Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article
Special Issue

Robust Estimation Methods in the Presence of Extreme Observations

View this Special Issue

Research Article | Open Access

Volume 2021 |Article ID 9931217 | https://doi.org/10.1155/2021/9931217

Showkat Ahmad Lone, Mir Subzar, Ankita Sharma, "Enhanced Estimators of Population Variance with the Use of Supplementary Information in Survey Sampling", Mathematical Problems in Engineering, vol. 2021, Article ID 9931217, 8 pages, 2021. https://doi.org/10.1155/2021/9931217

Enhanced Estimators of Population Variance with the Use of Supplementary Information in Survey Sampling

Academic Editor: Ishfaq Ahmad
Received25 Mar 2021
Revised11 Apr 2021
Accepted16 Apr 2021
Published26 Apr 2021

Abstract

In the present study, we propose the proficient class of estimators of the finite population mean, while incorporating the nonconventional location and nonconventional measures of dispersion with coefficient of variation of the auxiliary variable. Properties associated with the suggested class of improved estimators are derived, and an efficiency comparison with the usual unbiased ratio estimator and other existing estimators under consideration in the present study is established. An empirical study has also been provided to validate the theoretical results. Finally, it is established that the proposed class of estimators of the finite population variance proves to be more efficient than the existing estimators mentioned in this study.

1. Introduction

It is very quite often that utilization of supplementary information in survey sampling which has some sort of strong positive or negative correlation with the response variable is always found to be advantageous. So for the utilization of such information, various methods in survey sampling are presently used to increase precision, with the incorporation of these supplementary information, in estimating the population parameters. Various authors have put their sincere efforts to utilize supplementary information with different sampling designs in different situations in such a way that their estimation procedure becomes more proficient; for details see [112]. It is very often that some of the measures are so much affected by extreme observations and can give misleading results. In case of extreme values, using classical methods of estimation provides misleading results, but authors have also put their valuable efforts to come up with solutions to this situation, so that precise results should be obtained even with the presence of outliers in the data. Authors such as Subzar et al. [13] have proposed different robust ratio type estimators in simple random sampling without replacement (SRSWOR) while utilizing the Huber-M estimation technique. Subzar et al. [14] have also proposed different robust ratio type estimators in SRSWOR by utilizing the different robust regression techniques and compared with Ordinary Least Squares (OLS) and Huber-M estimation techniques. Subzar et al. [15] also proposed different robust ratio type estimators by comparing the generalized robust regression techniques with OLS and Huber-M estimation method. Recently, Almanjahie et al. [16] have proposed the generalized class of mean estimators with known measures for outlier’s treatment. Also, Shahzad et al. [12] have given a new class of L-Moments-based calibration variance estimators. So in the present study, we made the utilization of nonconventional location parameters, nonconventional measures of dispersion, and their functions with the coefficient of variation of the auxiliary variable in order to suggest the class of estimators for estimating the population variance. The properties of the proposed class of estimators are studied under large sample approximation. It has been shown theoretically as well as empirically that the proposed class of estimators is more efficient than existing estimators mentioned in this study.

2. Notations

Consider a finite population of units and let be the study and auxiliary variables defined on taking values , respectively, on . It is desired to estimate the population variance of the study variable using the information on an auxiliary variable . Let a simple random sample of size be drawn without replacement from the finite population . In this paper, we shall ignore the finite population correction (fpc) term. We denote the following.: the population mean of the study variable : the population mean of the auxiliary variable : the population mean square/variance of the study variable : the population mean square/variance of the auxiliary variable : tri-mean of the auxiliary variable : Hodges–Lehmann estimator of the auxiliary variable : mid-range of the auxiliary variable : Gini’s mean difference of the auxiliary variable : Downton’s method of the auxiliary variable : probability weighted moments of the auxiliary variable :where are nonnegative integers.

3. The Proposed Class of Estimators

In the present study, we have made the incorporation of nonconventional location parameters, nonconventional measures of dispersion, and their function with the coefficient of variation, whose generalized class for estimating population variance is given aswhere , and are either real constants or functions of known parameters of an auxiliary variable and is a constant such that which is a more flexible condition as given by Singh et al. [5] and Solanki et al. [17] over the constant in their estimators, and are constants such that the mean squared error (MSE) is minimum. Here, we note that . For suitable values of , the proposed class of estimators “” reduces to some known existing estimators based on quartiles and their functions given in Table 1.


S. no.EstimatorsValues of constants

1, usual unbiased estimator100
2, Das and Tripathi [1] and Isaki [4] estimator1010
3, Singh et al. [5] estimator101
4, Singh et al. [5] estimator101
5, Singh et al. [5] estimator101
6, Singh et al. [5] estimator101
7, Singh et al. [5] estimator101
8, Singh et al. [5] estimator101
9, Solanki et al. [17] estimator10
10, Solanki et al. [17] estimator10
11, Solanki et al. [17] estimator10
12, Solanki et al. [17] estimator10
13, Solanki et al. [17] estimator10
14, Solanki et al. [17] estimator10

From (2), we would like to remark the suitable values of and one can generate different estimators. For example, we have developed some new estimators from the proposed class of estimators “” which are listed in Table 2.


S. no.EstimatorsValues of constant

15
16
17
18
19
20

While obtaining the expression of bias and mean square error (MSE) for the suggested class of estimator “”, we write such that , and up to the first order of approximation, while fpc term is ignored, we have

Now, in terms of , the estimator “”, given in (2), is expressed aswhere

We assume that , so that is expendable. Expanding the right-hand side of (5) and multiplying out, we have

Taking expectation of both sides of (7), we get the bias of the estimators to the first degree of approximation as

Squaring both sides of (7), neglecting terms of having the power greater than two, we have

Taking expectation of both sides of (9), we get the MSE of the estimator to the first degree of approximation (ignoring fpc term) aswhere

Differentiating MSE () with respect to and and equating them to zero, we have

Simplifying (12), we get the optimum values of and as

Inserting (13) in (10), we get the resulting minimum MSE of “” given by

Thus we establish the following theorem.

Theorem 1. To the first degree of approximationwith equality holding ifwhere are given in (13).

Special Case 1:. for in (2), we get an alternative class of estimators for the population variance asPutting in (8) and (10), we get the bias and MSE of to the first degree of approximation, given byThe is minimum whenThus, the resulting minimum of is given byThis equals to the minimum MSE of the difference estimator envisaged by [1]

Special Case 2:. for in (2), we get another class of estimators for population variance asPutting in (8) and (10), we get the bias and MSE of , respectively, asThe is minimum forThus, the resulting minimum MSE of is given byThus, we state the following corollary.

Corollary 1. To the first degree of approximationwith equality holding if

Special Case 3:. if we set in (2), the class of estimators reduces to the estimatorwhich includes Solanki et al. [17] estimators to .
Putting in (8) and (10), we get the bias and MSE of to the first degree of approximation (ignoring fpc term), respectively, as

Special Case 4:. if we set in (2), the class of estimators reduces to the class of estimators of asInserting in (8) and (10), we get the bias and MSE of to the first degree of approximation (ignoring fpc term), respectively, aswhereThe at (34) is minimized forSubstitution of (36) in (34) yields the minimum MSE of and is given byThus, we established the following corollary.

Corollary 2. To the first degree of approximationwith equality holdings if

Special Case 5:. for in (2), the class of estimators “” reduces to the class of estimators of aswhich includes Singh et al. [5] estimators to .
Inserting in (8) and (10), we get the bias and MSE of to the first degree of approximation (ignoring fpc term), respectively, given by

4. Efficiency Comparison

To the first degree of approximation, the MSE of and are, respectively, given by

From (20), (43), and (44), we have

It follows from (45) and (46) that the proposed class of estimators or difference estimator is better than the usual unbiased estimator and the ratio type estimator due to [1, 4].

From (37) and (42), we have

This follows that the proposed class of estimators is more efficient than the proposed class of estimators according to Singh et al. [5]. Thus, the estimator is an improvement over the Singh et al. [5] estimators .

From (25) and (30), we have

Expression (48) clearly indicates that the proposed class of estimators is better than class of estimator recently proposed by Solanki et al. [17]. Thus, the estimators are an improvement over the estimator due to Solanki et al. [17].

Furthermore, from (14) and (20), we have

It follows from (49) that the proposed class of estimators “” is better than the class of estimators and the difference estimator .

Next, from (14) and (25), we have

Thus, from (48) and (50), we have the following inequality:

This clearly shows that the proposed family of estimators is better than families of estimators and .

Finally, from the above theoretical comparisons we conclude that the proposed family of estimators is better than the usual unbiased estimator , [4] ratio type estimator , and the estimators ( to ) and ( to ) recently proposed by Singh et al. [5] and Solanki et al. [17], respectively, and for other families of estimators (, ).

5. Empirical Study

The performances of the proposed estimators , which are members of the suggested class of estimators, , are evaluated against the usual unbiased estimators and the estimators proposed by [1, 4, 5, 10] for the population data set (source: [18]) summarized in Table 3.


800.35425.15009.318
208.456310.30010.405
51.82640.750716.97517.955
11.26462.866411.8259.0408
0.94132.26675.91258.0138
18.35692.220911.06257.9136

We have computed the percent relative efficiencies (PREs) of the suggested estimators with respect to the usual unbiased estimator in the range of by using the following formula:and the findings are summarized in Table 4.


PRE ()

−1.00257.0391259.3591291.8213258.2640256.4163253.0021
−0.75255.2481257.2658275.3213256.0011253.9286252.8256
−0.50254.2359256.9329273.3492255.3651253.6165252.4152
−0.25253.9139256.1231273.1029254.9606253.0259252.0081
0.00253.1029255.8569273.0051254.0436252.8169251.5621
0.25252.9320255.0091272.9321253.8186252.0411251.0561
0.50253.1001254.8563272.3649253.5646251.9103250.3961
0.75254.2569254.6561272.1536253.3963251.3851250.0011
1.00254.2031254.5341271.9806252.8646251.0091249.5651

For the purpose of comparison of the proposed estimators to with that of the usual unbiased estimator , [4] estimator , [5] estimators ( to ), and [17] estimators ( to ), in Table 5, we give the PREs of the estimators , and the PREs of the estimators to with respect to usual unbiased estimator along with the value of for which the PREs of the estimators to are maximum as given in [17].


EstimatorValue of PRE ()

100.00
183.23
1.00209.67
0.50230.60
0.30230.28
0.70230.57
1.00200.87
0.60230.80
1.00270.38
1.00270.61
1.00263.50
0.80270.58
1.00270.61
0.30270.58

It is observed from Tables 4 and 5 that all the estimators which are the members of the proposed class of estimators performed better than the usual unbiased estimator , ratio type estimators due to [4, 5] estimators ( to ) and [17] estimators ( to ), for .

6. Conclusion

In the present paper, we have suggested an improved class of estimators for population variance using the auxiliary information of nonconventional location parameters, nonconventional measures of dispersion, and their function with the coefficient of variation. The bias and mean square error (MSE) expressions of the proposed class of estimators are obtained and compared with the usual unbiased estimator, estimators in [4], [5], and [17]. We have also analyzed the performance of the proposed estimators by utilizing the data set of the known population and found that in convenient cases, the proposed estimators perform better than the other existing estimators. On comparison with the usual unbiased estimator, which is shown in Table 4, it is evident that upon increasing the values of either negative or positive, the percent relative efficiency also increases. From Table 5, we observe that our estimators are more proficient than the existing ones. Hence, we strongly recommend that our proposed estimators perform better than the existing estimators for use in practical applications.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. A. K. Das and T. P. Tripathi, “Use of auxiliary information in estimating the finite population variance,” Sankhya C, vol. 40, pp. 139–148, 1978. View at: Google Scholar
  2. C. Kadilar and H. Cingi, “Ratio estimators for the population variance in simple and stratified random sampling,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 1047–1059, 2006. View at: Publisher Site | Google Scholar
  3. C. N. Bouza and M. Subzar, “Sub sampling rules for item non response of an estimator based on the combination of regression and ratio,” Journal of King Saud Science, vol. 31, pp. 171–176, 2019. View at: Google Scholar
  4. C. T. Isaki, “Variance estimation using auxiliary information,” Journal of the American Statistical Association, vol. 78, no. 381, pp. 117–123, 1983. View at: Publisher Site | Google Scholar
  5. H. P. Singh, S. K. Pal, and R. S. Solanki, “Improved estimation of finite population variance using quartiles,” Istatistik Journal of The Turkish Statistical Association, vol. 6, no. 3, pp. 166–121, 2013. View at: Google Scholar
  6. H. P. Singh, S. K. Pal, and R. S. Solanki, “A new procedure for estimation of finite population variance using auxiliary information,” Journal of Reliability and Statistical Studies, vol. 7, no. 2, pp. 149–160, 2014. View at: Google Scholar
  7. M. Subzar, S. Maqbool, T. A. Raja, S. A. Mir, I. Jeelani, and M. A. Bhat, “Improved family of ratio type estimators for estimating population mean using conventional and non-conventional location parameters,” Revista Investigacion Operacional, vol. 38, no. 5, pp. 499–513, 2017. View at: Google Scholar
  8. M. Subzar, T. A. Raja, S. Maqbool, and N. Nazir, “New alternative to ratio estimator of population mean,” International Journal of Agricultural and Statistical Sciences, vol. 12, no. 1, pp. 221–225, 2016. View at: Google Scholar
  9. M. Subzar, T. A. Raja, S. Maqbool, S. A. Wani, A. B. Shikari, and N. Nazir, “Modified class of ratio estimators for population mean,” International Journal of Agricultural and Statistical Sciences, vol. 14, no. 1, pp. 313–317, 2018. View at: Google Scholar
  10. U. Shahzad, I. Ahmad, E. Oral, M. Hanif, and I. M. Almanjahie, “Estimation of the population mean by successive use of an auxiliary variable in median ranked set sampling,” Mathematical Population Studies, vol. 88, 2020. View at: Publisher Site | Google Scholar
  11. U. Shahzad, I. Ahmad, I. Mufrah Almanjahie, N. H. Al – Noor, and M. Hanif, “A new class of L-Moments based calibration variance Estimators,” Computers, Materials & Continua, vol. 66, no. 3, pp. 3013–3028, 2021. View at: Publisher Site | Google Scholar
  12. U. Shahzad, M. Hanif, I. Sajjad, and M. Anas, “Quantile regression-ratio-type estimators for mean estimation under complete and partial auxiliary information,” Scientia Iranica, vol. 21, 2020. View at: Publisher Site | Google Scholar
  13. M. Subzar, C. N. Bouza, S. Maqbool, T. A. Raja, and B. A. Para, “Robust ratio type estimators in simple random sampling using huber M estimation,” Revista Investigacion Operacional, vol. 40, no. 2, pp. 201–209, 2019. View at: Google Scholar
  14. M. Subzar, C. N. Bouza, and A. I. Al-Omari, “Utilization of different robust regression techniques for estimation of finite population mean in case of presence of outliers through ratio method of estimation,” Revista Investigacion Operacional, vol. 40, no. 5, pp. 600–609, 2019. View at: Google Scholar
  15. M. Subzar, A. Ibrahim Al-Omari, and A. R. A. Alanzi, “The robust regression methods for estimating of finite population mean based on SRSWOR in case of outliers,” Computers, Materials & Continua, vol. 65, no. 1, pp. 125–138, 2020. View at: Publisher Site | Google Scholar
  16. I. M. Almanjahie, A. I. Al-Omari, E. J. Ekpenyong, and M. Subzar, “Generalized class of mean estimators with known measures for outliers treatment,” Computer Systems Science and Engineering, vol. 38, no. 1, pp. 1–15, 2021. View at: Google Scholar
  17. R. S. Solanki, H. P. Singh, and S. K. Pal, “Improved ratio-type estimators of finite population variance using quartiles,” Hacettepe Journal of Mathematics and Statistics, vol. 44, no. 3, pp. 747–754, 2015. View at: Google Scholar
  18. M. N. Murthy, Sampling: Theory and Methods, Statistical Publishing Society, Calcutta, India, 1967.

Copyright © 2021 Showkat Ahmad Lone et al. 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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views400
Downloads347
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.