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Journal of Probability and Statistics
Volume 2014 (2014), Article ID 939701, 6 pages
http://dx.doi.org/10.1155/2014/939701
Research Article

An Improved Class of Chain Ratio-Product Type Estimators in Two-Phase Sampling Using Two Auxiliary Variables

Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826004, India

Received 13 September 2013; Accepted 23 January 2014; Published 6 March 2014

Academic Editor: Zhidong Bai

Copyright © 2014 Gajendra K. Vishwakarma and Manish Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a technique for estimating finite population mean of the study variable in the presence of two auxiliary variables using two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin. The properties of the proposed class of estimators are studied under large sample approximation. In addition, bias and efficiency comparisons are carried out to study the performances of the proposed class of estimators over the existing estimators. It has also been shown that the proposed technique has greater applicability in survey research. An empirical study is carried out to demonstrate the performance of the proposed estimators.

1. Introduction

The use of auxiliary information for estimating population mean of the study variable has greater applicability in survey research. It is utilized at the estimation stage and design stage to obtain an improved estimator compared to those not utilizing auxiliary information. The use of ratio and product strategies in survey sampling solely depends upon the knowledge of population mean of the auxiliary variable .

The ratio estimator was developed by Cochran [1] to estimate the population mean of the study variable by using information on auxiliary variable , positively correlated with . The ratio estimator is most effective when the relationship between and is linear through the origin and the variance of is proportional to. Robson [2] defined a product estimator that was revisited by Murthy [3]. The product estimator is used when the auxiliary variable is negatively correlated with the study variable .

When the population mean of the auxiliary variable is not known before the start of a survey, then a first-phase sample of size is selected from the population of size on which only the auxiliary variable is measured in order to furnish a good estimate of . And then a second-phase sample of size is selected from the first-phase sample of size on which both the study variable and the auxiliary variable are measured. This procedure of selecting the samples from the given population is known as two-phase sampling (or double sampling). The concept of double sampling was first introduced by Neyman [4]. Some contribution to two-phase sampling has been made by Sukhatme [5], Hidiroglou and Sarndal [6], Fuller [7], Hidiroglou [8], Singh and Vishwakarma [9], and Sahoo et al. [10].

We can use either one or two (or more than two) auxiliary variables while estimating population mean of the study variable; keeping this fact, Chand [11] introduced chain ratio estimators. This led various authors including Kiregyera [12], Singh and Upadhyaya [13], Prasad et al. [14], Singh et al. [15], Singh and Choudhury [16], and Vishwakarma and Gangele [17] to modify the chain type estimators and discuss their properties.

When the population mean of another auxiliary variable which has a positive correlation with X (i.e., ) is known and if , then it is advisable to estimate by , which would provide a better estimate of as compared to .

The usual chain type ratio and product estimators of under double sampling scheme using two auxiliary variables and are given, respectively, by

Singh and Choudhury [16] suggested the following exponential chain type ratio and product estimators of under double sampling scheme using two auxiliary variables and : where and are the sample means of and , respectively, based on the first-phase sample of size drawn from the population of size with the help of Simple Random Sampling Without Replacement (SRSWOR) scheme. Also, and are the sample means of and , respectively, based on the second-phase sample of size drawn from the first-phase sample of size with the help of SRSWOR scheme.

2. Proposed Estimator

It has been theoretically established that, in general, the linear regression estimator is more efficient than the ratio (product) estimator except when the regression line of on passes through the neighborhood of the origin, in which the efficiencies of these estimators are almost equal. However, owing to stronger intuitive appeal, survey statisticians favour the use of ratio and product estimators. Further, we note that, in many practical situations, the regression line does not pass through the neighborhood of the origin. In these situations, the ratio estimator does not perform well as the linear regression estimator. Considering this fact, Singh and Ruiz Espejo [18] made an attempt to improve the performance of these estimators and suggested the following ratio-product type estimator for population mean under double sampling scheme using single auxiliary variable : where is a real constant.

We propose the following exponential chain ratio-product type estimator for population mean under double sampling scheme using two auxiliary variables and : where is a real constant to be determined such that the Mean Square Error (MSE) of the proposed estimator is minimum. For , , whereas, for , .

Remark. It is noted that the proposed estimator in (4) is a special case of the class of estimators proposed by Srivastava [19], where is a parametric function such that and satisfies certain regularity conditions defined in Srivastava [19].

3. Bias and MSE of the Proposed Estimator

To obtain the Bias and Mean Square Error (MSE) of the proposed estimator , we consider such that where , , , .

Let , , and be the coefficients of variation of , , and , respectively. Also, let , , and be the correlation coefficients between and , and , and and , respectively. Then, we have where

Now, expressing the estimator in terms of , , , and and neglecting the terms of , , , and involving degree greater than two, we get

To the first degree of approximation, the Bias and Mean Square Error (MSE) of the proposed estimator are given by

To the first degree of approximation, the expressions for Bias and Mean Square Error (MSE) of the estimators , , , , and are, respectively, given by

3.1. Optimum Value of

As we know, is determined so as to minimize the Mean Square Error (MSE) of the estimators and . So, the optimum values of , for which and are minimum, are obtained by using the following conditions:

The optimum value of , which minimizes the Mean Square Error (MSE) of the estimator , is given by

The optimum value of , which minimizes the Mean Square Error (MSE) of the estimator , is given by

Substituting the value of from (15) in (13), we get the minimum MSE of as

Substituting the value of from (16) in (11), we get the minimum MSE of as

4. Efficiency Comparisons

It is well known that the Bias and variance of the usual unbiased estimator for population mean in SRSWOR are

From (11), (13), and (20), we have(i), if (ii), if (iii), if (iv), if (v), if (vi), if

The range of provides enough scope for choosing many estimators that are more efficient than the above considered estimators.

5. Empirical Study

To examine the merits of the proposed estimator of , we have considered the following natural population datasets.

Population I (source: Cochran [20]) is shown as follows:: number of “placebo” children,: number of paralytic polio cases in the “placebo” group,: number of paralytic polio cases in the “not inoculated” group, , , ,, and ,, , , , ,and .

Population II (source: Murthy [21]) is shown as follows:: area under wheat in 1964,: area under wheat in 1963,: cultivated area in 1961,, , , , , and ,, , , , , and .

Here, we have computed(i)the Absolute Relative Bias (ARB) of different suggested estimators of using the formula (ii)the Percentage Relative Efficiencies (PREs) of different suggested estimators of with respect to using the formula

6. Conclusion

It is observed from Table 1 that,(i)for population I, (ii)for population II,

tab1
Table 1: Absolute Relative Bias (ARB) of different estimators of  .

From Table 2, we see that the Percentage Relative Efficiency (PRE) of the proposed estimator , for populations I and II, is more as compared to all other existing estimators, that is, usual unbiased estimator , chain type ratio estimator , chain type product estimator , exponential chain type ratio estimator , exponential chain type product estimator , and ratio-product type estimator .

tab2
Table 2: Percentage Relative Efficiencies (PREs) of different estimators of   with respect to .

Finally, from Tables 1 and 2, we conclude that the proposed estimator (based on two auxiliary variables and ) is a more appropriate estimator in comparison to other existing estimators as it has appreciable efficiency as well as lower relative bias.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the editor Professor Zhidong Bai and the learned referee for their comments leading to the improvement of the paper.

References

  1. W. G. Cochran, “The estimation of the yields of the cereal experiments by sampling for the ratio of grain to total produce,” The Journal of Agricultural Science, vol. 30, pp. 262–275, 1940. View at Publisher · View at Google Scholar
  2. D. S. Robson, “Applications of multivariate polykays to the theory of unbiased ratio-type estimation,” Journal of the American Statistical Association, vol. 52, pp. 511–522, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. N. Murthy, “Product method of estimation,” The Indian Journal of Statistics A, vol. 26, pp. 69–74, 1964. View at Zentralblatt MATH · View at MathSciNet
  4. J. Neyman, “Contribution to the theory of sampling human populations,” Journal of American Statistical Association, vol. 33, pp. 101–116, 1938. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. B. V. Sukhatme, “Some ratio-type estimators in two-phase sampling,” Journal of the American Statistical Association, vol. 57, pp. 628–632, 1962. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. A. Hidiroglou and C. E. Sarndal, “Use of auxiliary information for two phase sampling,” Survey Methodology, vol. 24, pp. 11–20, 1998.
  7. W. A. Fuller, “Two-phase sampling,” in Proceedings of the Annual Meeting of the Survey Methods Section of the Statistical Society of Canada, pp. 23–30, 2000.
  8. M. A. Hidiroglou, “Double sampling,” Survey Methodology, vol. 27, pp. 143–154, 2000.
  9. H. P. Singh and G. K. Vishwakarma, “Modified exponential ratio and product estimators for finite population mean in double sampling,” Austrian Journal of Statistics, vol. 36, no. 3, pp. 217–225, 2007.
  10. L. N. Sahoo, G. Mishra, and S. R. Nayak, “On two different classes of estimators in two-phase sampling using multi-auxiliary variables,” Model Assisted Statistics and Applications, vol. 5, no. 1, pp. 61–68, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. L. Chand, Some ratio type estimators based on two or more auxiliary variables [Ph.D. dissertation], Iowa State University, Ames, Iowa, USA, 1975.
  12. B. Kiregyera, “A chain ratio-type estimator in finite population double sampling using two auxiliary variables,” Metrika, vol. 27, no. 4, pp. 217–223, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. N. Singh and L. N. Upadhyaya, “A class of modified chain-type estimators using two auxiliary variables in two phase sampling,” Metron, vol. 53, no. 3-4, pp. 117–125, 1995. View at Zentralblatt MATH · View at MathSciNet
  14. B. Prasad, R. S. Singh, and H. P. Singh, “Some chain ratio-type estimators for ratio of two population means using two auxiliary characters in two phase sampling,” Metron, vol. 54, no. 1-2, pp. 95–113, 1996. View at Zentralblatt MATH · View at MathSciNet
  15. S. Singh, H. P. Singh, and L. N. Upadhyaya, “Chain ratio and regression type estimators for median estimation in survey sampling,” Statistical Papers, vol. 48, no. 1, pp. 23–46, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. B. K. Singh and S. Choudhury, “Exponential chain ratio and product type estimators for finite population mean under double sampling scheme,” Global Journal of Science Frontier Research, vol. 12, no. 6, 2012.
  17. G. K. Vishwakarma and R. K. Gangele, “A class of chain ratio-type exponential estimators in double sampling using two auxiliary variates,” Applied Mathematics and Computation, vol. 227, pp. 171–175, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  18. H. P. Singh and M. Ruiz Espejo, “Double sampling ratio-product estimator of a finite population mean in sample surveys,” Journal of Applied Statistics, vol. 34, no. 1-2, pp. 71–85, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. K. Srivastava, “A generalized estimator for the mean of a finite population using multi-auxiliary information,” Journal of American Statistical Association, vol. 66, pp. 404–407, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. W. G. Cochran, Sampling Techniques, John Wiley & Sons, New York, NY, USA, 1977. View at MathSciNet
  21. M. N. Murthy, Sampling Theory and Methods, Statistical Publishing Society, Calcutta, India, 1967. View at MathSciNet