In this paper, a modified ratio cum product estimator for the estimation of finite population mean of the study variable using the known correlation coefficient of the auxiliary variable is introduced. The bias and mean squared error of the proposed estimator are also obtained. The relative performance of the proposed estimator along with some existing estimators is accessed for certain labeled and natural populations. The results show that the proposed estimator is to be more efficient than the existing estimators.

Keywords: bias, mean squared error, natural population, simple random sampling, linear regression estimator

In sampling theory, a wide variety of techniques is used to obtain efficient estimators for the population mean. The commonly used method to obtain the estimator for population mean is simple random sampling without replacement (SRSWOR) when there is no auxiliary variable available. There are methods that use the auxiliary information of the study characteristics. If there exists an auxiliary variable X which is correlated with the study variable Y, then a number of estimators such as ratio, product, modified ratio, modified product, regression estimators and their modifications are widely available for estimation of population mean of the study variable Y.

Consider a finite population $U=\left\{U_1,U_2,U_3\mathrm{....}{U}_N\right\}$ of N distinct and identifiable units. Let Y be the study variable which takes the values $Y=\left\{Y_1,Y_2,Y_3,\mathrm{...}{Y}_N\right\}$ . Here the problem is to estimate the population mean $\overline{Y}=\frac{1}{N}{\displaystyle {\sum }_{i=1}^N{Y}_i}$ on the basis of a random sample selected from the population U.

Before discussing further the various estimators, the notations to be used in this article are listed here.

N                             -               Population size
n                              -               Sample size         
$f=\frac{n}{N}$                    -               Sampling fraction
Y                              -               Study variable
X                              -               Auxiliary variable
$\overline{X},\overline{Y}$                        -               Population means
$\overline{x},\overline{y}$                         -               Sample means
$S_x,S_y$      -              Population standard deviations
$S_x,S_y$                     -               Sample standard deviations
$C_x,C_y$                    -               Coefficient of variations
$\rho$                            -               Correlation coefficient between x and y
${\beta }_1$                            -               Coefficient of skewness
${\beta }_2$                           -               Coefficient of kurtosis
B(.)                         -               Bias of estimators
MSE(.)                   -               Mean squared error of estimators

In simple random sampling without replacement, the estimator ${\overline{y}}_{srs}$ is an unbiased estimator for the population mean $\overline{Y}$ and its variance is given by                           

$V\left({\overline{y}}_{srs}\right)=\delta S_y^2$                                                                                                                   (1)        

Where $\delta =\left(\frac{1-f}{n}\right)$

Cochran,¹ use auxiliary information for the estimation of population mean of the variable under study and proposed the ratio estimator of the population mean $\overline{Y}$ of the study variable,

${\widehat{Y}}_R=\frac{\overline{y}}{\overline{x}}\overline{X}=\widehat{R}\overline{X}$

The bias and mean squared error of the ratio estimator are given by

$B\left({\stackrel{⌢}{\overline{Y}}}_R\right)=\delta \overline{Y}\left[C_{{}_x^2}-\rho C_x{C}_y\right]$

$MSE\left({\stackrel{⌢}{\overline{Y}}}_R\right)=\delta {\overline{Y}}^2\left[C_y^2+C_x^2-2\rho C_x{C}_y\right]$                                     (2)

The linear regression estimator and its variance are given by

${\overline{y}}_{lr}=\overline{y}+b\left(\overline{X}-\overline{x}\right)$

$V\left({\overline{y}}_{lr}\right)=\delta S_y^2\left(1-{\rho }^2\right)$                                                              (3)

where b is the regression coefficient Y on X

Murthy² proposed the product estimator to estimate the population mean of the study variable when there is a negative correlation between the study variable Y and auxiliary variable X as

${\stackrel{⌢}{\overline{Y}}}_p={\overline{y}}_{\overline{X}}^{\overline{x}}$

The bias and the mean squared error of the product estimator are given by

$B\left({\overline{Y}}_p\right)=\delta \overline{Y}\left[\rho C_x{C}_y\right]$     

$MSE\left({\stackrel{⌢}{\overline{Y}}}_p\right)=\delta {\overline{Y}}^2\left[C_y^2+C_x^2+2\rho C_x{C}_y\right]$                                                                                 (4)

Singh and Tailor³ introduced the modified ratio estimator for the population mean with known population correlation coefficient ρ of the auxiliary variable and is given by

${\stackrel{⌢}{\overline{Y}}}_{MR}=\overline{y}\left(\frac{\overline{X}+\rho }{\overline{x}+\rho }\right)$

The bias and mean squared error of this modified ratio estimator are given by

$B\left({\stackrel{⌢}{\overline{Y}}}_{MR}\right)=\delta \overline{Y}\left[{\theta }^2{C}_x^2-\theta \rho C_x{C}_y\right]$

$MSE\left({\stackrel{⌢}{\overline{Y}}}_{MR}\right)=\delta {\overline{Y}}^2\left[C_y^2+{\theta }^2{C}_x^2-2\theta \rho C_x{C}_y\right]$     

where     $\theta =\frac{\stackrel{\leftharpoonup }{X}}{\stackrel{\leftharpoonup }{X}+\rho }$

The modified product estimator with known correlation coefficient of the auxiliary variable when there is a negative correlation between the study variable Y and auxiliary variable X is given as

${\widehat{\stackrel{\leftharpoonup }{Y}}}_{Mp}=\stackrel{\leftharpoonup }{y}\left(\frac{\stackrel{\leftharpoonup }{x}+\rho }{\stackrel{\leftharpoonup }{X}+\rho }\right)$   

The bias and mean squared error of the modified product estimator are given by

$B \left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{Mp}\right)=\delta \stackrel{\leftharpoonup }{Y}\left[\theta \rho C_x{C}_y\right]$

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{Mp}\right)=\delta \widehat{\stackrel{\leftharpoonup }{Y}}\left[C_y^2+{\theta }^2{C}_x^2+2\theta \rho C_x{C}_y\right]$                                                (6)

where $\theta =\frac{\stackrel{\leftharpoonup }{X}}{\stackrel{\leftharpoonup }{X}+\rho }$

In literature, several estimators are available with auxiliary variables. However the problem is that the best estimator in terms of bias and efficiency are not fully addressed. In this paper, we attempt to solve such type of problems. The existing estimators are biased but the percentage relative efficiency is better than that of simple random sampling, ratio and product estimators. These points are motivated us to introduce a new class of improved ratio cum product estimators for the estimation of the population mean of the study variable.

For estimating population mean $\overline{Y}$  we have proposed a class of ratio cum product estimators⁴ for the population mean by using the known population correlation coefficient of the auxiliary variable and is given by

${\widehat{\stackrel{\leftharpoonup }{Y}}}_{pr}=\alpha {\lambda }_1\stackrel{\leftharpoonup }{y}\left(\frac{\stackrel{\leftharpoonup }{X}+\rho }{\stackrel{\leftharpoonup }{x}+\rho }\right)+\left(1-\alpha \right){\lambda }_2\stackrel{\leftharpoonup }{y}\left(\frac{\stackrel{\leftharpoonup }{x}+\rho }{\stackrel{\leftharpoonup }{X}+\rho }\right)$                                                                              (9)

Here, ${\lambda }_1=\frac{S_y}{S_y+{\gamma }_1{C}_y}$ and ${\lambda }_2=\frac{S_y}{S_y+{\gamma }_2{C}_y}$ , ${\gamma }_1=B\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{MR}\right),$ ${\gamma }_2=B\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{Mp}\right)$

Bias and mean squared error of the proposed estimators

The detailed derivation of the bias and mean squared error are given in the appendix whereas the procedures to obtain the bias and mean squared error of the proposed estimators are briefly outlined below:

Consider $e_{\theta }=\frac{\stackrel{\leftharpoonup }{y}-\stackrel{\leftharpoonup }{Y}}{\stackrel{\leftharpoonup }{Y}},e_1=\frac{\stackrel{\leftharpoonup }{x}-\stackrel{\leftharpoonup }{X}}{\stackrel{\leftharpoonup }{X}}$ , $\theta =\frac{\stackrel{\leftharpoonup }{X}}{\stackrel{\leftharpoonup }{X}+\rho }$

$E\left(e_{\theta }\right)=E\left(e_1\right)=0$  , $E\left(e_0{}^2\right)=\delta {\stackrel{\leftharpoonup }{Y}}^2{C}_y^2 , E\left(e_1{}^2\right) = \delta {\stackrel{\leftharpoonup }{X}}^2{C}_x^2, E\left(e_0{e}_1\right)= \delta \rho C_x{C}_y$

Substitute these values in equation (9) and neglecting the high order expressions, we get

$B\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)=E\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}-\stackrel{\leftharpoonup }{Y}\right)$

$B\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)=\stackrel{\leftharpoonup }{Y}\left(\alpha {\lambda }_1+\left(1-\alpha \right){\lambda }_2-1\right)+\delta \stackrel{\leftharpoonup }{Y}\left\{\alpha {\lambda }_1{\theta }^2{C}_x^2-\theta \rho C_x{C}_y\left(\alpha {\lambda }_1-\left(1-\alpha \right){\lambda }_2\right)\right\}$

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)={\stackrel{\leftharpoonup }{Y}}^2{\left(A-1\right)}^2+\delta {\stackrel{\leftharpoonup }{Y}}^2\left\{C_y{}^2{\left(\alpha {\lambda }_1+\left(1-\alpha \right){\lambda }_2\right)}^2+{\theta }^2{C}_x{}^2\left(3{\alpha }^2{\lambda }_1{}^2+{\left(1+\alpha \right)}^2{\lambda }_2{}^2-2\alpha {\lambda }_1\right)+2\theta \rho C_x{C}_y\left(\alpha {\lambda }_1-\left(1-\alpha \right){\lambda }_2\right)-2\left({\alpha }^2{\lambda }_1{}^2-{\left(1-\alpha \right)}^2{\lambda }_2{}^2\right)\right\}$

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)={\stackrel{\leftharpoonup }{Y}}^2{\left(A-1\right)}^2+\delta {\stackrel{\leftharpoonup }{Y}}^2\left\{A^2{C}_y{}^2+{\theta }^2{C}_x{}^2\left(A^2+\left(A+B\right)\left(B-1\right)\right)-2\theta \rho C_x{C}_{y}B\left(2A-1\right)\right\}$

where $A=\left(\alpha {\lambda }_1+\left(1-\alpha \right){\lambda }_2\right)$ , $B=\left(\alpha {\lambda }_1-\left(1-\alpha \right){\lambda }_2\right)$ and $\theta =\frac{\stackrel{\leftharpoonup }{X}}{\stackrel{\leftharpoonup }{X}+\rho }$

The optimal value of α is determined by minimizing the MSE ( ${\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}$ with respect to α. For this differentiate MSE with respect to α and equate to zero.⁵

$\frac{\partial MSE}{\partial \alpha }=0$ , and we get the value of α, as

$\alpha =\frac{\left({\lambda }_2-1\right)\left({\lambda }_2-{\lambda }_1\right)-\delta \left[C_y{}^2{\lambda }_2\left({\lambda }_1-{\lambda }_2\right)-{\theta }^2{C}_x{}^2\left({\lambda }_1+{\lambda }_2{}^2\right)+\theta \rho C_x{C}_y\left({\lambda }_1+{\lambda }_2-4{\lambda }_2{}^2\right)\right]}{{\left({\lambda }_1-{\lambda }_2\right)}^2+\delta \left[{\left({\lambda }_1-{\lambda }_2\right)}^2{C}_y{}^2+{\theta }^2{C}_x{}^2\left(3{\lambda }_{{{}_1}^2+{\lambda }_2{}^2}\right)4\theta \rho C_x{C}_y\left({\lambda }_2{}^2-{\lambda }_1{}^2\right)\right]}$

Efficiency comparison

The efficiencies of the proposed estimators with that of the existing estimators are obtained algebraically and are as follows:

Comparison of proposed estimator and simple random sampling (SRSWOR) estimator

The proposed estimator is more efficient than simple random sampling estimator,

$V\left({\stackrel{\leftharpoonup }{y}}_{lr}\right)\ge MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)$ if

$C_y^2\ge \frac{\left\{{\left(A-1\right)}^2+{\delta }^2\left\{{\theta }^2{C}_x^2\left(A^2+\left(A+B\right)\left(B-1\right)\right)-2\theta \rho C_x{C}_{y}B\left(2A-1\right)\right\}\right\}}{{\delta }^2\left(1-A^2\right)}$

Comparison of proposed estimator and linear regression estimator

The proposed estimator is more efficient than linear regression estimator,

$V\left({\stackrel{\leftharpoonup }{y}}_{lr}\right)\ge MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)$ if

$C_y^2\ge \frac{\left\{{\left(A-1\right)}^2+\delta \left[{\theta }^2{C}_x{}^2\left(A^2+\left(A+B\right)\left(B-1\right)\right)-2\theta C_x{C}_y\left(B\left(2A-1\right)-1\right)\right]\right\}}{\delta \left(1-{\rho }^2-A^2\right)}$

Comparison of proposed estimator and ratio estimator

The proposed estimator is more efficient than ratio estimator

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_P\right)\ge MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)$  if

$C_y^2\ge \frac{\left\{{\left(A-1\right)}^2+\delta \left\{C_x^2\left({\theta }^2\left(A^2+\left(A+B\right)\left(B-1\right)\right)-1\right)-2\rho C_x{C}_y\left(\theta B\left(2A-1\right)-1\right)+1\right\}\right\}}{\delta \left(1-A^2\right)}$

Comparison of proposed estimator and product estimator

The proposed estimator is more efficient than ratio estimator,⁶

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_P\right)\ge MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)$ if

$C_y^2\ge \frac{\left\{{\left(A-1\right)}^2+\delta \left\{C_x^2\left({\theta }^2\left(A^2+\left(A+B\right)\left(B-1\right)\right)-1\right)-2\rho C_x{C}_y\left(\theta B\left(2A-1\right)+1\right)\right\}\right\}}{\delta \left(1-A^2\right)}$

Comparison of proposed estimator and modified ratio estimator

The proposed estimator is more efficient than modified ratio estimator,⁷

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{MR}\right)\ge MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)$

$C_y^2\ge \frac{\left\{{\left(A-1\right)}^2+\delta \left\{{\theta }^2{C}_x^2\left(A^2+\left(A+B\right)\left(B-1\right)-1\right)-2\theta \rho C_x{C}_y\left(\theta B\left(A-1\right)-1\right)-1\right\}\right\}}{\delta \left(1-A^2\right)}$

Comparison of proposed estimator and modified product estimator

The proposed estimator is more efficient than modified product estimator,

$MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{Mp}\right)\ge MSE\left({\widehat{\stackrel{\leftharpoonup }{Y}}}_{\mathrm{Pr}}\right)$  if

$C_y^2\ge \frac{\left\{{\left(A-1\right)}^2+\delta \left\{{\theta }^2{C}_x^2\left(A^2+\left(A+B\right)\left(B-1\right)-1\right)-2\theta \rho C_x{C}_y\left(B\left(2A-1\right)-1\right)\right\}\right\}}{\delta \left(1-A^2\right)}$

Numerical study

In this section, we consider the four natural populations population 1 Khoshnevisan et al.,⁸ Population 2 Cochran<>9] (page 325) population 3 and 4 Singh and Chaudhary,¹⁰ (page 177) and are used to compare the percentage relative efficiency of proposed estimator with that of the existing estimators such as SRSWOR sample mean, linear regression estimator, ratio estimator, product estimator, modified ratio estimators, and modified product estimators.

None.

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