Some Improved Modified Ratio Estimators Based on Decile Mean of an Auxiliary Variable

The estimators developed so far regarding the study under consideration use the conventional measures of central tendencies, i.e. the mean, the median, the quartile mean etc., and comment on their properties. However, Sohel et al. (2012) have proposed decile mean as a measure of central tendency and have proved that it outperforms the conventional measures of central tendency. In this study, we have attempted to use the decile mean instead of the conventional measures suggested in previous studies. Also, we have used decile mean, population correlation coefficient, coefficient of variation and the linear combinations of auxiliary variable and investigated the properties associated with the proposed estimator. Theoretically, mean square error equations of all proposed ratio estimators are obtained and the efficiency conditions are derived. This study has been verified numerically.


Introduction
Now a day, the information obtained through auxiliary variables was largely discussed in survey sampling. The auxiliary variables are generally connected with the study variables and we may take up this information in different forms such as ratio, product and regression etc. The auxiliary information may be accessible from several sources such as similar studies in past, economic reports, national census etc.
The ratio and regression estimators are used to improve the efficiency of the simple random sampling without replacement (SRSWOR) sample mean when there is a positive correlation exist between study variable and an auxiliary variable under certain conditions (see for example Cochran (1977) and Murthy (1967)).

Let
* + be the different and particular units from a finite population and be the study variable with value obtained from , The purpose is to estimation population mean ̅ ∑ on the base of a random sample. The notations considered in this article are as follows:

Coefficient of kurtosis of auxiliary variable
Regression coefficient of on The ratio estimator based on the population mean, ̅ , is defined as The bias, related constant and the mean squared error (MSE) of the ratio estimator are respectively given by ).
The rest of the study is presented as follows: Section 2 gives a explanation of the existing estimators. The construction and the efficiency comparison of the suggested estimator with the existing estimators are obtainable in Section 3. Section 4 contains the numerical comparison of the proposed and existing estimators. Finally, conclusions of the study are given in Section 5. Cingi (2004, 2006) has been proposed some linear regression type ratio estimators as follows: ̅

The existing modified ratio estimators
Yan and Tian (2010) suggested some modified linear regression type ratio estimators as follows: The Subramani and Kumarapandiyan (2012a, 2012b, 2012c) suggested estimators based on population median, skewness, kurtosis and coefficient of variation of an auxiliary variable are shown below: The constants, biases and MSEs of the estimators developed by Kadilar  , .

The Suggested modified ratio estimators
In this section, we have suggested some modified ratio type estimators using the auxiliary information on population decile mean, population coefficient of variation and population correlation coefficient for the estimation of population mean of a variable of interest. Sohel et al. (2012) showed that the decile mean perform better than the conventional measures of locations such as mean, median, mode in the presence of extreme values. So, it is better to use decile mean instead of mean, median and mode when the extreme values are present in the data sets. The proposed estimators are shown below, whereas the bias, constant and MSE of these proposed estimators are respectively as under:

Efficiency Comparisons
In this section, the efficiency conditions for which the proposed modified ratio estimators performs better as compared to the existing modified ratio estimators have been derived algebraically.

Comparisons with existing ratio estimator
The proposed ratio estimators performed better than the existing ratio estimators if an only if, where .
If the above condition is fulfilled, then our suggested estimators perform better as compared to the existing estimators consider in this study.

Numerical Illustration
For the empirical study of the proposed and existing estimators, we have used 3 natural populations. The population 1 is obtained from Singh and Chaudhary (1986) page 177 and Population 2 and Population 3 are obtained from Murthy (1967) page 228.
The characteristics of the 3 populations are given in Table 1. The values of the constants and biases of the existing and proposed estimators are specified in Tables 2 and 3, respectively, whereas the mean square errors values of the existing and proposed estimators are given in Tables 4 and 5, respectively. From an analysis of Tables 2-5, several interesting observations can be made:  It is observed that the condition mentioned in equation (2) is satisfied because all the proposed estimators have smaller values of constants as compared to the existing estimators (cf Table 2 vs Table 3).


The biases of the suggested estimators are smaller than the existing estimators in literature (cf Table 2 vs Table 3).  Table 4 vs Table 5).     Table 4 vs Table 5).  It is noted that the proposed estimator, ̅ has a smaller MSE value i.e. (9644.04, 234368.4 and 144936.7) as compared to all the proposed estimators and existing estimator for three real populations consider in this study (cf Table 4 vs Table 5).
So in general, we can say that our suggested modified ratio estimators are perform more efficiently as compared to the existing modified ratio estimators.

Conclusions
In sample survey the availability of auxiliary information enhances the efficiency of the estimators. In this study, we have been proposed several modified ratio estimators using known value of population decile mean, coefficient of variation and population correlation coefficient by using the information on the auxiliary variable. It is observed that the mean squared errors values of the suggested estimators are smaller than existing estimators for all the selected three known populations. Also, as we know that the measures like the mean, median and mode are affected by the extreme values in the population, while decile mean is robust to extreme values. Hence, it can be claimed that our proposed estimators outperform than the exiting estimators for the practical consideration.