Improved ratio type estimator using two auxiliary variables under second order approximation

In this paper, we have proposed a new Ratio Type Estimator using auxiliary information on two auxiliary variables based on Simple random sampling without replacement (SRSWOR). The proposed estimator is found to be more efficient than the estimators constructed by Olkin (1958), Singh (1965), Lu (2010) and Singh and Kumar (2012) in terms of second order mean square error.


Introduction
In sampling survey, the use of auxiliary information is always useful in considerable reduction of the MSE of a ratio type estimator. Therefore, many authors suggested estimators using some known population parameters of an auxiliary variable. Hartley-Ross (1954), Quenouille's (1956) and Olkin (1958) have considered the problem of estimating the mean of a survey variable when auxiliary variables are made available. Jhajj and Srivastava (1983), Singh et al.(1995), Upadhyaya and Singh (1999), Singh and Tailor (2003), Kadilar and Cingi (2006), Khoshnevisan et al. (2007), Singh et al. (2007), Singh and Kumar (2011,), etc. suggested estimators in simple random sampling using auxiliary variable.
Moreover, when two auxiliary variables are present Singh (1965Singh ( ,1967 and Perri(2007) suggested some ratio -cum -product type estimators. Most of these authors discussed the Let U= (U 1 ,U 2 , U 3 , …..,U i , ….U N ) denotes a finite population of distinct and identifiable units.
For estimating the population mean Y of a study variable Y, let us consider X and Z are the two auxiliary variable that are correlated with study variable Y, taking the corresponding values of the units. Let a sample of size n be drawn from this population using simple random sampling without replacement (SRSWOR) and y i , x i and z i (i=1,2,…..n ) are the values of the study variable and auxiliary variables respectively for the i-th units of the sample.
When the information on two auxiliary variables are available Singh (1965Singh ( ,1967 proposed some ratio-cum-product estimators in simple random sampling without replacement to estimate the population mean Y of the study variable y, generalized version of one of these estimators is given by, This theorem will be used to obtain MSE expressions of estimators considered here. Proof of this theorem is straight forward by using SRSWOR ( see Sukhatme and Sukhatme (1970)).

First Order Biases and Mean Squared Errors
The expression of the biases of the estimators t 1, t 2 t 3 and t 4 to the first order of approximation are respectively, written as   where, The MSE of the estimator t 1 is minimized for where * 1  and * 2  are, respectively, partial regression coefficients of y on x and of y on z in simple random sampling.
The MSE of the estimator t 2 is minimum for 012 002 020 Differentiating (2.10) with respect to w 1 and w 2 partially, we get the optimum values of w 1 and w 2 respectively as 011 2 1 002 2 2 020 2 1 011 2 1 002 Estimators t 1, t 2 t 3 and t 4 at their respective optimum values attains MSE values which are equal to the MSE of regression estimator for two auxiliary variables.

Proposed Estimator
When auxiliary information on two auxiliary variables are known, we propose an estimator t 5 as where d and c are either real numbers or a function of the known parameters associated with auxiliary information . k 1 and k 2 are constants to be determined such that the MSE of estimator t 5 is minimum under the condition that k 1 +k 2 = 1 and 1  and 2  are integers and can take values -1, 0 and +1 .
Expressing the estimator 5 t in terms of e's we have are expandable. Expanding the right hand side of (3.2), and neglecting terms of e's having power greater than two we have Taking expectation on both sides we get bias of estimator t 5, to the first degree of approximation as   Squaring both sides of (3.3) and neglecting terms of e's having power greater than two we have   Taking expectation on both sides of (3.5) and using theorem 1.
Differentiating (3.6) with respect to k 1 and k 2 partially, equating them to zero and after simplification, we get the optimum values of k 1 and k 2 respectively, as Putting these values in (3.6) we get minimum MSE of estimator t 5 . The minimum MSE of the estimator t 1, t 2, t 3, t 4 and proposed estimator t 5 is equal to the MSE of combined regression estimator based on two auxiliary variables, which motivated us to study the properties of estimators up to the second order of approximation.

Numerical Illustration
For a natural population data, we have calculated the mean square error's of the estimator's and compared MSE's of the estimator's under first and second order of approximations.

Data Set
The data for the empirical analysis are taken from Book, "An Introduction to Multivariate