1. Introduction

In survey sampling, it is well known fact that efficiency of population parameters estimates can be improved by the use of auxiliary information, providing it is highly associated with the study variable. Auxiliary information plays a vital role in parameter selection and estimation to achieve efficient estimates of the unknown population parameters. Many authors have suggested estimators based on original auxiliary information. In fact, appropriate use of auxiliary variable in survey sampling results in significant decrease in the variance of the estimator of unknown population parameter(s). Variance is one of the features that can be used to describe a statistical population. Brilliant efforts have been taken to estimate this characteristic as accurately as possible. The goal is to reduce the scattering of the calculated estimate when it is obtained from many available samples. Furthermore, the importance of using information from one or more auxiliary variables is to improve estimator efficiency. Many attempts have previously been made in this direction, including use of the ratio and transformed ratio-product-type. By utilizing the association among the study variable and auxiliary variable, the survey statisticians have shaped effective ratio and product estimators for population variance. The estimation of population variance of the study variable is required in a variety of scenarios in application. In a variety of fields, including agriculture, medicine, biology, and business, where we encounter populations that are likely to be skewed, variance estimate for the population is important.

Variations can be found in our daily lives throughout the world. It is a common belief that no two things or people are exactly the same. For example, a farmer must have a good alertness of how climate geographies change over time to time in order to determine how to plant his lot. For proper treatment, a doctor must have a detailed realization of differences in human hypertension and fever. Many authors suggested different types of estimators for variance including Das [1], Prasad and Singh [2], Swain and Mishra [3], Ahmad and Shabbir [4], Kadilar and Cingi [5], Subramani and Kumarapandiyan [6], Singh and Solanki [7], Yadav et al. [8], Singh et al. [9], Yadav and Kadilar [10], Adichwal et al. [11,12], Yasmeen et al. [13], Zaman [14,15], Zaman and Bulut [16], Zaman et al. [17,18], Bhushan et al. [19], Shahzad et al. [20], Ahmad et al. [21], Ozturk [22], Ozturk and Demirel [23], Frey and Feeman [24], Zamanzade and Vock [25], Mahdizadeh and Zamanzade [26,27], Bhushan et al. [28], Bhushan et al. [29], Zaman and Bulut [30,31], Yadav and Zaman [32], Zaman and Kadilar [33], Zaman and Dunder [34] and Singh and Khalid [35]. We requisite to estimate population parameters for numerous variables in multidisciplinary surveys, such as house lengths, income level, and the number of unemployed people, in socio-economic surveys. The goal of this study is to offer a strategy for estimating population variance using a dual auxiliary variable in simple random sampling. We found no contribution on variance estimation using dual auxiliary variables under simple random sampling in the literature.

2. Notations and symbols

Consider ψ = (ψ₁, ψ₂,…,ψ_N) be a population comprise of size N elements, let a sample of size n is chosen from ψ by using simple random sampling without replacement (SRSWOR). Let Y, X and R_x represent the study variable, auxiliary variable and rank of the auxiliary variable respectively. Consider population variance of the study variable Y, auxiliary variable X and rank of the auxiliary variable R_x is denoted by and . Let y_i, x_i and r_xi, be the sample for the i^th units correspondingly. Let , be the sample variances. , of Y, X and R_x be the corresponding variances. To derive the properties of the suggested estimator, we have the following error terms.

Where r, s and t be the positive number and υ₂₀₀, υ₀₂₀, υ₀₀₂ are the second order moments about means of Y, X and rank of X, and ƛ_rst is the moment ratio.

3. Existing estimators

In this section, we discussed several well-known estimators of population variance.

1. The usual estimator, is given by: (1)
   The variance of , is given by: (2)
2. Isaki [36] suggested ratio estimator, is given by: (3)
   The properties of , is given as: and (4)
3. The difference estimator, is given by: (5)
   Where
   The minimum variance at the optimal value of Ψ₁₁, is given by: (6)
   Where
4. Rao [37] difference type estimator, is given by: (7) where
   The properties at the optimal value of Ϣ₁ and Ϣ₂, are given by: and (8)
5. Singh et al. [38] suggested the following estimators: (9)
   The properties of , are given by: (10) (11)
6. Grover and Kaur [39], suggested the following estimator, is given by: (12)
   Where Ϣ₃ and Ϣ₄ are the unknown constant
   The minimum MSE at the optimum values of Ϣ₃, Ϣ₄, are given by: (13)
7. Ahmad et al. [40], suggested the following estimator, given by: (14)
   After expanding the above equation, we have
   The bias and MSE of , are given by:
   The optimal values of Ϣ5, Ϣ6, and Ϣ₇, are given by: and
   The minimum mean square error at the optimal values of and , are given by: (15) where

4. Proposed improved ratio-in-regression type variance estimator

The need for auxiliary variable can boost estimator efficiency during the design and estimation stage. When the association among the study variable and auxiliary variables exist, then rank of the auxiliary variable is also associated with the study variable. Therefore, the rank of the auxiliary variable (that includes rank of the auxiliary variable) can be considered as a novel auxiliary variable, and this evidence may help us in increasing the precision of an estimator. Based on these ideas, we are encouraged to investigate dual use of the auxiliary variable for population variance because it has been examined that there has been no work before. The suggested estimator includes the auxiliary information in the form of an auxiliary variable and in the form of a rank auxiliary variable. There are several estimators for estimating population variance based on original auxiliary variables. The key advantage of our proposed estimator in simple random sampling is that it is more flexible and best than existing estimators. By taking motivation from Ahmad et al. [40] we proposed the following estimator: (16)

Simplifying (16), we have (17)

Simplify (17), we have (18)

The optimum values of Ϣ₈, Ϣ₉ and Ϣ₁₀ are given by:

Where,

The minimum MSE at the optimum values of Ϣ₈, Ϣ₉ and Ϣ₁₀ in (18), we have: (19)

5. Efficiency comparison

In this section, the theoretical comparison of the suggested estimator with existing estimators is considered:

1. From (2) and (19), where
2. From (4) and (19),
3. From (6) and (19),
4. From (8) and (19),
5. From (11) and (19),
6. From (13) and (19),
7. From (15) and (19),

6. Data description

In this part, we carry out a mathematical study to see the performances of proposed estimator in comparison of existing estimator. We considered two real data sets under SRS. The data descriptions of these populations are given in Table 1. We compare the production of our proposed estimator with existing counterparts in terms of PRE. To obtain the PRE, we used the following expression.

Where (u) = (0, 1, 2, 3, 4, 5, 6).

Population 1: [Source: Singh [41]]

Y: Estimated number of fish in year 1995.

X: Estimated number of fish in year 1994,

R_x = Rank of X variable

Population 2: [Source: Singh [41]]

Y: Estimated number of fish in year 1995.

X: Estimated number of fish in year 1993,

R_x = Rank of X variable

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Table 1. Summary statistic of Populations 1–2.

https://doi.org/10.1371/journal.pone.0276540.t001

7. Simulation study

We have generated two populations of size 5,000 from normal distribution with different parameters by using R language. The populations have different correlations, i.e. population I is positively correlated and population-II has strong positive correlation between X and Y variables. The population details are given below:

8. Discussion

To show the performances of our suggested improved variance estimator, we used two real populations. Data description of these mentioned populations are given in Table 1. The bias result using real data sets are given in Table 2. The MSE and PRE are presented in Table 3. From the numerical results, the proposed estimator is found to be more effective than its existing estimators. The gain in efficiency in population 2 is more as compared to population 1. As we increase the sample size, the MSE decreases and PRE increases, yielding the expected results. We also evaluated the efficiency of our proposed improved variance estimator in simple random sampling using simulation study. The bias result using simulated data sets are given in Table 4. From Table 5, it is concluded that our suggested improved variance estimator performs better than the standard estimator and other existing estimators for a number of correlation coefficient values. The outcome of the simulation study clearly determines that for the simulated data sets 1–2, the PRE of the proposed estimator is greater than the PRE of the existing estimate. Overall the gain in efficiency of our proposed estimator is the best as compared to all existing counterparts.

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Table 2. Bias result using Populations 1–2 using two real data sets.

https://doi.org/10.1371/journal.pone.0276540.t002

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Table 3. MSE and Percentage relative efficiency using Populations 1–2.

https://doi.org/10.1371/journal.pone.0276540.t003

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Table 4. Bias result using Populations 1–2 using simulated data sets.

https://doi.org/10.1371/journal.pone.0276540.t004

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Table 5. MSE and PRE of , w.r.t the existing counterparts using two simulated data.

https://doi.org/10.1371/journal.pone.0276540.t005

9. Conclusion

In this article, we proposed an improved ratio-in-regression type exponential estimator for the finite population variance using dual auxiliary variables under simple random sampling. Expressions for mean square error of the proposed estimator are derived up to first order of approximation. The proposed estimator is compared with several existing counterparts to judge their uniqueness and superiority using two real data sets. Moreover, simulation study is also conducted to check the robustness and generalizability of the proposed variance estimator. A numerical study is carried out to support the theoretical results. Based on the numerical findings, it is observed from the result of real data sets and simulation study, showing when compared to its existing counterparts; the suggested estimator produces well results. Therefore we recommend the use of our proposed estimators for efficiently estimating the finite population varaince under simple random sampling using dual auxiliary varaible. The current work can be extended to develop an improved class of estimators under measurement error, probability proportional to size (PPS), two-stage sampling using dual auxiliary variable for estimating the population variance.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R299), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia