Figures
Abstract
In sample surveys, it is usual to make use of auxiliary information to increase the precision of estimators. We propose a new exponential ratio-type estimator of a finite population mean using linear combination of two auxiliary variables and obtain mean square error (MSE) equation for proposed estimator. We find theoretical conditions that make proposed estimator more efficient than traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja and the estimator proposed by Abu-Dayeh et al. In addition, we support these theoretical results with the aid of two numerical examples.
Citation: Lu J, Yan Z, Peng X (2014) A New Exponential Ratio-Type Estimator with Linear Combination of Two Auxiliary Variables. PLoS ONE 9(12): e116124. https://doi.org/10.1371/journal.pone.0116124
Editor: Gang Han, Yale School of Public Health, United States of America
Received: July 10, 2014; Accepted: December 2, 2014; Published: December 26, 2014
Copyright: © 2014 Lu et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper.
Funding: This work was supported by the National Natural Science Foundation of China (No. 11361036 and No. 11461051). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
In the sampling theory, the use of supplementary information provided by auxiliary variables in survey sampling was extensively discussed. Such information is generally used in ratio, product and regression type estimators for the estimation of population mean of study variable. In literature, number of authors introduced many ratio and regression type estimators by using general linear transformation of the auxiliary variable [1]–[5]. For recent development, exponential estimators have been widely studied by several authors. Singh et al.[6] suggested the modified exponential ratio and product estimators in two phase sampling and analyzes their properties, these estimators were compared for their precision with simple mean per unit, usual double sampling ratio and product estimators. On base of the estimator of Singh et al., Ozgul and Cingi [7] suggested a class of exponential regression cum ratio estimator in two phase sampling, MSE of the proposed estimator were obtained. However, these estimators were considered using one auxiliary variate.
In this study, a new exponential ratio-type estimator using linear combination of two auxiliary variates is considered to estimate a finite population mean for the variable of interest. And we obtain mean square error (MSE) equation for the proposed estimator. We find theoretical conditions that make proposed exponential ratio-type estimator more efficient than traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja and the estimator proposed by Abu-Dayeh et al. We compared the traditional ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja, the estimator proposed by Abu-Dayeh et al. and proposed exponential ratio-type estimator using two statistic data sets. And we obtained the satisfactory results.
Materials and Methods
The existed estimators
The traditional multivariate ratio estimator using information of two auxiliary variables x1 and x2 to estimate the population mean, [8], as follows:
(1)
where denote the sample means of the variable y,
and
(i = 1,2) denote respectively the sample and the population means of the variable xi (i = 1,2);
are the weights that satisfy the condition,
The MSE of this traditional multivariate ratio estimator is given by (2)
where ; n and N are respectively the number of units in the sample and the population;
,
and
are the population variances of Y,
and
, respectively;
,
and
are the population covariances between
and
,
and
,
and
, respectively;
.
The optimum values of and
are given by
,
The minimum MSE of can be shown to be:
(3)
Bahl and Tuteja [9] proposed an exponential ratio-type estimator which is given by (4)
Abu-Dayeh et al.[10] proposed the estimator using two auxiliary variables given by(6)
where .
MSE of this estimator is given as follows:(7)
The proposed family of ratio estimators
We propose a new exponential ratio-type estimator using linear combination of two auxiliary variables as follows: (9)
where ,
;
are weights that satisfy the condition:
.
MSE of this estimator can be found using Taylor series method defined as (10)
where .
The MSE of this new multivariate exponential ratio-type estimator is given by (11)
The optimum values of and
are given by
The minimum MSE of can be shown to be:
where
Efficiency comparisons
We compare the MSE of the proposed exponential ratio-type estimator using linear combination of two auxiliary variables given in Eq. (12) with the MSE of traditional multivariate ratio estimator using information of two auxiliary variables given in Eq. (3) as follows:
We compare the MSE of the proposed exponential ratio-type estimator using linear combination of two auxiliary variables given in Eq. (12) with the MSE of the estimator of Bahl and Tuteja given in Eq. (5) as follows:
where denote
using auxiliary variable xi (i = 1,2).
We compare the MSE of the proposed exponential ratio-type estimator using linear combination of two auxiliary variables given in Eq. (12) with the MSE of the estimator of Abu-Dayeh et al given in Eq.(8) as follows:
Numerical illustration
To examine the merits of the proposed estimator, we have considered two natural population data sets. we apply the traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja,the estimator of Abu-Dayeh et al, given in Eqs. (1), (4), (6) and proposed exponential ratio-type estimator of a finite population mean using linear combination of two auxiliary variables, given in Eq. (9). The MSE of these estimators are computed as given in Eqs. (3), (5), (8), (12).
Example 1. In order to precisely estimate cotton output in one region, the sample size n = 8 villages were taken out from N = 18 villages using SRSWOR[8].
Y: Cotton output.
X1: The area of the plant.
X2: The proportion of good seed.
The statistics of example 1 are given in table 1
Example 2. The data set of this example can been seen in the reference [3].
Y: Number of ‘placebo’ children.
X1: Number of paralytic polio cases in the placebo group.
X2: Number of paralytic polio cases in the ‘not inoculated’ group.
The statistics of example 2 are given in table 2
Results and Discussion
MSE values of the traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja, the estimator of Abu-Dayeh et al and proposed exponential ratio-type estimator using linear combination of two auxiliary variables can be seen in Table 3 and Table 4.
From Table 3 and 4, we notice that our proposed exponential ratio-type estimator using linear combination of two auxiliary variables is more efficient than traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja and the estimator of Abu-Dayeh et al. We examine the conditions, determined in paper, for two data sets,
The examining of condition (13), (14) and (15) about example 1 can been seen as follows.
The examining of condition (13), (14) and (15) about example 2 can been seen as follows.
The result shows that the condition (13), (14) and condition (15) are satisfied. Therefore, we suggest that we should apply the proposed estimator to example1 and 2.
Conclusions
We develop a new exponential ratio-type estimator of a finite population mean using two auxiliary variables and find theoretical conditions that make proposed estimator more efficient than traditional multivariate ratio estimator using information of two auxiliary variables, the estimator of Bahl and Tuteja and the estimator proposed by Abu-Dayeh et al. These theoretical conditions are also satisfied by the results of two numerical examples.
Author Contributions
Conceived and designed the experiments: JL. Performed the experiments: JL. Analyzed the data: JL ZY XP. Contributed reagents/materials/analysis tools: JL ZY XP. Wrote the paper: JL.
References
- 1. Lu J (2013) The Chain Ratio Estimator and Regression Estimator with Linear Combination of Two Auxiliary Variables. Plos one 8(11):e81085.
- 2. Lu J, Yan Z (2014) A Class of Ratio Estimators of a Finite Population Mean Using Two Auxiliary Variables. Plos one 9(2):e89538.
- 3.
Choudhury S, Singh BK (2012) A Class of Product-cum-dual to Product Estimators of the Population Mean in Survey Sampling Using Auxiliary Information. Asian J Math Stat 6..
- 4. Kadilar C, Candan M, Cingi H (2007) Ratio estimators using robust regression. Hacet J Math Stat 36:181–188.
- 5. Al-Omari AI, Jemain AA, Ibrahim K (2009) New ratio estimators of the mean using simple random sampling and ranked set sampling methods. Revista Investigacion Operacional 30:97–108.
- 6. Sing HP, Vishwakarma GK (2007) Modified exponential ratio and product estimators for finite population mean in two phase sampling. Austrian Journal of Statistics 36(3):217–225.
- 7. Ozgul N, Cingi H (2014) A new class of exponential regression cum ratio estimator in two phase sampling. Hacettepe Journal of mathematics and statistics 43(1):131–140.
- 8.
Feng SY, Ni JX, Zou GH (1998) The Theory and Methods of Sampling Survey. Beijing: China Statistics Press, 145–150p. (in Chinese)
- 9. Bahl S, Tuteja RK (1991) Ratio and product type exponential estimators. J Inform Opti Scien 12:159–163.
- 10. Abu-Dayeh WA, Ahmed MS, Ahmed RA, Muttlak HA (2003) Some Estimators of a Finite Population Mean Using Auxiliary Information. Appl Math Comput 139:287–298.