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

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.


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][2][3][4][5][6][7][8][9][10][11][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. e 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.

Notations
Consider a finite population Z � (Z 1 , Z 2 , . . . , Z M ) of M units and let (z, r) be the study and auxiliary variables defined on Z taking values (z i , r i ), respectively, on Z i (i � 1, 2, . . . , M). It is desired to estimate the population variance W 2 z of the study variable z using the information on an auxiliary variable r. Let a simple random sample of size m be drawn without replacement from the finite population Z. In this paper, we shall ignore the finite population correction (fpc) term. We denote the following. 2 : the population mean square/variance of the auxiliary variable r TM � (Q 1 + 2Q 2 + Q 3 /4): tri-mean of the auxiliary variable r HL�median((R j +R k )/2,1≤j≤k≤M): Hodges-Lehmann estimator of the auxiliary variable r MR � (R (1) probability weighted moments of the auxiliary variable r: where

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 (cW 2 r + φQ 2 ) > 0, (cw 2 r + φQ 2 ) > 0, and (c, Q) are either real constants or functions of known parameters of an auxiliary variable r and φ is a constant such that |φ| ≤ 1 which is a more flexible condition as given by Singh et al. [5] and Solanki et al. [17] over the constant φ in their estimators, and (α 1 , α 2 ) are constants such that the mean squared error (MSE) is minimum. Here, we note that 0 ≤ φ ≤ 1. For suitable values of (α 1 , α 2 , c, φ), the proposed class of estimators "l" reduces to some known existing estimators based on quartiles and their functions given in Table 1.
From (2), we would like to remark the suitable values of (α 1 , α 2 , c, φ) and one can generate different estimators. For example, we have developed some new estimators from the proposed class of estimators "l" which are listed in Table 2.
While obtaining the expression of bias and mean square error (MSE) for the suggested class of estimator "l", we write such that E(ϑ 0 ) � E(ϑ 1 ) � 0, and up to the first order of approximation, while fpc term is ignored, we have Now, in terms of ϑ ′ s, the estimator "l", given in (2), is expressed as We assume that |θ * ϑ 1 | < 1, so that (1 + θ * ϑ 1 ) − 1 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 l to the first degree of approximation as

Mathematical Problems in Engineering
Squaring both sides of (7), neglecting terms of ϑ ′ s having the power greater than two, we have Taking expectation of both sides of (9), we get the MSE of the estimator l to the first degree of approximation (ignoring fpc term) as where Differentiating MSE (l) with respect to α 1 and α 2 and equating them to zero, we have Simplifying (12), we get the optimum values of α 1 and α 2 as � α 10 (say), � α 20 (say).

(13)
Inserting (13) in (10), we get the resulting minimum MSE of "l" given by us we establish the following theorem.

Theorem 1. To the first degree of approximation
with equality holding if where α i0 ′ s(i � 1, 2) are given in (13).
Special Case 1: for α 1 � 1 in (2), we get an alternative class of estimators for the population variance W 2 z as Putting α 1 � 0 in (8) and (10), we get the bias and MSE of l * to the first degree of approximation, given by us, the resulting minimum MSE of (l * ) is given by (20) is equals to the minimum MSE of the difference estimator l d envisaged by [1] Special Case 2: for α 2 � 0 in (2), we get another class of estimators for population variance W 2 z as Putting α 2 � 0 in (8) and (10), we get the bias and MSE of l * 1 , respectively, as e MSE(l * ) is minimum for us, the resulting minimum MSE of l * is given by us, we state the following corollary.

Corollary 1. To the first degree of approximation
with equality holding if Special Case 3: if we set (α 1 , α 2 ) � (1, 0) in (2), the class of estimators l reduces to the estimator which includes Solanki et al. [17] estimators l 9 to l 14 .
Putting (α 1 , α 2 ) � (1, 0) in (8) and (10), we get the bias and MSE of l * * 1 to the first degree of approximation (ignoring fpc term), respectively, as Special Case 4: if we set (α 1 , α 2 , c) � (α 1 , 0, 1)in (2), the class of estimators l reduces to the class of estimators of W 2 z as (32) (8) and (10), we get the bias and MSE of l * 2 to the first degree of approximation (ignoring fpc term), respectively, as where e MSE(l * 2 ) at (34) is minimized for  For the purpose of comparison of the proposed estimators l 15 to l 20 with that of the usual unbiased estimator l 1 � w 2 z , [4] estimator l 2 , [5] estimators (l 3 to l 8 ), and [17] estimators (l 9 to l 14 ), in Table 5, we give the PREs of the estimators l 1 , l 2 and the PREs of the estimators l 3 to l 14 with respect to usual unbiased estimator w 2 z along with the value of δfor which the PREs of the estimators l 3 to l 14 are maximum as given in [17].
It is observed from Tables 4 and 5 that all the estimators l k (k � 15, 16, . . . , 20) which are the members of the proposed class of estimators l performed better than the usual unbiased estimator l 1 � w 2 z , ratio type estimators l 2 due to [4,5] estimators (l 3 to l 8 ) and [17] estimators (l 9 to l 14 ), for φ ∈ (− 1.0, 1.0).

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.