The present paper deals with some two parameter modified ratio estimators for the estimation of finite population means with known skewness and correlation coefficient on two auxiliary variables. It has been shown that the proposed modified ratio estimators perform better than the simple random sampling without replacement (SRSWOR) sample mean and some of the existing ratio estimators for certain natural populations available in the literature.

Keywords: mean squared error, natural populations, percentage relative efficiency, simple random sampling, ams subject classification: 62DJO5

Let ( $X$ ) be the study (auxiliary) variable taking values $Y_i$ ( $X_i$ ) respectively on the unit $U_i$ , $i=1, 2, \dots , N$ wherein $U=\{U_1,U_2, \dots ,U_N\}$ be the finite population of size $N$ . The information on auxiliary variables are effectively used to improve the efficiency of the simple random sampling without replacement (SRSWOR) estimator of the population mean. As the results, ratio, product and regression estimators are widely utilized in many situations, see for example Cochran¹ and Murthy.² Modified ratio estimators are developed to achieve further improvements on the ratio estimator with known parameters of the auxiliary variable, which include Sisodia & Dwivedi³ with known Co-efficient of Variation, Singh et.,⁴ with known Kurtosis, Yan & Tian⁵ with the known Skewness, Subramani and Kumarapandiyan^6-9 with the known median and its linear combinations with the other known parameters. This paper deals with the two parameter modified ratio estimators with known correlation coefficient and skewness of two auxiliary variables $\overline{Y}$ .

Several modified estimators have been proposed by linking together ratio, product and regression estimators in order to obtain more efficient estimators with two auxiliary variables. For a more detailed discussion about the estimators with two auxiliary variables one may refer to Abu-Dayyeh et al.,¹⁰ Bandyopadhyay,¹¹ Cochran,¹ Kadilar & Cingi,¹²,¹³ Khare et al.,¹⁴ Murthy,2 Naik & Gupta,¹⁵ Olkin I,¹⁶ Perri,¹⁷,¹⁸ Rao and Mudholkar,¹⁹ Raj,²⁰ Sahoo & Swain,²¹ Singh,⁴ Singh,²²,²³ Singh & Tailor,²⁴ Srivenkataramana,²⁵ Srivenkataramana & Tracy,²⁶ Tailor et al.,²⁷ Tracy et al.,²⁸ and the references cited there in.

The notations described below are used in this paper:

$N-$ Population size
$n-$Sample size
$f=n/N$, Sampling fraction
$Y-$Study variable
$X_{1}and X_2 -$Auxiliary variables
${\overline{X}}_1,{\overline{X}}_2, \overline{Y}-$Population means
${\overline{x}}_1,{\overline{x}}_2, \overline{y}-$Sample means
${\beta }_1\left(X_i\right)-$ Coefficient of Skewness of auxiliary variable $X_i, i=1,2$  

$R_1=\frac{\overline{Y}}{{\overline{X}}_1}and R_2=\frac{\overline{Y}}{{\overline{\underset{¯}{X}}}_2}$

$S_{x_1 },S_{x_2 }, S_y-$Population standard deviations
$S_{x_{1}y }-$Population covariance between $X_1$  and $Y$
$S_{x_{2}y }-$Population covariance between $X_2$  and $Y$
$S_{x_1{x}_2}-$ Population covariance between $X_1$  and $X_2$
$C_x,C_{x_1},C_{x_2},C_y-$ Coefficient of variations
${\rho }_{xy}-$ Coefficient of correlation between $X$  and $Y$
${\rho }_{y{x}_1}-$ Coefficient of correlation between $Y$ and $X_1$
${\rho }_{y{x}_2}-$ Coefficient of correlation between $Y$  and $X_2$
${\rho }_{x_1{x}_2}-$Coefficient of correlation between $X_1$  and $X_1$
$B_1=\frac{S_{xy}}{S_{x_1}^2}$, Regression coefficient of $Y$  on $X_1$
$B_2=\frac{S_{xy}}{S_{x_2}^2}$, Regression coefficient of $Y$  on $X_2$
$MSE(.)-$Mean squared error of the estimator
${\tilde{\overline{Y}}}_i- i^{th}$Existing modified ratio estimator of $\overline{Y}$
In SRSWOR, the estimator of $\overline{Y}$  is ${\overline{y}}_r$  and its variance is

$V\left({\overline{y}}_r\right)=\frac{(1-f)}{n}{S}_y^2$, where $S_y^2=\frac{1}{(N-1)}\sum _{i=1}^N{(Y_i-\overline{Y})}^2$   (1)

Cochran [3] introduced the classical ratio estimator for estimating the population mean $\overline{\text{Y}}$ of the study variable $\text{Y}$ using auxiliary variable $\text{X}$ as given below:

${\widehat{\overline{\text{Y}}}}_{\text{R}}\frac{\overline{\text{y\hspace{0.17em}}}}{\overline{\text{x}}}\overline{\text{X}}\hat{\text{R}}\overline{\text{X}}\text{\hspace{0.17em}where\hspace{0.17em}}\hat{\text{R}}\frac{\overline{\text{y\hspace{0.17em}}}}{\overline{\text{x}}}\frac{\text{\hspace{0.17em}y\hspace{0.17em}}}{\text{x}}$   (2)

The mean squared error of ${\widehat{\overline{\text{Y}}}}_{\text{R}}$ to the first order of approximation is given below:

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{R}}\right)\frac{(\text{1}-\text{f})}{\text{n}}{\overline{\text{Y}}}^{\text{2}}({\text{C}}_{\text{y}}^{\text{2}}+{\text{C}}_{\text{x}}^{\text{2}}-{\text{2}}_{\text{xy}}{\text{C}}_{\text{x}}{\text{C}}_{\text{y}})$   (3)

Singh [24] has suggested a ratio estimator with two auxiliary variables for estimating the population mean and is given below:

${\widehat{\overline{\text{Y}}}}_{\text{1}}\overline{\text{y}}\left(\frac{{\overline{\text{X}}}_{\text{1}}}{{\overline{\text{x}}}_{\text{1}}}\right)\left(\frac{{\overline{\text{X}}}_{\text{2}}}{{\overline{\text{x}}}_{\text{2}}}\right)$   (4)

The mean squared error of ${\widehat{\overline{\text{Y}}}}_{\text{1}}$ to the first order of approximation is given below:

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{1}}\right)\frac{(\text{1}-\text{f})}{\text{n}}{\overline{\text{Y}}}^{\text{2}}({\text{C}}_{\text{y}}^{\text{2}}+{\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}+{\text{C}}_{{\text{x}}_{\text{2}}}^{\text{2}}-{\text{2}}_{{\text{yx}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{\text{y}}-{\text{2}}_{{\text{yx}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{2}}}{\text{C}}_{\text{y}}+{\text{2}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{2}}})$   (5)

Using known correlation coefficient between auxiliary variables, Singh and Tailor¹⁹ have suggested the following modified ratio cum product estimator:

${\widehat{\overline{\text{Y}}}}_{\text{2}}\overline{\text{y}}\left(\frac{{\overline{\text{X}}}_{\text{1}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}{{\overline{\text{x}}}_{\text{1}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}\right)\left(\frac{{\overline{\text{x}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}{{\overline{\text{X}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}\right)$   (6)

The mean squared error of ${\widehat{\overline{\text{Y}}}}_{\text{2}}$ to the first order of approximation is given below:

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{2}}\right)\frac{\text{1}-\text{f}}{\text{n}}{\overline{\text{Y}}}^{\text{2}}\left[{\text{C}}_{\text{y}}^{\text{2}}{+}_{\text{1}}^{\text{*}}{\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}(-{\text{2K}}_{{\text{yx}}_{\text{1}}}){+}_{\text{2}}^{\text{*}}{\text{C}}_{{\text{x}}_{\text{2}}}^{\text{2}}(+\text{2}\left({\text{K}}_{{\text{yx}}_{\text{2}}}{-}_{\text{1}}^{\text{*}}{\text{K}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\right))\right]$   (7)

${{\text{where\hspace{0.17em}K}}_{{\text{yx}}_{\text{1}}}}_{{\text{yx}}_{\text{1}}}\frac{{\text{C}}_{\text{y}}}{{\text{C}}_{{\text{x}}_{\text{1}}}}{{\text{,\hspace{0.17em}K}}_{{\text{yx}}_{\text{2}}}}_{{\text{yx}}_{\text{2}}}\frac{{\text{C}}_{\text{y}}}{{\text{C}}_{{\text{x}}_{\text{2}}}}\mathrm{,}{\text{K}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}$

${}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\frac{{\text{C}}_{{\text{x}}_{\text{1}}}}{{\text{C}}_{{\text{x}}_{\text{2}}}}{\mathrm{,}}_{\text{1}}^{\text{*}}\frac{{\overline{\text{X}}}_{\text{1}}}{{\overline{\text{X}}}_{\text{1}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}{\text{\hspace{0.17em}and\hspace{0.17em}}}_{\text{2}}^{\text{*}}\frac{{\overline{\text{X}}}_{\text{2}}}{{\overline{\text{X}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}$

Kadilar & Cingi⁶ have proposed a new ratio estimator using two auxiliary variables and isgiven below:

${\widehat{\overline{\text{Y}}}}_{\text{3}}\overline{\text{y}}{\left(\frac{{\overline{\text{X}}}_{\text{1}}}{{\overline{\text{x}}}_{\text{1}}}\right)}^{}{\left(\frac{{\overline{\text{X}}}_{\text{2}}}{{\overline{\text{x}}}_{\text{2}}}\right)}^{}+{\text{b}}_{\text{1}}({\overline{\text{X}}}_{\text{1}}-{\overline{\text{x}}}_{\text{1}})+{\text{b}}_{\text{2}}({\overline{\text{X}}}_{\text{2}}-{\overline{\text{x}}}_{\text{2}})$   (8)

The mean squared error of ${\widehat{\overline{\text{Y}}}}_{\text{3}}$ to the first order of approximation is given below:

$\begin{array}{ll}\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{3}}\right)\cong & \frac{\text{1}-\text{f}}{\text{n}}{\text{S}}_{\text{y}}^{\text{2}}+{({\text{R}}_{\text{1}}+{\text{B}}_{\text{1}})}^{\text{2}}{\text{S}}_{{\text{x}}_{\text{1}}}^{\text{2}}+{({\text{R}}_{\text{2}}+{\text{B}}_{\text{2}})}^{\text{2}}{\text{S}}_{{\text{x}}_{\text{2}}}^{\text{2}}-\text{2}({\text{R}}_{\text{1}}+{\text{B}}_{\text{1}}){\text{S}}_{{\text{yx}}_{\text{1}}}\\ & -\text{2}({\text{R}}_{\text{2}}+{\text{B}}_{\text{2}}){\text{S}}_{{\text{yx}}_{\text{2}}}+\text{2}({\text{R}}_{\text{1}}+{\text{B}}_{\text{1}})({\text{R}}_{\text{2}}+{\text{B}}_{\text{2}}){\text{S}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\end{array}$   (9)

Perri¹³ has suggested some modified ratio cum product estimators using two auxiliary variables for estimating the population mean and are given below:

${\widehat{\overline{\text{Y}}}}_{\text{4}}\overline{\text{y}}\frac{{}_{\text{2}}}{{}_{\text{1}}}\frac{{\overline{\text{X}}}_{\text{1}}}{{\overline{\text{X}}}_{\text{2}}}\text{,\hspace{0.17em}}{\widehat{\overline{\text{Y}}}}_{\text{5}}\overline{\text{y}}\frac{{\overline{\text{X}}}_{\text{1}}}{{}_{\text{1}}}\frac{{\overline{\text{X}}}_{\text{2}}}{{}_{\text{2}}}\text{\hspace{0.17em}and\hspace{0.17em}}{\widehat{\overline{\text{Y}}}}_{\text{6}}\overline{\text{y}}\frac{{}_{\text{1}}}{{}_{\text{2}}}\frac{{\overline{\text{X}}}_{\text{2}}}{{\overline{\text{X}}}_{\text{1}}}$   (10)

${\text{where\hspace{0.17em}}}_{\text{1}}{\overline{\text{x}}}_{\text{1}}{+}_{\text{1}}({\overline{\text{X}}}_{\text{1}}-{\overline{\text{x}}}_{\text{1}}){\text{\hspace{0.17em}and\hspace{0.17em}}}_{\text{2}}{\overline{\text{x}}}_{\text{2}}{+}_{\text{2}}({\overline{\text{X}}}_{\text{2}}-{\overline{\text{x}}}_{\text{2}})$

The mean squared errors of ${\widehat{\overline{\text{Y}}}}_{\text{4}}\text{,\hspace{0.17em}}{\widehat{\overline{\text{Y}}}}_{\text{5}}\text{\hspace{0.17em}and\hspace{0.17em}}{\widehat{\overline{\text{Y}}}}_{\text{6}}$ to the first order of approximation are given below:

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{4}}\right)\frac{\text{1}-\text{f}}{\text{n}}[{\text{S}}_{\text{y}}^{\text{2}}{+}_{{\text{x}}_{\text{1}}}^{\text{2}}{+}_{{\text{x}}_{\text{2}}}^{\text{2}}-\text{2}\left({-}_{{\text{yx}}_{\text{2}}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\right)]$   (11)

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{5}}\right)\frac{\text{1}-\text{f}}{\text{n}}[{\text{S}}_{\text{y}}^{\text{2}}{+}_{{\text{x}}_{\text{1}}}^{\text{2}}{+}_{{\text{x}}_{\text{2}}}^{\text{2}}-\text{2}\left({+}_{{\text{yx}}_{\text{2}}}{-}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\right)]$   (12)

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{6}}\right)\frac{\text{1}-\text{f}}{\text{n}}[{\text{S}}_{\text{y}}^{\text{2}}{+}_{{\text{x}}_{\text{1}}}^{\text{2}}{+}_{{\text{x}}_{\text{2}}}^{\text{2}}+\text{2}\left({-}_{{\text{yx}}_{\text{2}}}{-}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\right)]$   (13)

${\text{where\hspace{0.17em}}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{R}}_{\text{1}}{\text{R}}_{\text{2}}{\text{S}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\left(\text{1}{-}_{\text{1}}\right)\left(\text{1}{-}_{\text{2}}\right){\mathrm{,}}_{{\text{x}}_{\text{1}}}{\text{R}}_{\text{1}}{\text{S}}_{{\text{x}}_{\text{1}}}\left(\text{1}{-}_{\text{1}}\right){\text{,\hspace{0.17em}}}_{{\text{x}}_{\text{2}}}{\text{R}}_{\text{2}}{\text{S}}_{{\text{x}}_{\text{2}}}\left(\text{1}{-}_{\text{2}}\right)$   

${\text{R}}_{\text{1}}{\text{S}}_{{\text{yx}}_{\text{1}}}\left(\text{1}{-}_{\text{1}}\right){\text{\hspace{0.17em}and\hspace{0.17em}}}_{{\text{yx}}_{\text{2}}}{\text{R}}_{\text{2}}{\text{S}}_{{\text{yx}}_{\text{2}}}\left(\text{1}{-}_{\text{2}}\right)$

This paper is organized as follows: Section 2 introduces two parameter modified ratio estimators using the information of correlation coefficient and skewness of two auxiliary variables and the expressions for their bias and mean square error up to the first order of approximation have been derived. Section 3 is devoted to the analysis of the efficiencies of the proposed modified ratio estimators. In Section 4, an empirical analysis is carried out with some natural populations. Section 5 is ended with some concluding remarks.

In this Section, two parameter modified ratio estimators with known correlation coefficient and skewness and their linear combinations of two auxiliary variables have been proposed and are given below:

${\widehat{\overline{\text{Y}}}}_{\text{JS1}}\overline{\text{y}}\left(\frac{{\overline{\text{X}}}_{\text{1}}{+}_{\text{2}}{\overline{\text{X}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}{{\overline{\text{x}}}_{\text{1}}{+}_{\text{2}}{\overline{\text{x}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}}\right)$   (14)

${\widehat{\overline{\text{Y}}}}_{\text{JS2}}\overline{\text{y}}\left(\frac{\text{(}{\overline{\text{X}}}_{\text{1}}{+}_{\text{1}}\left({\text{X}}_{\text{1}}\right)\text{)}{+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}}{+}_{\text{1}}\left({\text{X}}_{\text{2}}\right)\text{)}}{\text{(}{\overline{\text{x}}}_{\text{1}}{+}_{\text{1}}\left({\text{X}}_{\text{1}}\right)\text{)}{+}_{\text{2}}\text{(}{\overline{\text{x}}}_{\text{2}}{+}_{\text{1}}\left({\text{X}}_{\text{2}}\right)\text{)}}\right)$   (15)

${\widehat{\overline{\text{Y}}}}_{\text{JS3}}\overline{\text{y}}\left(\frac{{\text{(}}_{\text{1}}\left({\text{X}}_{\text{1}}\right){\overline{\text{X}}}_{\text{1}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\text{)}{+}_{\text{2}}{\text{(}}_{\text{1}}\left({\text{X}}_{\text{2}}\right){\overline{\text{X}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\text{)}}{{\text{(}}_{\text{1}}\left({\text{X}}_{\text{1}}\right){\overline{\text{x}}}_{\text{1}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\text{)}{+}_{\text{2}}{\text{(}}_{\text{1}}\left({\text{X}}_{\text{2}}\right){\overline{\text{x}}}_{\text{2}}{+}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\text{)}}\right)$   (16)

${\widehat{\overline{\text{Y}}}}_{\text{JS4}}\overline{\text{y}}\left(\frac{{\text{(}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\overline{\text{X}}}_{\text{1}}{+}_{\text{1}}\left({\text{X}}_{\text{1}}\right)\text{)}{+}_{\text{2}}{\text{(}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\overline{\text{X}}}_{\text{2}}{+}_{\text{1}}\left({\text{X}}_{\text{2}}\right)\text{)}}{{\text{(}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\overline{\text{x}}}_{\text{1}}{+}_{\text{1}}\left({\text{X}}_{\text{1}}\right)\text{)}{+}_{\text{2}}{\text{(}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\overline{\text{x}}}_{\text{2}}{+}_{\text{1}}\left({\text{X}}_{\text{2}}\right)\text{)}}\right)$   (17)

In general the estimators proposed in (14) to (17) can be defined as given below:

${\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{y}}\left(\frac{\text{(}{\overline{\text{X}}}_{\text{1}}+{\text{T}}_{\text{1}}\text{)}{+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}}+{\text{T}}_{\text{2}}\text{)}}{\text{(}{\overline{\text{x}}}_{\text{1}}+{\text{T}}_{\text{1}}\text{)}{+}_{\text{2}}\text{(}{\overline{\text{x}}}_{\text{2}}+{\text{T}}_{\text{2}}\text{)}}\right)\text{,\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$

The mean squared errors of the proposed estimators are derived as given below:

${\text{Let\hspace{0.17em}e}}_0\frac{\overline{\text{y}}-\overline{\text{Y}}}{\overline{\text{Y}}}{\text{,\hspace{0.17em}e}}_{\text{1}}\frac{{\overline{\text{x}}}_{\text{1}}-{\overline{\text{X}}}_{\text{1}}}{{\overline{\text{X}}}_{\text{1}}}{\text{\hspace{0.17em}and\hspace{0.17em}e}}_{\text{2}}\frac{{\overline{\text{x}}}_{\text{2}}-{\overline{\text{X}}}_{\text{2}}}{{\overline{\text{X}}}_{\text{2}}}\text{.\hspace{0.17em}Further\hspace{0.17em}we\hspace{0.17em}can\hspace{0.17em}write\hspace{0.17em}}\overline{\text{y}}\overline{\text{Y}}(\text{1}+{\text{e}}_0)$

${\overline{\text{x}}}_{\text{1}}{\overline{\text{X}}}_{\text{1}}(\text{1}+{\text{e}}_{\text{1}})$ and ${\overline{\text{x}}}_{\text{2}}{\overline{\text{X}}}_{\text{2}}(\text{1}+{\text{e}}_{\text{2}})$ and from the definition of ${\text{e}}_0{\text{\hspace{0.17em}and\hspace{0.17em}e}}_{\text{1}}$ we obtain:

$\text{E}\left[{\text{e}}_0\right]\text{E}\left[{\text{e}}_{\text{1}}\right]\text{=0}$

$\text{E}\left[{\text{e}}_0^{\text{2}}\right]\frac{(\text{1}-\text{f})}{\text{n}}{\text{C}}_{\text{y}}^{\text{2}}\text{,\hspace{0.17em}E}\left[{\text{e}}_{\text{1}}^{\text{2}}\right]\frac{\text{1}-\text{f}}{\text{n}}{\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}\text{and\hspace{0.17em}E}\left[{\text{e}}_{\text{2}}^{\text{2}}\right]\frac{\text{1}-\text{f}}{\text{n}}{\text{C}}_{{\text{x}}_{\text{2}}}^{\text{2}}$

${\text{E(e}}_0{\text{e}}_{\text{1}}{\frac{\text{1}-\text{f}}{\text{n}}}_{{\text{yx}}_{\text{1}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{,\hspace{0.17em}E(e}}_0{\text{e}}_{\text{2}}{\frac{\text{1}-\text{f}}{\text{n}}}_{{\text{yx}}_{\text{2}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{2}}}{\text{\hspace{0.17em}and\hspace{0.17em}E(e}}_{\text{1}}{\text{e}}_{\text{2}}{\frac{\text{1}-\text{f}}{\text{n}}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{2}}}$

The proposed estimators ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$ can be written in terms of ${\text{e}}_0$, ${\text{e}}_{\text{1}}$ and ${\text{e}}_{\text{2}}$ as given below:

${\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{Y}}(\text{1}+{\text{e}}_0)\left(\frac{\text{(}{\overline{\text{X}}}_{\text{1}}+{\text{T}}_{\text{1}}\text{)}{+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}}+{\text{T}}_{\text{2}}\text{)}}{({\overline{\text{X}}}_{\text{1}}(\text{1}+{\text{e}}_{\text{1}})+{\text{T}}_{\text{1}}){+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}}(\text{1}+{\text{e}}_{\text{2}})+{\text{T}}_{\text{2}}\text{)}}\right)$

$\Rightarrow {\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{Y}}(\text{1}+{\text{e}}_0)\left(\frac{{\overline{\text{X}}}_{\text{1}}{+}_{\text{2}}{\overline{\text{X}}}_{\text{2}}{+}_{\text{1}}{\text{T}}_{\text{1}}{+}_{\text{2}}{\text{T}}_{\text{2}}}{{\overline{\text{X}}}_{\text{1}}{+}_{\text{2}}{\overline{\text{X}}}_{\text{2}}{+}_{\text{1}}{\text{T}}_{\text{1}}{+}_{\text{2}}{\text{T}}_{\text{2}}{+}_{\text{1}}{\overline{\text{X}}}_{\text{1}}{\text{e}}_{\text{1}}{+}_{\text{2}}{\overline{\text{X}}}_{\text{2}}{\text{e}}_{\text{2}}}\right)$

$\Rightarrow {\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{Y}}(\text{1}+{\text{e}}_0)\left(\frac{\text{1}}{\text{1}{+}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{+}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}\text{)}}\right){\text{,\hspace{0.17em}\hspace{0.17em}}}_{\text{1}}^{\text{'}}\frac{{\overline{\text{X}}}_{\text{1}}}{\text{(}{\overline{\text{X}}}_{\text{1}}+{\text{T}}_{\text{1}}\text{)}{+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}}+{\text{T}}_{\text{2}}\text{)}}\text{\hspace{0.17em}and\hspace{0.17em}}$

$\frac{{\overline{\text{X}}}_{\text{2}}}{\text{(}{\overline{\text{X}}}_{\text{1}}+{\text{T}}_{\text{1}}\text{)}{+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}}+{\text{T}}_{\text{2}}\text{)}}$

$\Rightarrow {\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{Y}}(\text{1}+{\text{e}}_0){\left(\text{1}{+}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{+}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}\right)}^{-\text{1}}$

$\Rightarrow {\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{Y}}(\text{1}+{\text{e}}_0)(\text{1}{-}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{-}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}+{\left({\text{e}}_{\text{1}}{+}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}\right)}^{\text{2}})$

$\Rightarrow {\widehat{\overline{\text{Y}}}}_{\text{JSj}}\overline{\text{Y}}(\text{1}+{\text{e}}_0)(\text{1}{-}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{-}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}{+}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}^{\text{2}}{+}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}^{\text{2}}+{\text{2}}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{\text{e}}_{\text{2}})$

Neglecting the terms of higher order, we have

${\widehat{\overline{\text{Y}}}}_{\text{JSj}}-\overline{\text{Y}}\overline{\text{Y}}{\text{e}}_0-{\overline{\text{Y}}}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}-{\overline{\text{Y}}}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}+{\overline{\text{Y}}}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{\text{e}}_{\text{2}}-{\overline{\text{Y}}}_{\text{1}}^{\text{'}}{\text{e}}_0{\text{e}}_{\text{1}}-{\overline{\text{Y}}}_{\text{2}}^{\text{'}}{\text{e}}_0{\text{e}}_{\text{2}}$

Squaring and taking expectations on both sides, we have

$\text{MSE(}{\widehat{\overline{\text{Y}}}}_{\text{JSj}}E\text{(}{\widehat{\overline{\text{Y}}}}_{\text{JSj}}-\overline{\text{Y}}{\text{)}}^{\text{2}}{\overline{\text{Y}}}^{\text{2}}{\text{E(e}}_0{-}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{-}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}{\text{)}}^{\text{2}}$

$\Rightarrow \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\text{=\hspace{0.17em}}{\overline{\text{Y}}}^{\text{2}}\text{E}({\text{e}}_0^{\text{2}}{+}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}^{\text{2}}{+}_{\text{2}}^{\text{'}}{\text{e}}_{\text{2}}^{\text{2}}-{\text{2}}_{\text{1}}^{\text{'}}{\text{e}}_0{\text{e}}_{\text{1}}-{\text{2}}_{\text{2}}^{\text{'}}{\text{e}}_0{\text{e}}_{\text{2}}\text{2}{+}_{\text{1}}^{\text{'}}{\text{e}}_{\text{1}}{\text{e}}_{\text{2}})$

$\Rightarrow \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right){\overline{\text{Y}}}^{\text{2}}\{{\text{E(e}}_0^{\text{2}}\text{)}{+}_{\text{1}}^{\text{'}}{\text{E(e}}_{\text{1}}^{\text{2}}\text{)}{+}_{\text{2}}^{\text{'}}{\text{E(e}}_{\text{2}}^{\text{2}}\text{)}-{\text{2}}_{\text{1}}^{\text{'}}{\text{E(e}}_0{\text{e}}_{\text{1}}\text{)}-{\text{2}}_{\text{1}}^{\text{'}}{\text{E(e}}_0{\text{e}}_{\text{2}}+2_{\text{1}}^{\text{'}}{\text{E(e}}_{\text{1}}{\text{e}}_{\text{2}}\text{)}\}$

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\frac{\text{1}-\text{f}}{\text{n}}{\overline{\text{Y}}}^{\text{2}}\{{\text{C}}_{\text{y}}^{\text{2}}{+}_{\text{1}}^{\text{'}}{\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}{+}_{\text{2}}^{\text{'}}{\text{C}}_{{\text{x}}_{\text{2}}}^{\text{2}}$

$-{\text{2}}_{\text{1}}^{\text{'}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{1}}}-{\text{2}}_{\text{2}}^{\text{'}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{2}}}+{{\text{2}}_{\text{1}}^{\text{'}}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{2}}}\}$   (18)

In this Section the conditions for which the proposed estimators will have minimum mean squared error compared to SRSWOR sample mean and other existing estimators discussed in Section 2 for estimating the finite population mean have been derived algebraically.

From the expressions given in (18) and (1) the conditions for which the proposed estimator, is more efficient than the SRSWOR sample mean have been derived and are given below:

$\text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{V}\left({\overline{\text{y}}}_{\text{r}}\right){\text{\hspace{0.17em}if\hspace{0.17em}}}_{\text{1}}^{\text{'}}{\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}{+}_{\text{2}}^{\text{'}}{\text{C}}_{{\text{x}}_{\text{2}}}^{\text{2}}\le \text{2}{(}_{{\text{yx}}_{\text{1}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{1}}}{+}_{\text{2}}^{\text{'}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{2}}}{{-}_{\text{1}}^{\text{'}}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{2}}})$   (19)

From the expressions given in (18) and (5) the conditions for which the proposed estimator ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$, $\text{j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$ is more efficient than the existing ratio estimator ${\widehat{\overline{\text{Y}}}}_{\text{1\hspace{0.17em}}}$ have been derived and are given below:

$\begin{array}{ll}& \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{1}}\right)\text{\hspace{0.17em}if\hspace{0.17em}}{(}^{\text{2}}-\text{1}){\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}+{\text{(}}_{\text{1}}^{\text{'}}-{\text{1)C}}_{{\text{x}}_{\text{2}}}^{\text{2}}\\ & \le \text{2}\{{\text{(}}_{\text{1}}^{\text{'}}-{\text{1)}}_{{\text{yx}}_{\text{1}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{1}}}+{(-\text{1})}_{{\text{yx}}_{\text{2}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{2}}}-{\text{(}}_{\text{1}}^{\text{'}}-{\text{1)}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{2}}}\}\end{array}$   (20)

From the expressions given in (18) and (7) the conditions for which the proposed estimator ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$, $\text{j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$ is more efficient than the existing ratio estimator ${\widehat{\overline{\text{Y}}}}_{\text{2\hspace{0.17em}}}$ have been derived and are given below:

$\begin{array}{ll}& \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{2}}\right)\text{\hspace{0.17em}if\hspace{0.17em}}{(}^{\text{2}}{-}_{\text{1}}^{\text{*}}){\text{C}}_{{\text{x}}_{\text{1}}}^{\text{2}}+{\text{(}}_{\text{2}}^{\text{'}}{-}_{\text{2}}^{\text{*}}{\text{)C}}_{{\text{x}}_{\text{2}}}^{\text{2}}\\ & \le \text{2}\{{\text{(}}_{\text{1}}^{\text{'}}{-}_{\text{1}}^{\text{*}}{\text{)}}_{{\text{yx}}_{\text{1}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{1}}}+{\left({+}_{\text{2}}^{\text{*}}\right)}_{{\text{yx}}_{\text{2}}}{\text{C}}_{\text{y}}{\text{C}}_{{\text{x}}_{\text{2}}}-{\text{(}}_{\text{1}}^{\text{'}}{+}_{\text{1}}^{\text{*}}{\text{)}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{\text{C}}_{{\text{x}}_{\text{1}}}{\text{C}}_{{\text{x}}_{\text{2}}}\}\end{array}$   (21)

From the expressions given in (18) and (9) the conditions for which the proposed estimator ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$, $\text{j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$ is more efficient than the existing ratio estimator have been derived and are given below:

$\begin{array}{ll}& \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{3}}\right){\text{\hspace{0.17em}if\hspace{0.17em}}}_{\text{1}}^{\text{2}}({\text{R}}_{\text{js}}^{\text{'}}-{\text{R}}_{\text{1}}^{\text{2}}){\text{S}}_{{\text{x}}_{\text{1}}}^{\text{2}}{+}_{\text{2}}^{\text{2}}({\text{R}}_{\text{js}}^{\text{'}}-{\text{R}}_{\text{2}}^{\text{2}}){\text{S}}_{{\text{x}}_{\text{2}}}^{\text{2}}+{\text{B}}_{\text{1}}{S}_{{\text{yx}}_{\text{1}}}+{\text{B}}_{\text{2}}{S}_{{\text{yx}}_{\text{2}}}\\ & \le \text{2}\{{\text{R}}_{\text{js}}^{\text{'}}\left(S_{{\text{yx}}_{\text{1}}}{+}_{\text{2}}{S}_{{\text{yx}}_{\text{2}}}\right)-{[}_{\text{2}}{\text{R}}_{\text{js}}^{\text{'}}-{\text{(}}_{\text{1}}{\text{R}}_{\text{1}}+{\text{B}}_{\text{1}}{\text{)(}}_{\text{2}}{\text{R}}_{\text{2}}+{\text{B}}_{\text{2}}\text{)}]{\text{S}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}\}\end{array}$    (22)

From the expressions given in (18) and (11) the conditions for which the proposed estimator ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$, $\text{\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$ is more efficient than the existing ratio estimator ${\widehat{\overline{\text{Y}}}}_{\text{4\hspace{0.17em}}}$ have been derived and are given below:

$\begin{array}{ll}& \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{4}}\right)\text{\hspace{0.17em}if\hspace{0.17em}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{1}}{\text{)}}^{\text{2}}{\text{R}}_{\text{1}}^{\text{2}}]{\text{S}}_{{\text{x}}_{\text{1}}}^{\text{2}}+[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{2}}{\text{)}}^{\text{2}}{\text{R}}_{\text{2}}^{\text{2}}]{\text{S}}_{{\text{x}}_{\text{2}}}^{\text{2}}\\ & \le \text{2}\{S_{{\text{yx}}_{\text{1}}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{1}}{\text{)R}}_{\text{1}}]+S_{{\text{yx}}_{\text{2}}}[{\text{R}}_{\text{js}}^{\text{'}}+\text{(1}{-}_{\text{2}}{\text{)R}}_{\text{2}}]-{\text{S}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{[}_{\text{2}}{\text{R}}_{\text{js}}^{\text{'}}+{\text{R}}_{\text{1}}{\text{R}}_{\text{2}}\text{(1}{-}_{\text{1}}\text{)(1}{-}_{\text{2}}\text{)}]\}\end{array}$   (23)

From the expressions given in (18) and (12) the conditions for which the proposed estimator ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$, $\text{j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$ is more efficient than the existing ratio estimator have been derived and are given below:

$\begin{array}{ll}& \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{5}}\right)\text{\hspace{0.17em}if\hspace{0.17em}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{1}}{\text{)}}^{\text{2}}{\text{R}}_{\text{1}}^{\text{2}}]{\text{S}}_{{\text{x}}_{\text{1}}}^{\text{2}}+[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{2}}{\text{)}}^{\text{2}}{\text{R}}_{\text{2}}^{\text{2}}]{\text{S}}_{{\text{x}}_{\text{2}}}^{\text{2}}\\ & \le \text{2}\{S_{{\text{yx}}_{\text{1}}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{1}}{\text{)R}}_{\text{1}}]+S_{{\text{yx}}_{\text{2}}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{2}}{\text{)R}}_{\text{2}}]-{\text{S}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{[}_{\text{2}}{\text{R}}_{\text{js}}^{\text{'}}-{\text{R}}_{\text{1}}{\text{R}}_{\text{2}}\text{(1}{-}_{\text{1}}\text{)(1}{-}_{\text{2}}\text{)}]\}\end{array}$   (24)

From the expressions given in (18) and (13) the conditions for which the proposed estimator ${\widehat{\overline{\text{Y}}}}_{\text{JSj}}$, $\text{j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4}$ is more efficient than the existing ratio estimator have been derived and are given below:

$\begin{array}{ll}& \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{JSj}}\right)\le \text{MSE}\left({\widehat{\overline{\text{Y}}}}_{\text{6}}\right)\text{\hspace{0.17em}if\hspace{0.17em}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{1}}{\text{)}}^{\text{2}}{\text{R}}_{\text{1}}^{\text{2}}]{\text{S}}_{{\text{x}}_{\text{1}}}^{\text{2}}+[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{2}}{\text{)}}^{\text{2}}{\text{R}}_{\text{2}}^{\text{2}}]{\text{S}}_{{\text{x}}_{\text{2}}}^{\text{2}}\\ & \le \text{2}\{S_{{\text{yx}}_{\text{1}}}[{\text{R}}_{\text{js}}^{\text{'}}+\text{(1}{-}_{\text{1}}{\text{)R}}_{\text{1}}]+S_{{\text{yx}}_{\text{2}}}[{\text{R}}_{\text{js}}^{\text{'}}-\text{(1}{-}_{\text{2}}{\text{)R}}_{\text{2}}]-{\text{S}}_{{\text{x}}_{\text{1}}{\text{x}}_{\text{2}}}{[}_{\text{2}}{\text{R}}_{\text{js}}^{\text{'}}+{\text{R}}_{\text{1}}{\text{R}}_{\text{2}}\text{(1}{-}_{\text{1}}\text{)(1}{-}_{\text{2}}\text{)}]\}\end{array}$   (25)

${\text{\hspace{0.17em}where\hspace{0.17em}R}}_{\text{js}}^{\text{'}}\frac{\overline{\text{Y}}}{\text{(}{\overline{\text{X}}}_{\text{1}}+{\text{T}}_{\text{1}}\text{)}{+}_{\text{2}}\text{(}{\overline{\text{X}}}_{\text{2}+}{\text{T}}_{\text{2}}\text{)}}$

The performance of the proposed modified ratio estimators are assessed with that of the SRSWOR sample mean and the existing modified ratio estimators for certain natural populations. In this connection, we have considered two natural populations for the assessment of the performance of the proposed estimators with that of the existing estimators. The population 1 is taken from Singh & Chaudhary²⁹ given in page 177 and population 2 is taken from taken from the Cingi & Kadilar³⁰ given in page 117. The description of the study variable and auxiliary variable for the two populations are given below: (Table 1-4).

The population parameters and constants computed for the above two populations are given below:

From the values of Table 5-12, it is observed that the proposed modified ratio estimators perform better than SRSWOR sample mean and the existing modified ratio estimators.^30–35