A class of ratio estimators of a finite population mean using two auxiliary variables.

In sample surveys, it is usual to increase the efficiency of the estimators by the use of the auxiliary information. We propose a class of ratio estimators of a finite population mean using two auxiliary variables and obtain mean square error (MSE) equations for the class of proposed estimators. We find theoretical conditions that make proposed family estimators more efficient than the traditional ratio estimator and the estimators proposed by Abu-Dayeh et al. using two auxiliary variables. In addition, we support these theoretical results with the aid of a numerical example.

Introduction

Use of auxiliary information has been in practice to increase the efficiency of the estimators. Such information is generally used in ratio, product and regression type estimators for the estimation of population mean of study variable. When correlation between study variable and auxiliary variable is positive ratio method of estimation is used. On the other hand if the correlation is negative, product method of estimation is preferred. Some research works have been done in ratio, product and regression type estimators by using an auxiliary variable [1]–[10].

In this study, a class of ratio estimators using two auxiliary variables is considered to estimate a finite population mean for the variable of interest. We considered several special estimators of the suggested estimators. The comparisons between the traditional multivariate ratio estimators and the estimators proposed by Abu-Dayeh et al. [11] with the proposed family of estimators using information of two variables are considered. We compared the traditional ratio estimator, the estimators proposed by Abu-Dayeh et al. and proposed several special estimators using the statistic data given in Table 1. And we obtained the satisfactory results.

Table 1

Data Statistics.

Materials and Methods

The Existed Estimators

The traditional multivariate ratio estimator using information of two auxiliary variables x 1 and x 2 to estimate the population mean,, as follows:where and (i = 1,2) denote respectively the sample and the population means of the variable x; and are the weights that satisfy the condition: [12].

The MSE of this estimator is given bywhere and denote the coefficient of variation of Y, X 1 and X 2 respectively 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

Abu-Dayeh et al. proposed the estimators using two auxiliary variables given by where .

MSE of these estimators are given as follows:

The optimum values of and are given by

The optimum values of and are given by

The Proposed Family of Ratio Estimators

We propose a class of multivariate ratio estimators using information of two auxiliary variables as follows:where and are weights that satisfy the condition: , are either real numbers or functions of known parameters.

MSE of these estimators can be found using Taylor series method defined aswhere where

MSE of the class of estimators are given as follows: where

The optimal values of and to minimize (12) can easily be found as follows:

Some Members of the Proposed Class of Ratio Estimators

The following are the proposed class of ratio estimators :

(The traditional ratio estimator)

The suitable choices of constants and are given in Table 2.

Table 2

The suitable choices of constants and .

Estimators
	1	0	1	0
	1		1
	1		1
	1		1

Efficiency Comparisons

We compare the MSE of the proposed class of ratio estimators given in Eq. (13) with the MSE of the traditional ratio estimator given in Eq.(3)as follows:

When this condition is satisfied, the proposed class of ratio estimators will be more efficient than the traditional ratio estimator.

We compare the MSE of the proposed class of ratio estimators given in Eq. (13) with the MSE of the estimators proposed by Abu-Dayeh et al. given in Eq. (8) and Eq. (9) as follows:

When these conditions are satisfied, the proposed class of ratio estimators will be more efficient than the estimators proposed by Abu-Dayeh et al.

Numerical Illustration

In this section, we apply the traditional ratio estimator, the estimators proposed by Abu-Dayeh et al. and some members of the proposed class of estimators , to data whose statistics are given in Table 1 [13]. We assume to take the sample size n = 70, from N = 180 using SRSWOR. The MSE of these estimators are computed.

Results and Discussion

MSE of the traditional ratio estimator , the estimators proposed by Abu-Dayeh et al. and some members of the proposed ratio estimators can be seen in Table 3.

Table 3

MSE Values of Ratio Estimators.

Estimators	MSE
	0.157645
	0.157421
	0.157526
	0.157911
	0.157601
	0.162033
	0.157489
	0.157427
	0.157463
	0.157581
	0.159698
	0.157421 ()

From Table 3, we understand that the proposed ratio estimators , , , , , and are more efficient than the traditional ratio estimator using two auxiliary variables. When we examine the condition (14), for this data set, we see that all of them are satisfied as follows:

(1) the proposed ratio estimator

(2) the proposed ratio estimator

(3) the proposed estimator

(4) the proposed ratio estimator

(5) the proposed ratio estimator .

(6) the proposed ratio estimator

(7) the proposed ratio estimator

The result shows that the condition (14) is satisfied.

From Table 3, we understand that the efficiency of proposed ratio estimator is as good as the estimator proposed by Abu-Dayeh et al. We also understand that the proposed ratio estimators are more efficient than the estimator proposed by Abu-Dayeh et al. When we examine the condition (16), for this data set, we see that all of them are satisfied as follows:

(8) the proposed ratio estimator

(9) the proposed ratio estimator

(10) the proposed ratio estimator

(11) the proposed ratio estimator

(12) the proposed ratio estimator

(13) the proposed ratio estimator

(14) the proposed ratio estimator

(15) the proposed ratio estimator

(16) the proposed ratio estimator

(17) the proposed ratio estimator

From Table 3, we also understand that the most efficient estimator is the proposed ratio estimator and the estimator proposed by Abu-Dayeh et al. Therefore, we suggest that we should apply the proposed ratio estimator and the estimator proposed by Abu-Dayeh et al. to this data set.

Conclusions

We develop a class of ratio estimators of a finite population mean using two auxiliary variables and theoretically show that the proposed family ratio estimators are more efficient than the traditional ratio estimator and the estimators proposed by Abu-Dayeh et al. in certain conditions. These theoretical conditions are also satisfied by the results of a numerical example.