Efficient Estimator of a Finite Population Mean Using Two Auxiliary Variables and Numerical Application in Agricultural, Biomedical, and Power Engineering

To improve the efficiency of an estimator with two auxiliary variables, we propose a new estimator of a finite population mean under simple random sampling. The bias and mean square error expressions of the proposed estimator have been obtained. In a comparison study, we found that the new estimator was consistently better than those of Abu-Dayyeh et al., Kadilar and Cingi, and Malik and Singh, as well as the regression estimator using two auxiliary variables, and that theminimumMSE values of the previous three above reported estimators were equal. We used four numerical examples in agricultural, biomedical, and power engineering to support these theoretical results, thus enriching the theory of survey samples by the development of new estimators with two auxiliary variables.


Introduction
In sampling theory, it is a well-established phenomenon that supplementary information provided by auxiliary variables or auxiliary attributes often improves the accuracy of estimators of unknown population parameters.Ratio-, product-, and regression-type estimators are three such methods.For this reason, some authors have exploited the use of auxiliary variables and attributes at the estimation stage to increase estimator efficiency.For example, the planting area and the proportion of good seeds in agricultural engineering are two important auxiliary variables when estimating average cotton output.Similarly, the breed of cow in animal husbandry engineering is an important auxiliary attribute when estimating average milk yield.Thus, auxiliary information can be used in the field of education, biostatistics, the medical research, agricultural and biomedical engineering, and so on.
In the literature, some authors have proposed many efficient ratio-, product-, and regression-type estimators using one auxiliary variable or attribute, including Singh and Vishwakarma [1], Grover and Kaur [2,3], Singh et al. [4], Singh and Solanki [5], and Gupta and Shabbir [6].More recently, several authors have proposed efficient estimators of finite population mean using two variables or attributes, including, Abu-Dayyeh et al. [7], Kadilar and Cingi [8], Malik and Singh [9], Sharma and Singh [10], and Muneer et al. [11].Although these studies are detailed and elaborated, the formulas of minimum MSE are not given, and the difference of minimum MSE values between these studies seems not to have been noticed.
In this paper, we compare the estimators reported by Abu-Dayyeh et al. [7], Kadilar and Cingi [8], and Malik and Singh [9] and introduce a new estimator with two auxiliary variables to estimate a finite population mean for the variable of interest.We obtained bias and mean square error (MSE) equations for the proposed estimator, and we compared the new estimator against those with relatively high efficiencies.An empirical study using four datasets in agricultural, biomedical, and power engineering was conducted, and we obtained satisfactory results, both theoretically and numerically.The analysis of these issues is of great significance for understanding agricultural, biomedical, and power engineering.Therefore, the proposed estimator could be applied across a broad spectrum of sampling survey.

Abu-Dayyeh
Estimator.Abu-Dayyeh et al. [7] proposed the following estimator of population mean when the population means  1 and  2 of the auxiliary variables were known: where  denotes the sample means of the variable y,   and   ( = 1, 2) denote, respectively, the sample and the population means of the variable   ( = 1, 2), and  1 and  2 are real numbers.
The MSE of  AD is given by where  = /;  and  are, respectively, the number of units in the sample and the population;  To minimize MSE( AD ), the optimum values of  1 and  2 are given by The minimum MSE of  AD can be shown as where

Kadilar and Cingi
Estimator.Kadilar and Cingi [8] proposed an estimator using two auxiliary variables,  1 and  2 , to estimate the population mean , as follows: where To minimize MSE( KC ), the optimum values of  1 and  2 are given by The minimum MSE of  KC can be shown as

Malik and Singh
Estimator.Malik and Singh [9] proposed an estimator to estimate the population mean , as follows: where  1 and  2 are real numbers.The MSE of  MS is given by To minimize MSE( MS ), the optimum values of  1 and  2 are given by The minimum MSE of  KC can be shown as 2.4.The Regression Estimator.Rao [12] proposed an estimator using one auxiliary variable,  1 , to estimate the population mean , as follows: Similarly, following Rao, a regression estimator of  using two auxiliary variables,  1 and  2 , is given by where  1 ,  2 , and  3 are real constants.
The MSE of  RE is given by The optimum values of  1 ,  2 , and  3 , obtained by minimizing (15), respectively, are given by The minimum MSE of  RE can be shown as 2.5.The Proposed Estimator.Singh and Espejo [13] proposed an estimator using one auxiliary variable, , to estimate the population mean , as follows: Inspired by this work, we propose a new estimator with two auxiliary variables, as follows: where  1 ,  2 , and  3 are real constants.
Let  0 = /−1,  1 =  1 / 1 −1, and  2 =  2 / 2 −1.Under simple random sampling without replacement (SRSWOR), we have the following expectations: The proposed estimator  pr can be rewritten as By rewriting  pr , we have By retaining only the terms up to the second degree of 's, we have The bias of the proposed estimator is given by The MSE of this new estimator with two auxiliary variables is given by The optimum values of  1 ,  2 , and  3 are given by The minimum MSE of  pr can be shown as 2.6.Comparison of  pr with Some Existing Estimators.We compared the MSE of the proposed estimator with two auxiliary variables given in ( 27) with the MSE of the estimator reported by Abu-Dayyeh et al. [7], as given in (4), Kadilar and Cingi [8], as given in ( 8), Malik and Singh [9], as given in (12), and the regression estimator, as given in (17), as follows: always. (28) Proof.

Numerical Application in Engineering.
To examine the merits of the proposed estimator, we considered four natural population datasets in agricultural, biomedical, and power engineering.We used the following formula to calculate the percent of relative efficiency of different estimators: where  =  RE or  KC or  MS or  pr .
1 : the area of the plant.fractions.From this viewpoint, the proposed estimator can save survey cost.In some sampling yields, the sample fraction is not very large due to the irreversibility or the high cost of the test.Then the accuracy of the proposed estimator is higher.Haq and Shabbir [15] also reported on estimators of finite population mean using two auxiliary attributes and found that their MSE values were reduced when sample size increased.Therefore, the findings of the present are consistent with that study.

Table 1 :
MSE and PRE values of different estimators about population I.

Table 2 :
MSE and PRE values of different estimators about population II.

Table 3 :
MSE and PRE values of different estimators about population III.

Table 4 :
MSE and PRE values of different estimators about population IV.