Comparison of Various Methods for Estimating Finite Population Total in Survey Sampling When Study Variable and Auxiliary Variable are Inversely Related

In the present paper, a model based calibration estimator of population total has been developed when study variable y and auxiliary variable x are inversely related. The relative performance of the proposed model based calibration estimator in comparison to model based estimator, the usual regression estimator and calibration based regression estimator have been examined by conducting a limited simulation study. In view of the results of the simulation study, it has been found that model based calibration estimator has outperformed the other estimators. However, calibration based regression estimator was found to be close to the model based calibration estimator.


Introduction
The auxiliary information is used to improve the precision of the estimates of the population parameters such as population mean, population total, population variance etc. in finite population survey sampling.Various estimation approaches for estimating finite population total using information on auxiliary variables have been resorted.Most common methods of estimation are ratio and regression estimators, model based estimator by Royall and Herson [2], calibration estimator by Deville and Särndal [5] and model based calibration estimator by Wu and Sitter [6].Recently, some research workers like Sud et al. [11], Mourya et al. [12] and Sandeep Kumar et al. [13,14] have contributed significantly in calibration approach based estimation in finite population survey sampling.Following Royall and Herson [2], a model based unbiased estimator of , (1) is given by where


, i y are realized values of independent random variables s ' y i and y is sample mean for given sample s of size n .The estimator in ( 2) is in fact the usual regression estimator of Y when i x 1 is considered as auxiliary variable instead of Sud et al. [11] developed calibration based regression type estimator of finite population total of the study variate y , when study variable is inversely related to an auxiliary variable x .Their estimator and it's variances etc. are briefly presented here.Consider that a sample s of size n is drawn from the population


according to sampling design (.) P .Let i  and ij  be the inclusion probability of i th unit and joint inclusion probability of i th and j th unit, respectively, in the sample s .Suppose that the information on an auxiliary variable x related to the study variate y is known for all x X  They developed calibration based regression type estimator as Under simple random sampling without replacement (SRSWOR) design, say SI, the estimator in (4) reduces to where Note that the usual regression estimator of Y as per Cochran [3] under SRSWOR is given by where Up to the first order of approximation following Sukhatme et al. [4], the variance of  is the correlation coefficient between y and x 1 , and x y,  is the correlation coefficient between y and x , and Obviously, it can be remarked from the expression ( 7) and ( 8) that if An estimate of variance of up to the first order of approximation according to Sud et al. [11], is given by where up to the first order of approximation is given by where x y r , is the estimate of  1) following Wu and Sitter [6] in section-2.A limited simulation study has been conducted to make the comparison of relative performance of various estimators described in preceding section and proposed model based calibration estimator in section-3.

Proposed Model Based Calibration Estimator
Following Wu and Sitter [6], we develop model based calibration estimator of Y under the following model We propose a model based calibration estimator of Y under the model (11) as where i w is calibrated weight, i w is obtained by minimizing a distance measure function where i Y ˆ is fitted value of i Y by least square technique.The following function is minimized with respect to i w , where 1  and 2  are Langrangian multipliers.This where ˆˆ, An approximate variance of MC Y ˆ is obtained following Wu and Sitter [6] as follows where , where For simple random sampling without replacement (SRSWOR), , we get the model based calibration estimator under SRSWOR, denoted as

Simulation Study
A limited simulation study has been conducted to examine the performance of the various estimators of population total i.e. usual regression estimator, calibration based regression type estimator due to Sud et al. [11], model based estimator and proposed model based calibration estimator.The performance of the estimators has been examined by their average estimates of variances obtained.
To examine the performance of the estimators through simulation, we generate hypothetical population using the following super population model We assume the value of

Conclusion
The proposed model based calibration estimator has been compared with model based estimator, calibration based regression estimator and the usual regression estimator by conducting a limited simulation study.The overall results indicate that model based calibration estimator has outperformed other estimators, and it can be recommended for use in practice.

1 
in the Horvitz-Thompson estimator of Y i.e.


obtained from the data contained in the sample s .An attempt has been made in the present paper to first develop a model based calibration estimator under model ( 17) into(13), the model based calibration estimator of Y is obtained as

Table 1 .
Average estimate of variance of the estimators.
[6]can be observed from the results of the Table1that the model based calibration estimator has outperformed the other estimators.The calibration based regression type estimator has performed better than the model based estimator and usual regression estimator.It may also be noted that the performance of the calibration based regression estimator is close to model based calibration estimator.This result justifies the argument of Wu and Sitter[6]that the calibration based regression estimator and model based calibration estimator perform almost equally if model is linear.However, in view of the results in the Table1, proposed model based calibration estimator in practice for estimating population total when the study variate y and the auxiliary variable x are inversely related.