A Regression Type Estimator with Two Auxiliary Variables for Two-Phase Sampling

This paper is an extension of Hanif, Hamad and Shahbaz estimator [1] for two-phase sampling. The aim of this paper is to develop a regression type estimator with two auxiliary variables for two-phase sampling when we don’t have any type of information about auxiliary variables at population level. To avoid multi-collinearity, it is assumed that both auxiliary variables have minimum correlation. Mean square error and bias of proposed estimator in two-phase sampling is derived. Mean square error of proposed estimator shows an improvement over other well known estimators under the same case.


Introduction
It is fact that precision of estimators of the mean of study variable "y" is increased by proper attachment of highly correlated auxiliary variables.In some situations where auxiliary information is available at population level and cost per unit of collecting study variable "y" is affordable then single-phase sampling is more appropriate.But in a situation where prior information of auxiliary variable is lacking then it is neither practical nor economical to conduct a census for this purpose.The appropriate technique used to get estimates of those auxiliary variables on the basis of samples is two-phase sampling.In such cases we take large preliminary sample and from that auxiliary variables are computed.The main sample is independently sub-sampled from that large sample.Two-phase sampling is a powerful technique which was firstly introduced by Neyman [2] for the stratification purpose.Two-phase sampling is based on the idea of a sampling design in which nature (specifically the size) of sampling units does not differ at any phase of sampling."Two-phase sampling is generally employed when number of units, required to give the desired precision on different items, is widely different.This technique is employed to utilize the information collected at the first phase in order to improve the precision of the information to be collected at the second phase" [3].
In two-phase sampling, regression and ratio estimation techniques are used to estimate the finite population mean.Ratio estimator incorporates the prior information closely related to study variable and regression technique is used when relation between study variable and auxiliary variable(s) is linear.Regression estimator is considered to be more useful than ratio estimator except when regression line does not pass through origin otherwise these two estimators have almost same significance and analyst has to decide intuitively.
Let the population consist of units, i i and .
Cochran [4] appears to be the first to use auxiliary information in Ratio estimator when there is highly positive correlation between study variable and auxiliary variables.Hansen and Hurwitz [5] were first to suggest the use of auxiliary information in selecting the population with varying probabilities.Robson [6] gave the idea of product estimator when there is highly negative correlation.Two-phase sampling version of [6] is: Sukhatme [3] used auxiliary variable in his ratio type estimators for two-phase sampling.One of his estimators was: Raj [7] proposed a method of using information on several variates to achieve higher precision in two-phase sampling.The two-phase sampling version of [7] is: where and "w" is a suitably chosen constant.
Mohanty [8] demonstrated that precision of study variable in two-phase sampling can be increased by combining the regression and ratio estimators using two auxiliary variables. . (1.9) Srivastava [9] developed a following class of ratio type estimators: Mukerjee et al. [10] developed three regression type estimators.One was for the situation when no auxiliary information was available.
Samiuddin and Hanif [11] developed two-phase sampling version of Sukhatme et al. [12] regression estimator when population means X and Z are not known.
and y xz  is the partial correlation coefficient of given Singh and Espejo [13] extended their own work of single phase sampling ratio-product estimator suggested in (2003) to two-phase sampling.
Hanif et al. [14] developed regression type estimators of population mean in two-phase sampling.One of those estimators was: .

Proposed Regression Type Estimator for Two-Phase Sampling
We propose following estimator using two auxiliary Copyright © 2013 SciRes.OJS variables for two-phase sampling when we don't have any information of auxiliary variables i.e. both 1 x and 1 z are unknown.
Putting the notations of (1.1) in (2.1), squaring and taking expectation, we can obtain mean square as: In order to get optimum value of K 1 and K 2 we differentiate (2.2) with respect to K 1 and equating to zero we get: Putting the value of (2.3) in (2.2) and differentiating with respect to K 1 , we get: where yx z   is the partial regression coefficient of y on x keeping constant.z Putting the value of (2.4) in (2.3) we get: Putting the values of (2.4) and (2.5) in (2.2) and on simplification we have: Expressing the proposed estimator in terms of (1.1) and taking the assumption that  is very small and ex- gree, we obtain bias of above estimator as follows Putting (2.4) and (2.5) in (2.7) and after simplific th (2.8)

Mathematical Comparison of Proposed
In osed esti- Estimator over Other Estimators this section, an improvement of our prop mator is shown over well-known estimators of two-phase sampling.In each case no information about population characteristics of auxiliary variables is available.It is proved through mathematical comparison that our proposed estimator outperforms the other estimators.We have compared our estimator with [3,[6][7][8][9][10][11]13,14] estimators.The mathematical efficiency of our proposed estimator is given as: a) Comparison with Robson [6] Estimator

Conclusion
ave proposed a regression type esti- values of the i-th unit of the character and Z respectively.Here i is our variable of interest, i