The chain ratio estimator and regression estimator with linear combination of two auxiliary variables.

In sample surveys, it is usual to make use of auxiliary information to increase the precision of the estimators. We propose a new chain ratio estimator and regression estimator of a finite population mean using linear combination of two auxiliary variables and obtain the mean squared error (MSE) equations for the proposed estimators. We find theoretical conditions that make proposed estimators more efficient than the traditional multivariate ratio estimator and the regression estimator using information of two auxiliary variables.

Introduction

The use of supplementary information provided by auxiliary variables in survey sampling was extensively discussed [1]–[10]. The ratio estimator and regression estimator are among the most commonly adopted estimators of the population mean or total of study variable of a finite population with the help of two auxiliary variables when the correlation coefficient between the two variables is positive. It is well known that these estimators are more efficient than the usual estimator of the population mean based on the sample mean of a simple random sampling.

In this study, we proposed a new chain ratio estimator and regression estimator using linear combination of two auxiliary variates, and obtain the mean squared error (MSE) equations for the two proposed estimators. The proposed estimators, the traditional multivariate ratio estimator and the regression estimator using information of two auxiliary variables were compared at theoretical conditions. And we obtained the satisfactory results.

Materials and Methods

The existed estimators

The classical ratio estimator and regression estimator for the population mean of the variate of interest y using one auxiliary information are defined by where it is assumed that the population mean of the auxiliary variate x is known.

Herewhere n is the number of units in the sample[11], and is the regression coefficient for of Y on X. and are the population variances of the y and x, respectively. is the population covariance between y and x [11].

The MSE of the classical ratio estimator iswhere ; N is the number of units in the population; is the population ratio, and are the population means of the y and x respectively.

The MSE of the regression estimator iswhere is the population correlation coefficient between and.

Kadilar and Cingi[12] proposed the chain ratio estimator using one auxiliary information for as where is real number.

MSE of this estimator is given as follows:

The traditional multivariate ratio estimator and regression estimator using information of two auxiliary variables x 1 and x 2 to estimate the population mean, [13], as follows: where and (i = 1,2) denote respectively the sample and the population means of the variable x (i = 1,2); and are the regression coefficients of on and on, respectively, here and are the variances of and , respectively, and and are the covariances between Y and , Y and , respectively., are the weights that satisfy the condition, respectively: and .

The MSE of this traditional multivariate ratio estimator is given by where and denote the correlation coefficient between Y and X 1, Y and X 2, X 1 and X 2 respectively. .

The optimum values of and are given by

The minimum MSE of can be shown to be:

The MSE of this traditional multivariate regression estimator is given by

The optimum values of and are given by

The minimum MSE of can be shown to be:

The suggested estimators

We propose the multivariate chain ratio estimator and regression estimator using linear combination of two auxiliary variables as follows: where is a arbitrary constant, ,,and is the regression coefficient on .

and are weights that satisfy the condition: and .

The MSE of this new multivariate ratio estimator is given by where

The optimum values of and are given by

The minimum MSE of can be shown to be:where

The MSE of this new multivariate regression estimator is given by

Where ,.

The optimum values of and are given by

The minimum MSE of can be shown to be:

Where

Efficiency comparison

We compare the MSE of the proposed multivariate ratio estimator using information of two auxiliary variables given in Eq. (17) with the MSE of traditional multivariate ratio estimator using information of two auxiliary variables given in Eq.(11) as follows:

We compare the MSE of the proposed regression estimators given in Eq. (19) with the MSE of the traditional multivariate regression estimator using information of two auxiliary variables given in Eq.(13) as follows:

Numerical illustration

The comparison among these estimators is given by using a data set whose statistics are given in Table 1 [14]. we apply the traditional multivariate ratio estimator and regression estimator using information of two auxiliary variables, given in Eqs.(8) and (9) and proposed chain ratio estimator and regression estimator of a finite population mean using linear combination of two auxiliary variables, given in Eqs. (14) and (15), to data whose statistics are given in Table 1. We assume to take the sample size n = 70, from N = 180 using SRSWOR. The MSE of these estimators are computed as given in Eqs.(11), (13), (17) and (19).

Table 1

Data Statistics.

Results and Discussion

MSE values of the traditional multivariate ratio estimator and regression estimator using information of two auxiliary variables and proposed chain ratio estimator and regression estimator using linear combination of two auxiliary variables can be seen in Table 2.

Table 2

MSE Values of Estimators.

Estimators	MSE
	0.1576
	0.1574
	0.1766
	0.1574

From Table 2, we notice that our proposed chain ratio estimator using linear combination of two auxiliary variables is more efficient than traditional multivariate ratio estimator using information of two auxiliary variables and our proposed regression estimator using linear combination of two auxiliary variables is more efficient than traditional multivariate regression estimator using information of two auxiliary variables. We examine the conditions for this data set,

The result shows that the condition (20) and condition (21) are satisfied. Therefore, we suggest that we should apply the proposed estimators to this data set.

Conclusions

We develop a new chain ratio estimator and a new regression estimator of a finite population mean using two auxiliary variables and theoretically show that the proposed estimators are more efficient than the traditional ratio estimator and traditional regression estimator using two auxiliary variables in certain condition.