Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 9931217, 8 pages
https://doi.org/10.1155/2021/9931217

Research Article

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

Showkat Ahmad Lone,¹ Mir Subzar,² and Ankita Sharma³

¹Department of Basic Sciences, Saudi Electronic University, Jeddah Campus, Riyadh, Saudi Arabia
²Division of Agricultural Statistics SKUAST-K, Shalimar 190025, India
³Department of Statistics, University of Jammu, Jammu, India

Correspondence should be addressed to Ankita Sharma; ankitareasi@gmail.com

Received 25 March 2021; Revised 11 April 2021; Accepted 16 April 2021; Published 26 April 2021

Academic Editor: Ishfaq Ahmad

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.

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 [1–12]. 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 as

where , 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.

Table 1: Some existing known estimators of the class of estimators .

S. no.
Estimators
Values of constants

1
, usual unbiased estimator
1
0
0
—
—

2
, Das and Tripathi [1] and Isaki [4] estimator
1
0
1
—
0

3
, Singh et al. [5] estimator
1
0
1

4
, Singh et al. [5] estimator
1
0
1

5
, Singh et al. [5] estimator
1
0
1

6
, Singh et al. [5] estimator
1
0
1

7
, Singh et al. [5] estimator
1
0
1

8
, Singh et al. [5] estimator
1
0
1

9
, Solanki et al. [17] estimator
1
0

10
, Solanki et al. [17] estimator
1
0

11
, Solanki et al. [17] estimator
1
0

12
, Solanki et al. [17] estimator
1
0

13
, Solanki et al. [17] estimator
1
0

14
, Solanki et al. [17] estimator
1
0

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.

Table 2: Some new members of the class of estimators .

S. no.
Estimators
Values 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 as

where

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) as

where

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 approximation

with equality holding if

where are given in (13).

Special Case 1:. for in (2), we get an alternative class of estimators for the population variance as

Putting in (8) and (10), we get the bias and MSE of to the first degree of approximation, given by

The is minimum when

Thus, the resulting minimum of is given by

This 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 as

Putting in (8) and (10), we get the bias and MSE of , respectively, as

The is minimum for

Thus, the resulting minimum MSE of is given by

Thus, we state the following corollary.

Corollary 1. To the first degree of approximation

with equality holding if

Special Case 3:. if we set in (2), the class of estimators reduces to the estimator

which 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 as

Inserting in (8) and (10), we get the bias and MSE of to the first degree of approximation (ignoring fpc term), respectively, as

where

The at (34) is minimized for

Substitution of (36) in (34) yields the minimum MSE of and is given by

Thus, we established the following corollary.

Corollary 2. To the first degree of approximation

with equality holdings if

Special Case 5:. for in (2), the class of estimators “” reduces to the class of estimators of as

which 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.

Table 3: The parameters of population data set.

80

0.3542

5.1500

9.318

20

8.4563

10.300

10.405

51.8264

0.7507

16.975

17.955

11.2646

2.8664

11.825

9.0408

0.9413

2.2667

5.9125

8.0138

18.3569

2.2209

11.0625

7.9136

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.

Table 4: PRE of estimators with respect to .

PRE ()

−1.00
257.0391
259.3591
291.8213
258.2640
256.4163
253.0021

−0.75
255.2481
257.2658
275.3213
256.0011
253.9286
252.8256

−0.50
254.2359
256.9329
273.3492
255.3651
253.6165
252.4152

−0.25
253.9139
256.1231
273.1029
254.9606
253.0259
252.0081

0.00
253.1029
255.8569
273.0051
254.0436
252.8169
251.5621

0.25
252.9320
255.0091
272.9321
253.8186
252.0411
251.0561

0.50
253.1001
254.8563
272.3649
253.5646
251.9103
250.3961

0.75
254.2569
254.6561
272.1536
253.3963
251.3851
250.0011

1.00
254.2031
254.5341
271.9806
252.8646
251.0091
249.5651

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].

Table 5: PREs of the estimators , and .

Estimator
Value of
PRE ()

—
100.00

—
183.23

1.00
209.67

0.50
230.60

0.30
230.28

0.70
230.57

1.00
200.87

0.60
230.80

1.00
270.38

1.00
270.61

1.00
263.50

0.80
270.58

1.00
270.61

0.30
270.58

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.

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