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Abstract
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.
Citation: Lu J (2013) The Chain Ratio Estimator and Regression Estimator with Linear Combination of Two Auxiliary Variables. PLoS ONE 8(11): e81085. https://doi.org/10.1371/journal.pone.0081085
Editor: Raya Khanin, Memorial Sloan Kettering Cancer Center, United States of America
Received: July 26, 2013; Accepted: October 9, 2013; Published: November 18, 2013
Copyright: © 2013 Jingli Lu. 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.
Funding: The authors has no support or funding to report.
Competing interests: The author has declared that no competing interests exist.
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
(1)
(2)where it is assumed that the population mean
of the auxiliary variate x is known.
Here(3)where 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 yi and xi, respectively.
is the population covariance between yi and xi [11].
The MSE of the classical ratio estimator is(4)where
; N is the number of units in the population;
is the population ratio,
and
are the population means of the yi and xi respectively.
The MSE of the regression estimator is(5)where
is the population correlation coefficient between
and
.
Kadilar and Cingi[12] proposed the chain ratio estimator using one auxiliary information for as
(6)where
is real number.
MSE of this estimator is given as follows:(7)
The traditional multivariate ratio estimator and regression estimator using information of two auxiliary variables x1 and x2 to estimate the population mean, [13], as follows:
(8)
(9)where
and
(i = 1,2) denote respectively the sample and the population means of the variable xi (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 (10)where
and
denote the correlation coefficient between Y and X1, Y and X2, X1 and X2 respectively.
.
The optimum values of and
are given by
The minimum MSE of can be shown to be:
(11)
The MSE of this traditional multivariate regression estimator is given by(12)
The optimum values of and
are given by
The suggested estimators
We propose the multivariate chain ratio estimator and regression estimator using linear combination of two auxiliary variables as follows: (14)
(15)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 (16)where
The optimum values of and
are given by
The minimum MSE of can be shown to be:
(17)where
The MSE of this new multivariate regression estimator is given by(18)
Where ,
.
The optimum values of and
are given by
The minimum MSE of can be shown to be:
(19)
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:(20)
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:(21)
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).
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.
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.
Author Contributions
Conceived and designed the experiments: JL. Performed the experiments: JL. Analyzed the data: JL. Contributed reagents/materials/analysis tools: JL. Wrote the paper: JL.
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