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 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.
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.
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].
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 .
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.
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.
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].
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.