Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 8704734, 7 pages
https://doi.org/10.1155/2017/8704734

Research Article

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

Jingli Lu

College of Science, Inner Mongolia University of Technology, Hohhot, Inner Mongolia, China

Correspondence should be addressed to Jingli Lu; lujingli2004@163.com

Received 23 January 2017; Revised 21 July 2017; Accepted 25 July 2017; Published 23 August 2017

Academic Editor: Guido Ala

Copyright © 2017 Jingli Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 the minimum MSE 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.

1. 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.

2. Materials and Methods

2.1. Abu-Dayyeh Estimator

Abu-Dayyeh et al. [7] proposed the following estimator of population mean when the population means and of the auxiliary variables were known:

where denotes the sample means of the variable y, and () denote, respectively, the sample and the population means of the variable (), and and are real numbers.

The MSE of is given by

where ; and are, respectively, the number of units in the sample and the population; , , and are the coefficients of variation of , , and , respectively; and , , and are the correlation coefficients between and , and , and and , respectively.

To minimize , the optimum values of and are given by

The minimum MSE of can be shown as

where ; ; .

2.2. Kadilar and Cingi Estimator

Kadilar and Cingi [8] proposed an estimator using two auxiliary variables, and , to estimate the population mean , as follows:

where and ; and are the variances of , , and , respectively; and and are the covariance between and and and , respectively.

The MSE of is given by

To minimize , the optimum values of and are given by

The minimum MSE of can be shown as

2.3. Malik and Singh Estimator

Malik and Singh [9] proposed an estimator to estimate the population mean , as follows:

where and are real numbers.

The MSE of is given by

To minimize , the optimum values of and are given by

The minimum MSE of can be shown as

2.4. The Regression Estimator

Rao [12] proposed an estimator using one auxiliary variable, , to estimate the population mean , as follows:

Similarly, following Rao, a regression estimator of using two auxiliary variables, and , is given by

where , , and are real constants.

The MSE of is given by

The optimum values of , , and , obtained by minimizing (15), respectively, are given by

The minimum MSE of 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 , , and are real constants.

Let , , and . Under simple random sampling without replacement (SRSWOR), we have the following expectations:

The proposed estimator can be rewritten as

By rewriting , 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 , , and are given by

The minimum MSE of can be shown as

2.6. Comparison of 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:

Proof.

where

2.7. 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 or or or .

Population I (source in biomedical engineering [14])

 : number of “placebo” children. : number of paralytic polio cases in the placebo group. : number of paralytic polio cases in the “not inoculated” group. , , , , , , , , , and .

Population II (source in agricultural engineering)

 : cotton output. : the area of the plant. : the proportion of good seed. , , , , , , , , , and .

Population III (source in biomedical engineering [14])

 : weight measurement of children. : midarm circumference of children. : skull circumference of children. , , , , , , , , , .

Population IV (source in power engineering)

 : Electricity consumption by region in China in 2002. : Electricity consumption by region in China in 2001. : Electricity consumption by region in China in 2000. , , , , , , , , , and .

3. Results and Discussion

MSE and PRE values of different estimators about population I can be seen in Table 1.

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

MSE
10
0.294
0.738607
0.738607
0.738607
0.716737
0.433563

15
0.441
0.390406
0.390406
0.390406
0.384209
0.297354

20
0.588
0.211030
0.211030
0.211030
0.209207
0.182492

25
0.735
0.105515
0.105515
0.105515
0.105057
0.098166

10
0.294
100
100
100
103.05
170.36

15
0.441
100
100
100
101.61
131.29

20
0.588
100
100
100
100.87
115.64

25
0.735
100
100
100
100.44
107.47

MSE and PRE values of different estimators about population II can be seen in Table 2.

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

MSE
10
0.056
1.703302
1.703302
1.703302
1.688617
1.607564

50
0.278
0.260505
0.260505
0.260505
0.260159
0.258197

90
0.500
0.100194
0.100194
0.100194
0.100143
0.099852

130
0.722
0.038536
0.038536
0.038536
0.038529
0.038486

10
0.056
100
100
100
100.870
105.955

50
0.278
100
100
100
101.133
100.894

90
0.500
100
100
100
100.051
100.343

130
0.722
100
100
100
100.020
100.132

MSE and PRE values of different estimators about population III can be seen in Table 3.

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

MSE
10
0.182
0.15048093
0.15048093
0.15048093
0.15040335
0.15032137

20
0.364
0.05981617
0.05981617
0.05981617
0.05980391
0.05979094

30
0.545
0.02821517
0.02821517
0.02821517
0.02821245
0.0282096

40
0.727
0.01279088
0.01279088
0.01279088
0.01279032
0.01278973

10
0.182
100
100
100
100.0516
100.1061

20
0.364
100
100
100
100.0205
100.0422

30
0.545
100
100
100
100.0097
100.0199

40
0.727
100
100
100
100.0044
100.0090

MSE and PRE values of different estimators about population IV can be seen in Table 4.

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

MSE
10
0.333
21.30709
21.30709
21.30709
21.30557
17.16434

15
0.500
10.54174
10.54174
10.54174
10.54137
9.52763

20
0.667
6.70838
6.70838
6.70838
6.70823
6.29770

25
0.833
3.51391
3.51391
3.51391
3.51387
3.40123

10
0.333
100
100
100
100.0071
124.1358

15
0.500
100
100
100
100.0035
110.6438

20
0.667
100
100
100
100.0022
106.5210

25
0.833
100
100
100
100.0012
103.3129

The relative efficiency was studied based on the traditional regression- or ratio-type estimators in many literatures (Abu-Dayyeh et al. [7], Haq and Shabbir [15], and Verma et al. [16]). However, we studied the relative efficiency based on the higher efficient estimators using two auxiliary variables and found that the efficiency of the proposed estimator is higher than the estimators noted above under any conditions.

Under different sample sizes and different datasets, we notice from the data given in Tables 1, 2, 3, and 4 that the proposed estimator of a finite population mean using two auxiliary variables is always more efficient than the estimators , , , and . Although the expressions of the estimators reported by Abu-Dayyeh et al., Kadilar and Cingi, and Malik and Singh are different, we note from the comparative study above that the minimum MSE values for the estimators reported by Abu-Dayyeh et al. [7], Kadilar and Cingi [8], and Malik and Singh [9] are equal and have same expression. The regression estimator has a smaller MSE value than the three estimators noted above. With increase in sample fraction, the MSE and PRE values of the proposed estimator decrease. Therefore, a smaller sampling fraction yields better results relative to MSE and PRE values when compared to a larger sampling fraction. Moreover, with a large sampling fraction, the efficiency differential among all estimators in the present study is very small. Therefore, it is suggested that our estimators be used with small sampling 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.

4. Conclusions

In this paper, we proposed the improved estimator of a finite population mean by utilizing information on two auxiliary variables in SRS. Bias and MSE expressions of the proposed estimator, , were obtained. We clearly proved that the new estimator is always more efficient than the estimators reported by Abu-Dayyeh et al. [7], Kadilar and Cingi [8], and Malik and Singh [9], as well as the regression estimator using two auxiliary variables. These theoretical conditions are also satisfied by the results of four numerical examples in agricultural, biomedical, and power engineering. It should be noted that a smaller sample size yields better results relative to MSE and PRE values when compared to a larger sample size. Thus, for use with small sample size, the suggested estimator would be cost-saving in actual practice and are, therefore, recommended for efficient estimation of finite population mean.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11461051) and Natural Science Foundation of Inner Mongolia Autonomous Region of China (no. 2017MS(LH)0101), and the author gratefully acknowledges this support.

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