A note on a difference-type estimator for population mean under two-phase sampling design

In this manuscript, we have proposed a difference-type estimator for population mean under two-phase sampling scheme using two auxiliary variables. The properties and the mean square error of the proposed estimator are derived up to first order of approximation; we have also found some efficiency comparison conditions for the proposed estimator in comparison with the other existing estimators under which the proposed estimator performed better than the other relevant existing estimators. We show that the proposed estimator is more efficient than other available estimators under the two phase sampling scheme for this one example; however, further study is needed to establish the superiority of the proposed estimator for other populations.

x i of the study as well as auxiliary variable respectively.
Also let S 2 y = 1 N − 1 N i=1 y i −Ȳ 2 and S 2 x = 1 N − 1 N i=1 x i −X 2 be the population variances of the study and the auxiliary variable respectively and let C y and C x be the coefficient of variation of the study as well as auxiliary variable respectively, and ρ yx be the correlation coefficient between x and y. Let y and x be the study and the auxiliary variable in the sample with corresponding values y i and x i respectively for the i-th unit i = {1, 2, 3…, n} in the sample with unbiased means ȳ = 1 n n i=1 y i and x = 1 n n i=1 x i respectively. Also let Ŝ 2 y = 1 n − 1 n i=1 y i −ȳ 2 and Ŝ 2 x = 1 n − 1 n i=1 (x i −x) 2 be the corresponding sample variances of the study as well as auxiliary variable respectively.
be the co-variances between their respective subscripts respectively. Similarly b yx(n) =Ŝ yx S 2 x is the corresponding sample regression coefficient of y on x based on a sample of size n. Also C y = S ȳ Y , C x = S x X and C z = S z Z are the coefficients of variations of the study and auxiliary variables respectively. Also

Some existing estimators
Let us consider a finite population U = {U 1 , U 2 , U 3 , …, U N } of size N units. To estimate the population mean Ȳ of the variable of interest say y taking values y i , in the existence of two auxiliary variables say x and z taking values x i and z i for the ith unit U i . We assume that there is a high correlation between y and x as compared to the correlation between y and z, (i.e. ρ yx > ρ yz > 0). When the population X of the auxiliary variable x is unknown, but information on the other cheaply auxiliary variable say z closely related to x but compared to x remotely to y, is available for all the units in a population. In such a situation we use a two phase sampling. In the two phase sampling scheme a large initial sample of size n′ (n′ < N) is drawn from the population U by using simple random sample without replacement sampling (SRSWOR) scheme and measure x and z to estimate X . In the second phase, we draw a sample (subsample) of size n from first phase sample of size n′, i.e. (n < n′) by using (SRSWOR) or directly from the population U and observed the study variable y. The variance of the usual simple estimator t 0 =ȳ = 1 n n i=1 y i up to first order of approximation is, given by The classical ratio and regression estimators in two-phase probability sampling and their mean square errors up to first order of approximation are, given by Chand (1975), suggested the following chain ratio-type estimator the suggested estimator is, given by The mean square error of the suggested estimator is, given as Khare et al. (2013), proposed a generalized chain ratio in regression estimator for population mean, the recommended estimator is given by where α is the unknown constant, and the minimum mean square error at the optimum value of α = ρ yz C x ρ yx c z is, given by Recently Singh and Mahji (2014), suggested a chain-type exponential estimators for Ȳ given by The mean square errors of the suggested estimators, up to first order of approximation are, given as follows

The proposed estimator
On the lines of Khare et al. (2013), we propose a difference-type estimator for population mean under two-phase sampling scheme using two auxiliary variables; the suggested estimator is, given by where k 1 and k 2 are the unknown constants, To obtain the properties of the proposed estimator we define the following relative error terms and their expectations.
Rewriting (16), in terms of e's, we have Expanding the right hand side of the above equation, and neglecting terms of e's having power greater than two, we have On squaring and taking expectation on both sides of Eq. (17), and keeping terms up to second order, we have Further simplifying, we get Now to find the minimum mean squared error of t m , we differentiate Eq. (18) with respect to k 1 and k 2 respectively and putting it equal to zero, that is On substituting the optimum values of k 1 and k 2 in Eq. (18) we get the minimum mean square error (MSE) of the proposed estimator t m up to order one is, given as

Efficiency comparison
In this section, we have compare the propose estimator with the other existing estimators.