-C_{\mathit{yr}}{C}_x{A}_5$$$$B_4=C_x{A}_3{A}_6-C_{\mathit{xr}}{A}_3{A}_5,B_5=C_x\left(4A_2{A}_3-A_5^2\right),B_6=4C_{\mathit{xr}}{A}_2{A}_3-C_x{A}_5{A}_6$$$$C_1=B_1{B}_2-B_3{B}_4,C_2=B_2{B}_5-B_4{B}_6$$$$D_1={A_{1}B}_2{C}_2-4{A_{2}B}_2{C}_1-A_6\left(B_3{C}_2-B_6{C}_1\right),D_2=\left(B_3{C}_2-B_6{C}_1\right)$$

4 Results and discussion

In this section, we evaluated the performance of proposed class of estimators as compared to other unbiased/almost unbiased estimators. For this purpose, we selected five real life data sets with different correlation coefficients (first three with positive and last two with negative) between study variable and auxiliary variable. The descriptions of the populations are given below.

Population 1: [Source: Singh and Mangat (1996), p. 369]$$y=\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{of}\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$$$x=\mathrm{N}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\left(\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{s}\right)\mathrm{f}\mathrm{o}\mathrm{r}69\mathrm{v}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{D}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{o}\mathrm{f}$$$$\mathrm{P}\mathrm{u}\mathrm{n}\mathrm{j}\mathrm{a}\mathrm{b},\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{a}$$$$N=69,n=10,\stackrel{-}{Y}=135.2609,\stackrel{-}{X}=345.7536,\stackrel{-}{R}=35,$$$$C_y=0.8422,C_x=0.8422,C_r=0.5732,{\rho }_{\mathit{yx}}=0.9224,{\rho }_{\mathit{yr}}=0.7136,$$$${\rho }_{\mathit{xr}}=0.8185,{\beta }_{2(x)}=7.2159,{\beta }_{1(x)}=2.3808$$

Population 2: [Source: Cochran (1977), p.152]$$y=\mathrm{P}\mathrm{o}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{i}\mathrm{n}1930$$$$x=\mathrm{P}\mathrm{o}\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{i}\mathrm{n}1920$$$$N=49,n=12,\stackrel{-}{Y}=127.7959,\stackrel{-}{X}=103.1429,\stackrel{-}{R}=25,$$$$C_y=0.9634,C_x=1.0122,C_r=0.5715,{\rho }_{\mathit{yx}}=0.9817,{\rho }_{\mathit{yr}}=0.7207,$$$${\rho }_{\mathit{xr}}=0.7915,{\beta }_{2(x)}=5.1412,{\beta }_{1(x)}=2.2553$$

Population 3: [Source: Singh and Mangat (1996), p. 369]$$y=\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$$$x=\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}69\mathrm{v}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{D}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{o}\mathrm{f}\mathrm{P}\mathrm{u}\mathrm{n}\mathrm{j}\mathrm{a}\mathrm{b},\mathrm{I}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{a}$$$$N=69,n=10,\stackrel{-}{Y}=135.2609,\stackrel{-}{X}=21.2319,\stackrel{-}{R}=35,$$$$C_y=0.8422,C_x=0.7969,C_r=0.5726,{\rho }_{\mathit{yx}}=0.9119,{\rho }_{\mathit{yr}}=0.7364,$$$${\rho }_{\mathit{xr}}=0.8616,{\beta }_{2(x)}=3.7653,{\beta }_{1(x)}=1.8551$$

Population 4: [Source: Gujarati (2004), p. 433]$$y=\mathrm{A}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{s}$$$$x=\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\left(\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\right)\mathrm{o}\mathrm{f}81\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{s}$$$$N=81,n=16,\stackrel{-}{Y}=33.8346,\stackrel{-}{X}=112.4568,\stackrel{-}{R}=41,$$$$C_y=0.2972,C_x=0.1256,C_r=0.5728,{\rho }_{\mathit{yx}}=-0.6908,{\rho }_{\mathit{yr}}=-0.7298,$$$${\rho }_{\mathit{xr}}=0.8456,{\beta }_{2\left(x\right)}=4.1454,{\beta }_{1\left(x\right)}=1.9016$$

Population 5: [Source: Gujarati (2004), p. 433]$$y=\mathrm{A}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{s}$$$$x=\mathrm{C}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{c}\mathrm{f}\mathrm{e}\mathrm{e}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{c}\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}81\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{s}$$$$N=81,n=18,\stackrel{-}{Y}=33.8346,\stackrel{-}{X}=98.7654,\stackrel{-}{R}=41,$$$$C_y=0.2972,C_x=0.2258,C_r=0.5727,{\rho }_{\mathit{yx}}=-0.3683,{\rho }_{\mathit{yr}}=-0.4732,$$$${\rho }_{\mathit{xr}}=0.9245,{\beta }_{2\left(x\right)}=0.9202,{\beta }_{1\left(x\right)}=-0.5902$$

We calculated the variances of all the estimators i.e. ${\stackrel{-}{y}}_0^{\left(u\right)},{\stackrel{-}{y}}_{\mathit{HR}}^{\left(u\right)},{\stackrel{-}{y}}_{S1}^{\left(u\right)},{\stackrel{-}{y}}_{S2}^{\left(u\right)},{\stackrel{-}{y}}_{CK1}^{\left(u\right)},{\stackrel{-}{y}}_{CK2}^{\left(u\right)}\mathrm{a}\mathrm{n}\mathrm{d}{\stackrel{-}{y}}_P^{\left(u\right)}$for the populations 1–5. Expressions for the variances of all the existing and proposed estimators are given in section 1 & section 3 in detail. All empirical results are summarized in Tables 2-6.