Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 693782 | 6 pages | https://doi.org/10.1155/2014/693782

Efficiency of Ratio, Product, and Regression Estimators under Maximum and Minimum Values, Using Two Auxiliary Variables

Academic Editor: Bernard J. Geurts
Received04 Dec 2013
Accepted09 Feb 2014
Published13 Apr 2014

Abstract

To obtain the best estimates of the unknown population parameters have been the key theme of the statisticians. In the present paper we have suggested some estimators which estimate the population parameters efficiently. In short we propose a ratio, product, and regression estimators using two auxiliary variables, when there are some maximum and minimum values of the study and auxiliary variables, respectively. The properties of the proposed strategies in terms of mean square errors (variances) are derived up to first order of approximation. Also the performance of the proposed estimators have shown theoretically and these theoretical conditions are verified numerically by taking four real data sets under which the proposed class of estimators performed better than the other previous works.

1. Introduction

In the literature of survey sampling, the use of ancillary information provided by auxiliary variables was discussed by various statisticians in order to improve the efficiency of their constructed estimators or to obtain improved estimators for estimating some most common population parameters, such as population mean, population total, population variance, and population coefficient of variation. In such a situation, ratio, product, and regression estimators provide better estimates of the population parameters. The work of Neyman [1] is considered as the early works where auxiliary information has been used. After that a lot of work has been done for estimating finite population mean and other population parameters using auxiliary information and for improving their efficiency. For a more related work one can go through Das and Tripathi [2, 3], Upadhyaya and Singh [4], Singh [5], and so forth. Sisodia and Dwivedi [6] have proposed ratio estimator using coefficient of variation of an auxiliary variable. Kadilar and Cingi [7] have suggested an estimator for population mean using two auxiliary variables. Khan and Shabbir [8] have introduced the idea of ratio type estimator or the estimation of population variance using quartiles of an auxiliary variable. Mouatasim and Al-Hossain [9] have studied reduced gradient method for minimax estimation of a bounded poisson mean in which concept of auxiliary variables can be easily placed and study. Further Al-Hossain [10] has studied inference on compound Rayleigh parameters with progressively type II censored samples wherein censored samples can be chosen as to auxiliary variables. Recently Khan and Shabbir [11] suggested different estimators of finite population mean using maximum and minimum values.

Let us consider a finite population of size of different units . Let , , and be the study and the auxiliary variables with corresponding values , , and , respectively, for the th unit defined on a finite population . Let , , and be the population means of the study as well as auxiliary variables, respectively, let , , and be the corresponding population variances of the study as well as auxiliary variables, respectively, let , , and be the coefficient of variation of the study as well as auxiliary variables, respectively, and let , , and be the population correlation coefficient among , , and between and , respectively.

In order to estimate the unknown population mean, we take a random sample of size units from the finite population by using simple random sample without replacement. Let , , and be the study and the auxiliary variables with corresponding values , , and , respectively, for the th unit in the sample. Let , , and be the sample means of the study as well as auxiliary variables, respectively, and let , , and be the corresponding sample variances of the study as well as auxiliary variables, respectively. Also let , , and be the sample coefficient of variation of the study variable as well as auxiliary variables and , respectively, and let , , and be the sample covariances between , , and and between and , respectively.

The usual unbiased estimator to estimate the population mean of the study variable is The variance of the estimator up to first order of approximation is given as follows: where .

In many real data sets there exist some large () or small values () and to estimate the unknown population parameters without considering this information is very sensitive in case the result will be either overestimated or underestimated. In order to handle this situation Sarndal [12] suggested the following unbiased estimator for the estimation of finite population mean using maximum and minimum values: where is a constant, which is to be found for minimum variance.

The minimum variance of the estimator up to first order of approximation is given as where the optimum value of is

The ratio estimator for estimating the unknown population mean of the study variable using two auxiliary variables is given by

The mean square error of the estimator up to first order of approximation is given by

The product estimator for estimating the unknown population mean of the study variable using two auxiliary variables is given by

The mean square error of the estimator up to first order of approximation is given by

When there are two auxiliary variables, then the regression estimator to estimate the finite population mean is given by where and are the sample regression coefficients between and and between and , respectively.

The variance of the estimator up to first order of approximation is given as

2. Proposed Estimators

On the lines of Sarndal [12], we propose a ratio, product, and regression estimators using two auxiliary variables when there are some maximum and minimum values of the study variables and the auxiliary variables, respectively.

Case 1. When the correlation between the study variable and the auxiliary variable is positive, the selection of the larger value of the auxiliary variable the larger the value of study variable is to be expected, and the smaller the value of auxiliary variable the smaller the value of study variable is to be expected, and using such type of information the ratio estimator using two auxiliary variables becomes where if the sample contains and . if the sample contains and . And for all other samples.

Case 2. Similarly when the correlation is negative the selection of the larger value of the auxiliary variable the smaller the value of study variable is to be expected, and the smaller the value of auxiliary variable the larger the value of study variable is to be expected, and using such type of information the product estimator using two auxiliary variables becomes where if the sample contains and ,  if the sample contains and , and for all other samples. Also , , and are unknown constants, whose value is to be determined for optimality conditions.

To obtain the properties of the proposed estimators in terms of bias and mean square error, we define the following relative error terms and their expectations.

, , and , such that . Consider also Rewriting (12), in terms of ’s, we have Expanding the right hand side of above equation and including terms up to second powers of ’s, that is, up to first order of approximation, we have

On squaring both sides of (17) and keeping ’s powers up to first order of approximation, we have

Taking expectation on both sides of (18), we get mean square error up to first order of approximation, given as

To find the minimum mean squared error of , we differentiate (19) with respect to , , and , respectively; that is,

On differentiating (19), with respect to , , and , respectively, we get one equation with three unknowns and so unique solution is not possible; so let

On substituting the optimum value of , , and from (21) in (19), we get the minimum mean square error of the proposed estimator, given as where .

Similarly the mean square error of the product estimator, up to first order of approximation, is given by where .

Now the minimum variance of the regression estimator in the case of positive correlation, up to first order of approximation, is given by where .

Similarly for the case of negative correlation, the minimum variance of the regression estimator, up to first order of approximation, is given by

But when there is positive and negative correlation, the regression estimator gives us better result, and so for both cases (positive and negative correlation) we write the variance as

3. Comparison of Estimators

In this section, we have compared the proposed estimators with the ratio, product, and regression estimators and some of their efficiency comparison condition has been carried out under which the proposed estimators perform better.(i)By (7) and (22), if (ii)By (9) and (23), if (iii)By (11) and (26), if

From (i), (ii), and (iii) we have observed that the proposed estimators performed better than the other existing estimators because the conditions are in the form of a square and greater than zero which is always true.

4. Numerical Illustration

In this section we demonstrate the performance of the suggested estimators over various other estimators, through four real data sets. The description and the necessary data statistics of the populations are given as follows.

Population 1 (source: Agricultural Statistics (1999) [13], Washington, DC.): estimated number of fish caught during 1995;: estimated number of fish caught during 1994;: estimated number of fish caught during 1993;, , , , , , , , , , , , , , , , , , , , , and .

Population 2 (source: Agricultural Statistics (1998) [14], Washington, US.): season average price per pound during 1996;: season average price per pound during 1995;: season average price per pound during 1994;, , , , , , , , , , , , , , , , , , , , , and .

Population 3 (source: Agricultural Statistics (1999) [13], Washington, DC.): estimated number of fish caught during 1995;: estimated number of fish caught during 1994;: estimated number of fish caught during 1992;, , , , , , , , , , , , , , , , , , , , , , and .

Population 4 (source: Agricultural Statistics (1999) [13], Washington, DC.): estimated number of fish caught during 1995;: estimated number of fish caught during 1993;: estimated number of fish caught during 1992;, , , , , , , , , , , , , , , , , , , , , , and .

The mean squared error of the proposed and the existing estimators is shown in Table 1.


EstimatorPopulation 1Population 2Population 3Population 4
MSE(·)MSE(·)MSE(·)MSE(·)

1671738.9470.00041513620.2531440668.498
12164963.920.002911772394.8211494548.32
1254819.8850.00031217975.5631231973.764
Proposed
1293565.5570.0003957662.57911989.0387
9654413.7600.00218830692.2388616039.070
971781.54010.00025753868.3966769477.0316

5. Conclusion and Future Work

We have developed some ratio, product, and regression estimators under maximum and minimum values using two auxiliary variables. The proposed estimators under certain efficiency conditions are shown to be more efficient than the ratio, product, and regression estimators using two auxiliary variables. The results are shown numerically in Table 1 where we observed that the performance of the proposed estimators is better than the usual ratio, product, and the regression estimators using two auxiliary variables. We can easily implement the concept of auxiliary variables minimax or maximin estimation of a bounded Poisson (respectively some other distribution) mean and censors samples. Thus the proposed estimators may be preferred over the existing estimators for the use of practical applications.

Conflict of Interests

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

References

  1. J. Neyman, “Contributions to the theory of sampling human populations,” Journal of the American Statistical Association, vol. 33, no. 201, pp. 101–116, 1938. View at: Publisher Site | Google Scholar
  2. A. K. Das and T. P. Tripathi, “Sampling strategies for population mean when the coefficient of variation of an auxiliary character is known,” Sankhya, vol. 42, pp. 76–86, 1980. View at: Google Scholar
  3. A. K. Das and T. P. Tripathi, “A class of sampling strategies for population mean using information on mean and variance of an auxiliary character,” in Proceedings of the Indian Statistical Institute Golden Jubilee International Conference on Statistics: Applications and New Directions, pp. 174–181, Calcutta, India, December 1981. View at: Google Scholar
  4. L. N. Upadhyaya and H. P. Singh, “Use of transformed auxiliary variable in estimating the finite population mean,” Biometrical Journal, vol. 41, no. 5, pp. 627–636, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. S. Singh, “Golden and Silver Jubilee year-2003 of the linear regression estimators,” in Proceedings of the American Statistical Association, Survey Method Section, pp. 4382–4389, American Statistical Association, Toronto, Canada, 2004. View at: Google Scholar
  6. B. V. S. Sisodia and V. K. Dwivedi, “A modified ratio estimator using coefficient of variation of auxiliary variable,” Journal of the Indian Society of Agricultural Statistics, vol. 33, no. 1, pp. 13–18, 1981. View at: Google Scholar
  7. C. Kadilar and H. Cingi, “A new estimator using two auxiliary variables,” Applied Mathematics and Computation, vol. 162, no. 2, pp. 901–908, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. M. Khan and J. Shabbir, “A ratio type estimator for the estimation of population variance using quartiles of an auxiliary variable,” Journal of Statistics Applications and Probability, vol. 2, no. 3, pp. 157–162, 2013. View at: Google Scholar
  9. A. E. Mouatasim and A. Al-Hossain, “Reduced gradient method for minimax estimation of a bounded poisson mean,” Journal of Statistics: Advances in Theory and Applications, vol. 2, no. 2, pp. 185–199, 2009. View at: Google Scholar
  10. Y. Al-Hossain, “Inferences on compound rayleigh parameters with progressively type-II censored samples,” World Academy of Science, Engineering and Technology, vol. 76, pp. 885–892, 2013. View at: Google Scholar
  11. M. Khan and J. Shabbir, “Some improved ratio, product and regression estimators of finite population mean when using minimum and maximum values,” The Scientific World Journal, vol. 2013, Article ID 431868, 7 pages, 2013. View at: Publisher Site | Google Scholar
  12. C. E. Sarndal, “Sample survey theory vs general statistical theory: estimation of the population mean,” International Statistical Institute, vol. 40, pp. 1–12, 1972. View at: Google Scholar
  13. http://www.nass.usda.gov/Publications/Ag_Statistics/1999/index.asp.
  14. http://www.nass.usda.gov/Publications/Ag_Statistics/.

Copyright © 2014 Abdullah Y. Al-Hossain and Mursala Khan. 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.


More related articles

1877 Views | 628 Downloads | 4 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.