An exponential and log ratio estimator of population mean using auxiliary information in double sampling

In this study an improved version of ratio type exponential estimator is been proposed for estimating average of study variable when the population parameter(s) information of second auxiliary variable is available. The proposed estimator compared with usual unbiased estimator and conventional ratio estimators numerically and hypothetically. The mean square error is also obtained and checked the efficiency of the proposed estimator with usual ratio, Singh and Vishwakarma (2007), Singh et al. (2008), Noor-ul-Amin and Hanif (2012), Yadav et al. (2013) and Sanaullah et al. (2015) estimators.


Introduction
The use of auxiliary variables often increases estimator's efficiency when ratio and regression methods are considered for estimation in survey sampling. The product estimation method may be applied for negative linear relationship and the ratio method of estimation should be applied for the positive linear relationship between response and auxiliary variables. The ratio method for estimating population mean of study variable was introduced by Cochran (1940) using auxiliary variable in single phase sampling design for estimating population mean. Single-phase sampling was used by Robson (1957)-and- Murthy (1964) for estimating the population mean using product estimators. Further contributions under single phased sampling design on the use of auxiliary information include Srivastava (1967), Walsh (1970)- Gupta (1978)- Vos (1980), Kaur (1983), Bahl and Tuteja (1991), Samiuddin and Hanif (2006), Upadhyaya et al. (2011) and Hanif et al. (2009). Sometimes, estimation is done from large sample to observe the auxiliary variable x when the mean value X is unknown and later on a smaller sample is chosen from already drawn large sample for estimating y (study variate). This method is named as two phase or double sampling. Hidiroglou and Särndal (1998) proposed twophase sampling design using two auxiliary variables, which was very cost effective.
Usually the ratio estimators do not work efficiently under situations when the relationship is not linear between the auxiliary and study variables. In such situations the exponential product and ratio estimators proposed by (Bahl & Tuteja, 1991). The exponential ratio estimators were proposed by Singh and Vishwakarma (2007) followed the work of Bahl and Tuteja (1991) in double sampling design. The auxiliary information was used by Samiuddin and Hanif (2007) in two-phase sampling and presented three situations for estimating population parameter i.e. full, partial and no information case. Singh et al. (2008) proposed an unbiased estimator when the information of population parameters was available in two phase sampling. Hanif et al. (2009) extended the work of ratio estimators under two and multi phase sampling schemes. Furthermore, Ahmad et al. (2009) did the comprehensive contribution of ratio estimators. An improved exponential ratio type estimators using two auxiliary variables has been proposed by Sanaullah et al.
Consider the usual ratio estimator A ratio type exponential estimator suggested by Bahl and Tuteja (1991) in two phase sampling is modified by Singh and Vishwakarma (2007) as: The mean square error of Singh and Vishwakarma (2007) is The ratio-cum-product exponential estimator in double sampling is proposed by Noor-ul-Amin and Hanif (2012) as: The min (MSE) of the Y NH a is given as: Noor-ul-Amin and Hanif (2012) also proposed chain ratio-type estimator as: Inspired by the above estimators Sanaullah et al. (2012) proposed modified exponential estimator using two auxiliary variables as: Yadav et al. (2013) proposed ratio type exponential estimator given as: The estimators cited above have been extensively used for estimation of population mean in varied situations. The core objective of the study is to proposed an improved ratio type exponential estimator and analyzes its properties. We have used auxiliary information of two variables and proposed a log function estimator in the following section. Efficiency of the proposed estimator has been checked mathematically with some existing estimators i.e.

SA and Y
Yad in section three, while empirical study and conclusion is discussed in section four and five respectively.

The Proposed Estimator
Following Srivastava (1971), we proposed an log in exponential type estimator in two double sampling using x and z auxiliary variables. Let To acquire the properties of the proposed estimator, such as the mean square error we consider the following as, On further simplification of (2.2) up to the order O n -1 ( ) , we may get as, We may get the optimum values of  Applying expectation under both phases when the information from both phases is used, then MSE of (2.7) given as: iv.

MSEŶ SA
If the conditions above in (3.1) to (3.6) are met then proposed estimator Y g 1 ( ) are more efficient than the estimators

Empirical Study
Real

Conclusion
An exponential type ratio estimator in double sampling has been proposed in this study. Its mean square error is obtained up-to the first degree of approximation. In some situations the suggested estimator