On The Efficiency of Ratio Estimators of Finite Population Mean using Auxiliary Information

Ratio estimation is technique that usages available auxiliary information which is certainly correlated with study variable. In this study, class of ratio-type estimators of finite population mean has been anticipated to solve delinquent of estimation of population mean. Properties of anticipated estimators namely Bias & Mean Square Error were acquired up to first order of approximation & condition for their efficiency over some existing estimators was also established. The results show that anticipated estimators are enhanced & proficient (minimum mean square errors) than other estimators with the highest precision


Introduction
Usage of auxiliary information is made through the ratio & product techniques of estimation to enhance estimates of population mean.Estimation of population mean of variable of interest with higher precision is unremitting issue in sample survey.So, precision could be increased by the used of apposite estimation procedure which consumes auxiliary information which is meticulously associated to variable of interest.In ratio method of estimation, auxiliary information is available which is linearly related to the variable of study.The population parameters such as populations' median, coefficient of kurtosis, skewness, coefficient of variation, decile, quartile, correlation, etc are auxiliary variables.Efficiency of estimators of population parameters can be increased by suitable usage of auxiliary information in relationship with auxiliary variable.Cochran 1 came up with what is known as ratio-type estimator for estimation of population mean which is more competent than sample mean.Many authors have used different auxiliary information inorder to enhance the precision of the estimates by using prior knowledge of population parameters.Researchers in sample survey like Kadilar and Cingi 2,3 developed classes of ratio estimators using known auxiliary information on coefficients of variation, & kurtosis.Abid et al. 4 also suggested set of ratio-type estimators for the population mean using nonconventional location parameters like mid-range, and tri-mean as auxiliary information.Other researchers are Upadhyaya and Singh, 5 Yan and Tian, 6 Subramani and Kumarapadiyan, 7 Subramani and Kumarapadiyan, 8 Subramani and Kumarapadiyan, 9 Subramani and Kumarapadiyan, 10 Jeelani et al, 11 and Nasir et al. 12 The objective of this study is to develop innovative set of ratio-type estimators to increase precision of estimates of population mean using known auxiliary information.

Proposed Estimator
Motivated by the work of Subzar et al 13 , we proposed ratio-type estimators for estimating population mean using value of hodges-lehmann as: ... (

Empirical Study
To evaluate performance of anticipated estimators, following real populations are used.

Future Scope
The future scope ofstudy is to transform the sampling technique from simple random sampling to other sampling techniques like stratified sampling, two stage sampling, cluster sampling or successive sampling.

Conclusion
In Table 3

size n drawn without replacement. and is population mean square of study variable and is population mean square of auxiliary variable. Following are other symbols used in this study.
,3,...,N has pair of values.Y is study variable & X is auxiliary variable which is associated (correlated) with Y, in which x = (x 1 ,x 2 ,...x n ) & y = (y 1 , y 2 ,...y n ) are the n sample values.& ȳ is sample mean of variable of interest

Table 3 : shows the mean square error (MSE) & percentage relative efficiency (PRE) for the three populations.
Bias, Mean Square Error (MSE) & Percentage Relative Efficiency (PRE) of anticipated & existing estimators considered in study for all populations used.Outcomes also discovered that anticipated estimators have least MSE and advanced PRE than other estimators.The outcomes also show that average dispersion of anticipated estimators gives better estimates on the average compare to other estimators considered.
, anticipated estimators performed better than prevailing estimators considered in study.So, it is clear that anticipated estimators performed superior than other estimators having minimum Mean Square Error (MSE) & highest Percentage Relative Error (PRE).We therefore conclude that anticipated estimators are relatively efficient and better than other estimators for estimation of population mean.