Alternative Ratio Estimators for Estimating Population Mean in Simple Random Sampling Using Auxiliary Information

Alternative ratio estimators are proposed for a finite population mean of a study variable in simple random sampling using the information on the mean of the auxiliary variable, which is positively correlated with the study variable. The properties associated with the proposed estimators are assessed by mean square error and bias and the expressions for bias and mean square for proposed estimators are also obtained. Both analytical and numerical comparisons have shown the proposed alternative estimators are more efficient than the classical ratio and the existing estimators under consideration.


INTRODUCTION
In Sample Surveys, auxiliary information is always used to improve the precision of the estimates of the population parameters. This can be done at either estimation or selection stage or both stages. The commonly used estimators, which make use of auxiliary variables, include ratio estimators, regression estimator, product estimator and difference estimator. The classical ratio estimator is preferred when there is a high positive correlation between the variable of interest, Y and the auxiliary variable, X with the regression line passing through the origin. The classical product estimator, on the other hand is mostly preferred when there is a high negative correlation between Y and X while the linear regression estimator is most preferred when there is high positive correlation between the two variables and the regression line of the study variable on the auxiliary variable has intercept on Y axis.
Ratio estimation has gained relevance in estimation theory because of its improved precision in estimating the population parameters.
It has been widely applied in Agriculture to estimate the mean yield of crops in a certain area and in Forestry, to estimate with high precision, the mean number of trees or crops in a forest or plantation. Other areas of relevance include Economics and Population studies to estimate the ratio of the income to family size.
So Cochran (1940) initiated the use of auxiliary information at estimation stage and proposed ratio estimator for population mean. It is well established fact that ratio type estimators provide better efficiency in comparison to simple mean estimator if the study variable and auxiliary variable are positively correlated and the regression line pass through origin and if on the other side correlation between the study variable and auxiliary variable is positive and does not pass through origin, but makes an intercept, in that case regression method provide better efficiency than ratio, simple mean and product type estimator and if the correlation between the study variable and auxiliary variable is negative, product estimator given by Robson (1957) is more efficient than simple mean estimator.
Further, Subramani and Kumarapandiyan (2012) had taken initiative by proposing modified ratio estimator for estimating the population mean of the study variable by using the population deciles of the auxiliary variable.
Recently Subzar et al. (2016) had proposed some estimators using population deciles and correlation coefficient of the auxiliary variable, also  had proposed some modified ratio type estimators using the quartile deviation and population deciles of auxiliary variable and  had also proposed an efficient class of estimators by using the auxiliary information of population deciles, median and their linear combination with correlation coefficient and coefficient of variation and  also proposed some modified ratio estimators for estimating population mean using the auxiliary information of quartiles and their linear combination with correlation coefficient and coefficient of variation.
In this paper we have envisaged an alternative ratio estimator's for estimation of population mean of the study variable using the information of non-conventional location parameters, non-conventional measures of dispersion, coefficient of variation and median. Let The classical Ratio estimator for the population mean Y of the study variable Y is defined as: Where y is the sample mean of the study variable Y and x is the sample mean of the auxiliary variable X . It is assumed that the population mean X of the auxiliary variable X is known. The bias and mean squared error of R Yˆto thefirst degree of approximation are given below; Before discussing about the proposed estimators, we will mention the estimators in Literature using the notations given in the next sub-section.
Decile mean

Subscript
For existing estimators For proposed estimators The biases, related constants and the mean square error (MSE) for Abid et al. (2016) estimators are respectively given by: The biases, related constants and the mean square error (MSE) for Abid et al (2016) estimators are respectively given by:

IMPROVED RATIO ESTIMATORS
Motivated by the mentioned estimators in Section 1.2, we propose Alternative ratio estimators using the linear combination of non-conventional location parameters, nonconventional measures of dispersion with coefficient of variation and median.
The bias, related constant and the MSE for the first proposed estimator can be obtained as follows: MSE of this estimator can be found using Taylor series method defined as As shown in Wolter (1981), (2.1) can be applied to the proposed estimator in order to obtain MSE equation as follows: Where . Note that we omit the difference of Similarly, the bias is obtained as Thus the bias and MSE of the proposed estimator is given below: Similarly, the bias, constant and the mean square error can be found using the Taylor series method and is given as below:

Efficiency Comparisons:
From the expressions of the MSE of the proposed estimators and the existing estimators, we have derived the conditions for which the proposed estimators are more efficient than the usual and existing modified ratio estimators are given as follows: Comparison with the classical ratio estimator Modified proposed ratio estimators are more efficient than that of the classical ratio

APPLICATIONS
The performances of the proposed ratio estimators are evaluated and compared with the mentioned ratio estimators in Section 1.2 by using the data of the natural population. For the population we use the data of Singh and Chaudhary 1986, page 177. We apply the proposed, classical ratio and existing estimators to this data set and the data statistics of this population is given in Table 1. From Table 2, we observe that the proposed estimators are more efficient than all of the estimators in literature as their Bias, Constant and Mean Square error are much lower than the existing estimators.
The percentage relative efficiency (PRE) of the proposed estimators (p), with respective to the existing estimators (e), is computed by These PRE values are given in Table 3     CONCLUSION From the above results it can be concluded that the proposed ratio estimators are more efficient than the existing estimators and the above estimators are also more efficient than the classical ratio estimator, thus providing better alternative estimators for use in practical situations.