1. Introduction
Massive literature is available for estimating finite population mean/total, coefficient of variation, median, variance and proportion in sample surveys e.g. [1–10], and many more.
Sample mean is the best suitable estimator to the estimation of finite population mean under different sampling designs. While it is biased, it has a substantial amount of variation. So, we are trying to find out such estimators that might be biased, but has minimum mean squared error (MSE) relative to the sample mean. For the purpose of least MSE, we use some auxiliary variables which are already known and some association with study variable. Auxiliary variable increases the efficiency of the estimators. There are certain types of estimators such as ratio estimator proposed by [11], product estimator proposed by [12], and regression estimator proposed by [13]. Ratio-type estimators are used when study variable and auxiliary variable are strongly positive correlated to each other and the regression line passes through or nearly passes through the origin. When auxiliary variable and study variable are strongly negative correlated, then product-type estimators are used. The regression-type estimators are used when the regression line passes away from the origin.
Several survey sampling strategies are employed in our daily lives. But the most commonly sampling method is simple random sampling (SRS). It is the type of probability sampling when the population is homogeneous. When the probability of sample selection is same for every unit of population, it is known as SRS. Considerable efforts have been carried out for the estimation of unknown finite population mean by utilizing bivariate auxiliary information under SRS. Moving along this direction, some remarkable contributions in SRS without replacement have been developed by several authors. For instance, [14] suggested two classes of ratio-type estimators to estimate unknown population mean. They considered a real-life application to check the supremacy of their suggested estimators over the competitors. They extended their work for more than two auxiliary variables. [15] proposed regression-type estimator by considering [14] estimator. A real-life application is also considered to confirm the proficiency of the proposed estimator over [14] estimator. [16] suggested chain ratio and regression estimators for improving mean estimation. He investigated mathematical properties including bias and MSE for the newly estimators. He compared the newly suggested estimators with the traditional multivariate ratio and regression estimators. [17] suggested new improved estimator for unknown population mean. Mathematical expressions in terms of bias and MSE are derived. He proved that his estimator is more reliable over the [14, 15] and traditional multivariate regression estimators. Moreover, four real-life applications taken from agriculture, biomedical and power engineering to check the supremacy of the theoretical findings. [18] proposed a new family of estimators for unknown population mean. Four real data sets considered to confirm the efficiency of the newly estimators over the competing estimators. Recently, [18] suggested class of exponential ratio-type estimators for the estimation of population mean. Mathematical properties have been derived. Various real-life applications are considered to confirm the supremacy of the suggested class of estimators over the competitors.
Stratified random sampling is used when population is heterogeneous. A stratified random sample divides the entire population into homogeneous groups known as strata, based on shared characteristics. Homogenous groups should be non-overlapping. Random samples are then selected from each stratum. Noteworthy efforts have been carried out to estimate unknown population mean by utilizing bivariate auxiliary variables in stratified random sampling for instance, [19] suggested a new ratio-cum-product exponential estimator for unknown mean. They compared their proposed estimator with combined ratio, combined product and other competing estimators. [20] suggested ratio-cum-product estimator for unknown population mean. Asymptotic expressions in terms of bias and MSE have been obtained up to first degree of approximation. An empirical study endorsed that newly ratio-cum-product estimator is more effective than combined ratio and product-type and other competing estimators. [17] suggested improved separate family of estimators for the estimation of population mean. Mathematical expressions of the suggested estimators have been obtained up to first order of approximation. Theoretically conditions and empirical study confirm that their suggested estimators are more proficient as compared to traditional ratio, product and other competing estimators under study. [18] proposed a family of estimators by the inspiration form [21]. They recommended that their suggested estimators are more efficient over the multivariate ratio, regression, difference and [21] estimators. Recently, [18] suggested difference-cum-exponential ratio-type estimators for the estimation of unknown population mean. Mathematical properties such as bias and minimum MSE have been derived. Two real-life applications are considered to confirm the supremacy of the suggested estimators over the traditional difference estimator using two auxiliary variables.
Theses authors suggested a number of estimators to estimate unknown population mean by using bivariate auxiliary information under SRS and stratified random sampling. Efficiency of theses estimators is suspicious in case of extreme values because all existing estimators based on only conventional measures of the auxiliary variables. None of them used non-conventional measures of the auxiliary variables. In the present study, the objective is to estimate unknown population mean by utilizing bivariate auxiliary information based on non-conventional measures (robust measures) under SRS and stratified random sampling schemes. The proposed generalized estimator can provide many estimators subject to the availability of the auxiliary information. Various real-life applications are used to confirm the superiority of the suggested estimators over the competitors.
2. Existing estimators under SRS
Consider "n"(sample size) is taken by SRS without replacement from a finite population Ω = {Ω1, Ω2,…ΩN} of N objects. For each object, Ωi(i = 1,…N) in the population, the values yi and (xi,zi) are collected corresponding to the study variable (y) and auxiliary variables (x,z), respectively. Following relative error terms are used to get the mathematical expressions in terms of bias, MSE and minimum MSE of the existing and proposed estimators.
such as E(ξi) = 0 for i = 0,1,2.where
Some well-known estimators for estimating unknown finite population mean using conventional measures based on bivariate auxiliary information are given below.
1. (1) Sample mean estimator is defined in Eq (1)
(1)
Variance of is given in Eq (2):(2)
1. (2) [22] proposed traditional multivariate ratio-type estimator, given by:
(3)
By substituting the optimum values of θ1 and θ2 in the expression of MSE of , we get minimum MSE of
(4)
1. (3) [23] proposed regression-type estimator, is given by:
(5)
After substituting the optimal values of θ3, θ4 and θ5 in the expression of MSE of , we get minimum MSE of (6)
1. (4) [14] suggested ratio-type estimator, is given by:
(7)
Minimum MSE of is given in Eq (8):(8)where
1. (5) [24] suggested general class of estimators is given in Eq (9):
(9)where θ8 and θ9 are the weights satisfying the condition: θ8+θ9 = 1,a1(≠0),a2(≠0), (b1 and b2) are either real numbers. Some combinations of their suggested estimators is presented in Table 1. After putting the optimum values of θ8 and θ9 in the expression of MSE, minimum MSE of is:(10)where and
[Figure omitted. See PDF.]
1. (6) The traditional difference estimator, is given by:
(11)
After putting the optimum values of θ10 and θ11 in the expression of MSE, minimum MSE of is given by:(12)where
1. (7) Recently, [18] proposed difference-cum-exponential type estimator, is given by:
(13)
Substituting the optimum values of θ12,θ13 and θ14 in the expression of MSE, the minimum MSE of is given in Eq (14):(14)
3. Suggested family of estimators under SRS
One noticeable shortcoming of existing estimators e.g. [14, 18, 24] and many others is that they execute unproductively in the presence of extreme observations. [18] suggested difference-cum-exponential type estimator for the estimation of unknown population mean under SRS. Enchanting inspiration from [18], we proposed a class of estimators for estimating population mean under SRS scheme using non-conventional measures related to auxiliary variables. The suggested general estimator is defined as:(15)
Here, m1,m2 and m3 are suitable weights to be selected in a way that τ1(≠0) and τ2 be any function of the known non-conventional parameters of the auxiliary variables.
Expressing in terms of relative error terms and subtract from both side of Eq (15), we get:(16)where
To get bias of , taking expectation on both sides of Eq (16)(17)
For MSE of , squaring and taking expectation on both sides of Eq (16)(18)
Differentiating Eq (18) w.r.t to m1,m2 and m3, we get optimal values of m1,m2 and m3 as given by:
Substituting optimal values in Eq (18), we get the minimum MSE of as given by:(19)where
4. Numerical study under SRS
In this segment, we investigate the supremacy of the proposed estimators under SRS using two real-life applications. Population I consider high and positive correlation and Population II consider low and negative correlation, among study and auxiliary variables, respectively. Description of each population is given in Tables 2 and 3.
[Figure omitted. See PDF.]
[Figure omitted. See PDF.]
Population I: (Source: [25])
y: Number of “placebo” children
x: Number of paralytic polio cases in the “not in occulted” group
z: Number of Paralytic polio cases in the “placebo” group
Population II: (Source: [26])
y: Wild cats
x: Price well head
z: Domestic output
We calculated MSE of the existing and proposed estimators i.e., Expressions for the MSE of all the estimators are prescribed below in section 2–3 in detail. Subsequently, empirical findings are summarized in Tables 4 and 5. Some valuable insights are drawn from Tables 4 and 5 as follows:
From Table 4, the existing estimator have minimum MSE over the all other existing estimators i,e.
From Table 5, the proposed estimators where i = 1,3,…,17 under SRS have minimum MSE over the all existing estimators. It reveals that the use of non-conventional measures has reduced the MSE.
[Figure omitted. See PDF.]
[Figure omitted. See PDF.]
4.1. A Simulation study
A Monte Carlo simulation study using R software were also performed to check the supremacy of the proposed estimators i.e., over the existing estimators i.e., . For this purpose, two real-life applications (Population I and Population II) are taken from Section 4. Important descriptive measures can be seen in Tables 2 and 3. Different sample sizes i.e. (n = 6,8 and 10) and i.e. (n = 5,6 and 8) are used for Population I and Population II, respectively. Following expression is used to get MSE/minimum MSE for all existing and proposed estimators under study:where
Following five steps are done to perform simulation study:
1. Choose a SRS without replacement of size n from N.
2. Obtain MSE of all the existing and proposed estimators by using step 1.
3. Repeat step 1 & 2, 10,000 times and get 10,000 values for MSE of each estimator i,e.
4. Use of step 3, take the average of 10,000 values and get the MSE of each estimators
under study.
Table 6 presents the values of MSE using different sample sizes. Simulation study, similarly in applications to real data sets tells that
1. It is worth pointing that has the least MSE as compared to all existing estimators.
2. performs better as compared to all the competing estimators i.e., and .
3. MSE of all the estimator’s i.e., decreases as sample size increases.
[Figure omitted. See PDF.]
5. Existing estimators under stratified random sampling
Let Ω = {Ω1,Ω2,…,ΩN} be a finite population of N units allocated into L strata with the hth stratum (h = 1,2,3,…L), taking Nh units. Let yhi and (xhi,zhi) are the study variable (Yh) and the auxiliary variables (Xh and Zh) individually for the ith population unit in the hth stratum. A sample of size nh is taken from each stratum i.e., . Following relative error terms and their expectations are used to develop the mathematical properties such as bias, MSE and minimum MSE of all the estimators.
such that E(ξih) = 0, for i = 0,1 and 2.where
In this section, we discussed well-known existing estimators by using two auxiliary variables for the estimation of unknown finite population mean under stratified random sampling.
1. (1) The usual sample mean estimator is given in Eq (20):(20)where
Variance of , is given by:(21)
2. (2) Traditional multivariate ratio estimator, is given by:(22)where θ1h and θ2h are the unknown constants.
MSE of is given by:(23)
3. (3)The chain ratio-ratio estimator suggested by [27], is given by:(24)
The MSE of , is given by:(25)
4. (4)The ratio- product estimator presented by [28], is given by:(26)
MSE of the , is given by:(27)
5. (5)The classical regression estimator proposed by [23], is given by:(28)where are sample regression coefficients.
MSE of the , is given by:(29)
6. (6)Traditional difference estimator utilizing bivariate information of the auxiliary variables, is given by:(30)where θ3h and θ4h are unknown constants.
Minimum MSE (, is given by:(31)
7. (7) [18] proposed difference-cum-exponential ratio type estimator, is given by:(32)
Minimum MSE of after substituting the optimal values of θ5h, θ6h and θ7h in the expression of MSE of , given as:(33)
6. Proposed estimator under stratified random sampling
Enchanting inspiration from [18], we proposed a generalized estimator to estimate unknown finite population mean in stratified random sampling based on non-conventional measures related to auxiliary variables. The proposed generalized estimator with the help of known auxiliary information is defined as:(34)where m1h, m2h and m3h are suitable weights to be selected in a way that τ1h(≠0) and τ2h be any constant values or function of the known non-conventional parameters of the auxiliary variables.
Solving Eq (34) in terms of relative error terms (ξoh,ξ1h and ξ2h)(35)where .
Taking expectation on both sides of Eq (35), we get bias of as(36)
MSE of the proposed estimators i,e. is given by:(37)
Differentiating Eq (37) w.r.t to m1h,m2h and m3h, we get optimal values of m1h,m2h and m3h as given by:
For minimum MSE, by substituting all the optimum values in Eq (37), we have(38)where
7. Numerical study under stratified random sampling
We considered a real life population (Population III) taken from [29] to check the proficiency of the proposed estimators over the existing estimators in terms of percent relative efficiency (PRE). Here, y: output for 80 factories, x: number of workers and z: fixed capital. Important descriptive measures are reported in Table 7. The percentage relative efficiencies (PRE) of all the estimators with respect to the usual unbiased estimator are calculated through the expression given below:
[Figure omitted. See PDF.]
Empirical findings are summarized in Tables 8 and 9, respectively. Some valuable insights are drawn from Tables 8 and 9, as follows:
1. ✓ From Table 8, the existing estimator have maximum PRE as compared to other existing estimators i,e. .
2. ✓ From Table 9, the proposed estimators where i = 2,4,…,16 under stratified random sampling have maximum PRE over the existing estimator .
[Figure omitted. See PDF.]
[Figure omitted. See PDF.]
8. Concluding remarks
Difference-cum-exponential-type estimators have been proposed for estimating unknown population mean by utilizing bivariate auxiliary information. Proposed estimators are based on non-conventional measures of the auxiliary variables under simple and stratified random sampling schemes. Theoretical findings in the form of bias and MSE were presented in section 3 and section 6. Proposed estimators are found to be more proficient over the competing estimators under both sampling schemes. Furthermore, empirical and simulation studies are performed with the integration of several real populations to assess the supremacy of the proposed estimators with the competing estimators in terms of MSE and PRE.
In the upcoming articles, new optimal estimators can be proposed under other sampling designs using non-conventional measures of the auxiliary variables.
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Citation: Javed M, Irfan M, C. Shongwe S, Hussain MA, Zico Meetei M (2025) Difference-Cum-Exponential-type estimators for estimation of finite population mean in survey sampling. PLoS ONE 20(1): e0313712. https://doi.org/10.1371/journal.pone.0313712
About the Authors:
Maria Javed
Roles: Conceptualization, Methodology, Visualization, Writing – original draft
Affiliation: Department of Statistics, Government College University, Faisalabad, Pakistan
Muhammad Irfan
Roles: Conceptualization, Investigation, Methodology, Project administration, Supervision, Writing – original draft, Writing – review & editing
Affiliation: Department of Statistics, Government College University, Faisalabad, Pakistan
Sandile C. Shongwe
Roles: Conceptualization, Funding acquisition, Validation, Writing – review & editing
Affiliation: Department of Mathematical Statistics and Actuarial Science, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa
Muhammad Ali Hussain
Roles: Writing – original draft, Writing – review & editing
E-mail: [email protected]
Affiliation: Business School, NingboTech University, Ningbo, China
ORICD: https://orcid.org/0000-0002-7716-7058
Mutum Zico Meetei
Roles: Resources, Writing – review & editing
Affiliation: Department of Mathematics, College of Science, Jazan University, Jazan, Kingdom of Saudi Arabia
[/RAW_REF_TEXT]
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1. Irfan M., Javed M., and Lin Z. (2018). Efficient ratio-type estimators of finite population mean based on correlation coefficient. Scientia Iranica, 25(4): 2361–2372.
2. Irfan M., Javed M., and Bhatti S. H. (2022). Difference-type-exponential estimators based on dual auxiliary information under simple random sampling. Scientia Iranica: Transactions on Industrial Engineering (E), 29(1): 343–354.
3. Javed M., Irfan M. and Pang T. (2019). Utilizing bivariate auxiliary information for enhanced estimation of population mean under simple and stratified random sampling schemes. Journal of the National Science Foundation of Sri Lanka, 47 (2): 199–211.
4. Cekim H. C. (2022). Modified unbiased estimators for population variance: An application for COVID-19 deaths in Russia. Concurrency and Computation Practice and Experience, 34 (22): e7169. pmid:35941888
5. Shahzad U., Hanif M., Sajjad I., and Anas M., M. (2022). Quantile regression-ratio-type estimators for mean estimation under complete and partial auxiliary information. Scientia Iranica: Transactions on Industrial Engineering (E), 29(3): 1705–1715.
6. Shahzad U., Ahmad I., Almanjahie I. M., Al-Noor N. H., and Hanif M. (2023). Mean estimation using robust quantile regression with two auxiliary variables. Scientia Iranica: Transactions on Industrial Engineering (E), 30(3): 1245–1254.
7. Ahmad S., Shabbir J., Zahid E., Aamir M., and Alqawba M. (2023). New generalized class of estimators for estimation of finite population mean based on probability proportional to size sampling using two auxiliary variables: A simulation study. Science Progress, 106 (4): 1–24. pmid:37885238
8. Hussain M, A, Javed M., Zohaib M., Shongwe S. C., Awais M., Zaagan A. A., and Irfan M. (2024). Estimation of Population Median Using Bivariate Auxiliary Information in Simple Random Sampling. Heliyon, 10 (2024) e28891. pmid:38601683
9. Pandey M, K, Singh G, N., Zaman T., Mutairi A, A., and Mustafa M, S. (2024). Improved estimation of population variance in stratified successive sampling using calibrated weights under non-respons. Heliyon, 10 (2024) e27738.
10. Koyuncu N. and Kadilar C. (2009). Efficient estimators for the population mean. Hacettepe Journal of Mathematics and Statistics, 38(2): 217–225.
11. Cochran W. G. (1940). The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce. The Journal of Agricultural Science, 30(2): 262–275.
12. Murthy M. N. (1964). Product method of estimation. Sankhya: The Indian Journal of Statistics, Series A, 69–74.
13. Watson D. J. (1937). The estimation of leaf area in field crops. Journal of Agriculture Science, 27: 474–483.
14. Abu-Dayyeh W. A., Ahmed M. S., Ahmed R. A., and Muttlak H. A. (2003). Some estimators of a finite population mean using auxiliary information. Applied Mathematics and Computation, 139(2–3): 287–298.
15. Kadilar C. and Cingi H. (2005). A new estimator using two auxiliary variables. Applied Mathematics and Computation, 162(2): 901–908.
16. Lu J. (2013). The chain ratio estimator and regression estimator with linear combination of two auxiliary variables. Plos one, 8(11): e81085. pmid:24260537
17. Lu J. (2017). Efficient estimator of a finite population mean using two auxiliary variables and numerical application in agricultural, biomedical, and power engineering. Mathematical Problems in Engineering.
18. Muneer S., Shabbir J. and Khalil A. (2017). Estimation of finite population mean in simple random sampling and stratified random sampling using two auxiliary variables. Communications in Statistics-Theory and Methods, 46(5): 2181–2192.
19. Tailor R., and Chouhan S. (2014). Ratio-Cum-Product type exponential estimator of finite population mean in stratified random sampling. Communications in Statistics-Theory and Methods, 43: 343–354.
20. Lone H. A., Tailor R., and Singh H. P. (2016). Generalized ratio-cum-product type exponential estimator in stratified random sampling. Communications in Statistics-Theory and Methods, 45(11): 3302–3309.
21. Singh V. K. and Singh R. (2014). Performance of an estimator for estimating population mean using simple and stratified random sampling. SOP Transactions on Statistics and Analysis, 1(1): 1–8.
22. Singh D. and Chaudhry F.S. (1986). Theory and analysis of sample survey design. New Age International Publisher. New Delhi, India.
23. Rao T. J. (1991). On certain methods of improving ratio and regression estimators. Communications in Statistics. Theory and Methods, 20(10): 3325–3340.
24. Lu J. and Yan Z. (2014). A class of ratio estimators of a finite population mean using two auxiliary variables. Journal.pone.0089538, 9(2). pmid:24586855
25. Cochran W.G. (1977). Sampling Techniques. John Wiley & Sons. Inc., New York, USA.
26. Gujrati D. N. (2004). Basic Econometrics, The McGraw-Hill Companies, New York, 2004.
27. Singh M. P. (1965). On the estimation of ratio and product of the population parameters. Sankhya B, 27: 231–328.
28. Singh M. P. (1967). Ratio-cum-product method of estimation. Metrika. 12: 34–72.
29. Murthy M. (1967). Sampling Theory and Methods, 1st Ed., Statistical Publishing Society, India.
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Abstract
Extensive research work has been done for the estimation of population mean using bivariate auxiliary information based on conventional measures. Conventional measures of the auxiliary variables provide suspicious results in the presence of outliers/extreme values. However, non-conventional measures of the auxiliary variables include quartile deviation, mid-range, inter-quartile range, quartile average, tri-mean, Hodge-Lehmann estimator etc. give efficient results in case of extreme values. Unfortunately, non-conventional measures are not used by survey practitioners to enhance the estimation of unknown population parameters using bivariate auxiliary information. In this article, difference-cum-exponential-type estimators for population mean utilizing bivariate auxiliary information based on non-conventional measures under simple and stratified random sampling schemes have been suggested. Mathematical properties such as bias and mean squared error are derived. To support theoretical findings, various real-life applications are used to confirm the superiority of the suggested estimators as compared to the competing estimators under study.
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