
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38582765
58841
10.1038/s41598-024-58841-x
Article
Improvement in variance estimation using transformed auxiliary variable under simple random sampling
Ali Hameed 1
Asim Syed Muhammad 1
Ijaz Muhammad 2
Zaman Tolga zamantolga@gmail.com

3
Iftikhar Soofia 4
1 https://ror.org/02t2qwf81 grid.266976.a 0000 0001 1882 0101 Department of Statistics, University of Peshawar, Peshawar, Pakistan
2 https://ror.org/05vtb1235 grid.467118.d 0000 0004 4660 5283 Department of Mathematics and Statistics, University of Haripur, Harīpur, Pakistan
3 https://ror.org/011y7xt38 grid.448653.8 0000 0004 0384 3548 Department of Statistics, Cankiri Karatekin University, Çankırı, Turkey
4 https://ror.org/00s2rk252 grid.449638.4 0000 0004 0635 4053 Department of Statistics, Shaheed Benazir Bhutto Women University, Peshawar, Pakistan
6 4 2024
6 4 2024
2024
14 81177 4 2023
3 4 2024
© The Author(s) 2024
2024
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This paper offers a novel approach to formulate efficient ratio estimator of the population variance using a transformed auxiliary variable. The impact of transformation on auxiliary information has also been discussed. It is observed that incorporating a transformed auxiliary variable result in a high gain in efficiency. Theoretical properties of the newly developed estimators have been derived. The empirical and simulation studies show that the suggested estimators outperformed the existing estimators.

Keywords

Auxiliary variable
Mean square error
Population variance
Percentage relative efficiency
Subject terms

Applied mathematics
Statistics
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Sampling is a crucial aspect of making well-informed decisions in various real-life domains. Inferences about statistical populations or data are drawn from samples, and it is imperative that a sample accurately represents every characteristic of the population of interest. Data is characterized by specific parameters, and estimating population parameters from a sample poses a challenging task. Two essential measures for specifying data are the measure of location and scale. This article focuses on the latter, specifically the estimation of variance, and a frequently used measure of data scale. Estimating variance is vital in processes where precise quantification of data variation is necessary. From economics and business to physical science, biological science, and environmental sciences, sophisticated tools are required to measure variation for making informed decisions. For instance, economists analyze the variation in commodity prices, manufacturers assess taste preferences for customer satisfaction, and agriculturists study variability in climate factors to optimize yields and minimize costs.

Extensive research has sought to enhance the efficiency of ratio and product estimators for finite population variance, considering the correlation between survey variables and auxiliary variables. Auxiliary variables are those correlated with the main study variable, either positively or negatively. In environmental studies, for example, auxiliary variables like wind speed or temperature are considered when estimating air quality variance. Economic surveys may involve estimating household income variance using employment rates as an auxiliary variable. In healthcare planning, patient demographics or medical histories serve as auxiliary variables when estimating variance in patient recovery times.

Integrating auxiliary variables with the main survey variable to estimate variance provides additional information, refining the accuracy of estimates. This additional information includes contextual data, environmental factors, or supplementary metrics correlated with main study variable. The use of auxiliary information often leads to more efficient and robust variance estimators. Similarly, the transformative role is notable in enhancing estimate efficacy. Transformations make estimators flexible, allowing them to better utilize additional information from auxiliary variables, such as mean, variance, skewness, kurtosis, quantiles, etc. This flexibility, along with generalization and optimization constants, enhances the robustness of estimates against variations in the sample.

That is why various sample survey statisticians preferred transformed auxiliary variables instead of considering them in their original form. Keeping in view the above, numerous researchers developed many efficient estimators of population variance. Some of the pioneer work on estimating variance of finite population using auxiliary variable are due to1 who proposed an unbiased estimator of variance and2 compares variance estimators under various sample designs with available auxiliary information. Illustrates improved bias and mean squared error over common estimators. Similarly3, introduces chain estimators for finite population variance using double sampling and two auxiliary variables. Compares estimators based on mean square estimator (MSE) criterion. Proposes a ratio-type exponential estimator for population variance, consistently more efficient than previous estimators4. Conducts efficiency comparisons mathematically and numerically. Introduces a family of estimators based on adaptations of previous work, showing efficiency through mean square error comparisons5. A generalized modified ratio-type estimator for population variance using known parameters of the auxiliary variable was suggested by6. Compares with existing estimators for simulated and real data to show the performance of the developed estimator against the competing estimators. Suggests a generalized class of finite population variance, deriving large sample bias and mean square error. Considers special cases and provides numerical examples for comparison7. Suggested estimator of population variance utilizing information on two auxiliary variables under SRSWOR scheme8. A generalized exponential estimator for estimating population variance using two auxiliary variables was proposed by9. Demonstrates efficiency through empirical and simulated studies using real and simulated data. Has suggested efficient formulation of population variance in simple random sampling using supplementary variable10. Using searl’s constants, develop an efficient estimator to estimate the population variance11. The bias and mean squared error of the proposed estimator is obtained up to the first degree of approximation. Suggested a chain ratio type and chain ratio type exponential estimator of variance of finite population using auxiliary information Improved version of the suggested class of estimators is also given along with its properties12. An empirical study is carried out in support of the findings of the study. Addresses estimation of current population variance in the presence of random non-response13. Examines proposed estimators through empirical studies, comparing with estimators for complete response situations. Suggested a class of estimators for finite population variance using an auxiliary attribute14. Developed variance estimator using the tri-mean and third quartile of the auxiliary variable, demonstrating its superior performance over various competing estimators based on sampling properties, bias, and mean squared error15. Suggested finite population variance estimator using unconventional measures16. Demonstrates efficacy and robustness through empirical and simulation studies considering real data sets form various domain of life. Similarly,17 explores the effect of distribution on suggested variance estimators. Compares twelve estimators across eight distributions through simulation studies. Formulate a Searl’s ratio-type estimator using tri-mean and third quartile of the auxiliary variable18. Demonstrates superiority through bias, mean squared error through theoretical comparison and empirical studies. A hybrid-type estimators of population variance developed by19, and demonstrated the efficiency over competing estimators through theoretical and empirical comparisons. A generalized family of estimators of population variance is formulated by20 and demonstrated the performance of the estimators through empirical and simulation study. A robust ratio-type estimator for finite population variance suggested by21, considering robust covariance matrices. Derive conditions for efficiency against competing estimators and demonstrated the performance of the suggested estimators through empirical and simulation study.

Novelty and significance

This work introduces innovative contributions in the field of survey sampling, with several key aspects. The paper puts forward three novel ratio estimators designed for finite population variance. These estimators consider a transformed auxiliary variable under simple random sampling without replacement. Demonstrated to outperform existing methods, these suggested estimators exhibit superior efficacy when there is a positive correlation between the survey variable and auxiliary variable. In addition to theoretical comparisons of mean squared errors (MSEs), the paper includes an empirical analysis using real data sets. Through a simulation study, it substantiates that the newly proposed estimators consistently outperform competing estimators across various scenarios, such as different correlation and sample size, affirming their high efficiency. The versatility of the proposed estimators is highlighted as they can seamlessly adapted into other sampling methodologies. This includes applications in stratified random sampling, non-response sampling, and adaptive cluster sampling, leading to the derivation of efficient versions of the estimators. The applicability of the proposed estimators extends to diverse fields such as environmental studies, agriculture, and economics, especially in situations where a positive correlation between the study and auxiliary variable exists. The heightened efficiency of these estimators can significantly enhance the accuracy of population variance estimates, offering practical implications for decision-making processes in real-world applications. The proposed estimator can significantly contribute in the estimation of parameters other than variance, such as mean, median, coefficient of variation etc.

In conclusion, this paper significantly contributes to the field of survey sampling by introducing novel estimators for finite population variance. The adaptability of these estimators to various sampling schemes and their practical implications underscores their potential impact on enhancing accuracy in real-world decision-making.

Methodology

Consider a population Ω=yi,xi,i=1,2,...N. of size N. Suppose a random sample yi,zi of size n is taken from a population under simple random sampling without replacement case, i-e (SRSWOR). Let yi,xi be the value of ith unit of the main study and auxiliary variable and zi=txi is the transformed auxiliary variable observed on the sample. The supplementary variate x is supposed to be correlated positively with the main study variable y. It is to be noted that the correlation between yi,zi and between yi,xi is same. Let y¯=1n∑i=1nyi, x¯=1n∑i=1nxiandz¯=1n∑i=1nzi represents the sample mean of study variable, ancillary variable and transformed ancillary variable.

Y¯=1N∑i=1Nyi and X¯=1N∑i=1NxiandZ¯=1N∑i=1Nzi be the population mean of variate y,x and z.

Let us define the random error due to sampling by ε0=sy2-Sy2Sy2 and ε1=sx2-Sx2Sx2, such that1 Eε0=Eε1=0andEε02=λϕ40-1=V40Eε12=λϕ04-1=V04Eε0ε1=λϕ22-1=V22whereλ=1/n,ϕrs=μrsμ20r/2μ02s/2μrs=1N∑i=1N(yi-Y¯)r(xi-X¯)s.

where r and s be the non-negative integer and μrs,μ20andμ02 are the second order moments and ϕrs is the moment’s ratio.2 Sy2=1N-1∑i=1NYi-Y¯2,sy2=1n-1∑i=1n(yi-y¯)2Sx2=1N-1∑i=1NXi-X¯2,Sy2=1N-1∑i=1N(xi-x¯)2.

where Cy2,Cx2 are the coefficient of variation of the survey variable and auxiliary variables Y and X respectively.ρyx is the correlation coefficient between main study variable Y and auxiliary variable X, and β(1)x,β(2)x are the coefficient of skewness and the coefficient of kurtosis of the auxiliary variables respectively.

In literature, some estimators of the population variance are given asThe usual classical estimator of population variance is given by3 τ1=sy2=1n-1∑i=1nyi-y¯2,

where τ1 is an unbiased estimator. Its variance is as under4 Varτ1=Sy4V40.

Developed the estimator of population variance, it is given by25 τ2=sy2Sx2sx2,

The MSE of τ2 is as below.6 MSEτ2≅Sy4V40+V04-2V22

3. Provide the estimator of population variance, it is given below227 τ3=sy2expSx2-sx2Sx2+sx2,

The MSE of τ3 is given by8 MSEτ3≈Sy4V40+V044-V22.

4. For ratio estimator of population variance, the linear regression estimator developed by2 is given by9 τ4=sy2+bSx2-sx2,

here b=sy2V22sx2V40 represents the sample regression coefficient.

The variance of τ4 is as below10 Varτ4=Sy4V401-V222V40V04.

5. The transformed estimator by23 is as follows:11 τ5=k1sy2+k2Sx2-sx2θaSx2+basx2+b+1-θexpaSx2-sx2aSx2+sx2+2b,

here k1 and k2 are optimization constants, θ is generalization constants which can takes value between 0 and 1 and “a” and “b” some function of auxiliary variable. The optimum value of k1 and k1 are as belowk1opt=1-18q21+3θ+4θ2V041-14q2θ1+3θV042+V401-V222V04V40

andk2opt=Sy2Sx21+q1+θ+k1optV22V04-q1+θ,

The minimum MSE at k1opt and k2opt is given by12 MSEτ5≃Sy41-14q21+θ2V04-1-18q21+3θ+4θ2V0421-14q2θ1+3θV042+V401-V222V04V40.

where q=aSx2aSx2+b.

The MSE of τ is minimum at θ,a,b=1,1,0.6. Introduced the following difference-cum-exponential estimator of population variance7,13 τ6=c11sy2+c22Sx2-sx2expSx2-sx2Sx2+sx2,

where c11 and c11 are optimization constants. The optimum MSE of τ6 is given by14 MSEτ6=MSEτ4-Sy4V40-1+8V041-V222V40V04641+V40-V222V04.

7. Developed the following ratio estimator of population variance2415 τ7=sy2k11Sx2sx2+k12sx2Sx2expSx2-sx2Sx2+sx2

The optimum value of k11andk12 are given byk11OPT=-18A1V40+V04+A2+16V22A3

k12OPT=18A1V40+21V04-3A2+16V22A3,

whereA1=16V04-V22,A2=83V04V22-2V222+V04andA3=V042-4V04V22-16V04V22+16V222-4V04.

The minimum MSE at k11optandk12opt is given by16 MSEτ7min≅Sy4A311616V22V224V40-3V04+4+3V042-8V40V04+9V043+64V04V04-1V40.

8. Advocated the following difference-cum-exponential type exponential estimators of population variance given by2517 τ8=sy22expSx2-sx2Sx2+sx2+expsx2-Sx2Sx2+sx2+k1Sx2-sx2+k2sy2expSx2-sx2Sx2+sx2,

18 τ9=sy2αexpSx2-sx2Sx2+sx2+1-αexpsx2-Sx2Sx2+sx2+k11Sx2-sx2+k22sy2δexpSx2-sx2Sx2+sx2+1-δexpsx2-Sx2Sx2+sx2,

where k1 and k2 are optimization constants that minimize the MSE of τ8.which is given by19 MSEτ8≅MSEτ4-Sy4V40-V222V04+14V0421+V40-V222V04.

20 MSEτ9≅Sy4V40+ΩΩV04+2V22-D1D52+D2D42-2D3D4D5D1D2-D32.

D1=R2V04,D2=1+V04+2V221-2ω+ω2V04,D3=RV22+1-2ωV04,D4=RV22+ΩV04

andD5=V40+2Ω+1-2ω2V22+αω+1-2ω2V04.

9. Suggested a general type of estimator of population variance given by2021 τa,b=t^a,b+kSx2-sx2expw1X¯-x¯X¯+ω1-1x¯+w2Sx2-sx2Sx2+ω2-1sx2,

The particular case of estimator τ9a,b for estimating variance is obtained by putting a = 0, b = 2,w1=0andw2=1 and ω2 =  2 as following22 τ9=sy2+kSx2-sx2expSx2-sx2Sx2+sx2,

With MSE given by23 MSEtmin=MSEt^(a,b)-t(a,b)2nf3a,b2-2f2a,bf3a,bδ03+δ04-1f2a,b2δ04-δ032-1

wheref2a,b=aρXYCγ+b2δ21f3a,b=aρ12Cγ+b2δ22-1.

Which in case of variance estimator, the MSE τ will induce the following particular case:24 MSEτ9=MSEτ1-f30,22-2f20,2V40+V04-1f20,22V04-V402-1

Proposed estimators

The first proposed estimator

Motivated by24 and using transformed auxiliary variable, the following class of transformed ratio-product type exponential estimator is suggestedorτP1=sy2c1Zz+c2zZexpZ-zZ+z

25 τP1=sy2c1γ1Sx2-γ2γ1sx2-γ2+c2γ1sx2-γ2γ1Sx2-γ2expγ1Sx2-sx2γ1Sx2+sx2-2γ2

where c1 and c2 are optimization constants and γ2 and γ2 are suitable constants or some function of auxiliary variables.

The second proposed estimator

Motivated by2, we can write the proposed estimator as a linear combination of usual ratio and exponential estimators as followingOrτP2=ψ1τ2+ψ2τ3

26 τP2=sy2ψ1Sx2sx2+ψ2expSx2-sx2Sx2+sx2

ψ1, and ψ2 are optimization constants whose value is to be obtained so that the MSE of τP2 is minimum.

The third proposed estimator

Applying transformation to the auxiliary variable in (26), we can write the third proposed estimator as following27 τP3=sy2ψ3Zz+ψ4expZ-zZ+z

OrτP3=sx2ψ3γ1Sx2-γ2γ1sx2-γ2+ψ4expγ1Sx2-sx2γ1Sx2+sx2-2γ2

ψ3 and ψ4 are optimization constants whose value is to be determined so that the MSE of τP3 is minimum. It is to be noted that for γ1=1andγ2=0.

The third proposed estimator τP3 given by (26) become equivalent to the second proposed estimator τP2 given by (26).

Similarly, the first proposed estimator τP1 given by (25) become equivalent to τP7 as suggested by Muneer et al.9 given by (13).

Theoretical properties of the proposed estimators

This unit aims at, deriving the theoretical properties of the new estimators using the notations given in (1) and (2). Rewriting (25), (26) and (27) respectively in term of error terms, as followingτP1=Sy21+ε0c11-πkε1+πk2ε12+⋯+c21+πkε11-12πkε1+38πk2ε12+⋯,

τP2=Sy21+ε0ψ11-ε1+ε12+⋯+ψ21-12ε1+38ε12+⋯,

andτP3=Sy21+ε0ψ31-πkε1+πk2ε12+⋯+ψ41-12πkε1+38πk2ε12+⋯,

where πk=γ1Sx2γ1Sx2-γ2. or28 τP1-Sy2≅Sy2c11+ε0-πkε1-32πkε1ε0+158πk2ε12+c21+ε0+12πkε1+12πkε1ε0-18πk2ε12-1

29 τP2-Sy2≅Sy2ψ1+ψ2-1+ψ1+ψ2ε0-ψ1+ψ22ε1+ψ1+3ψ28ε12-ψ1+ψ22ε0ε1,

and30 τP3-Sy2≅Sy2ψ3+ψ4-1+ψ3+ψ4ε0-πkψ3+ψ42ε1+ψ3+3ψ48πk2ε12-πkψ4+ψ42ε0ε1,

Taking expectation of both sides of (28), (29) and (30) respectively, and after simplification we get31 BiasτP1≅Sy2c1+c2-1+c1λ158πk2ϕ04-1-12πkϕ02-1c1λπk12πkϕ02-1-18πkϕ04-1,

32 BiasτP2≅Sy2ψ1+ψ2-1+ψ1+3ψ28λϕ04-1-λψ1+ψ22ϕ22-1,

33 BiasτP3≅Sy2ψ3+ψ4-1+ψ3+3ψ48πk2λϕ04-1-πkλψ3+ψ42ϕ22-1,

Squaring both sides of (31), (32) and (33) respectively and applying expectation to get the MSE of τP1,τP2andτP3, as following.34 EτP1-Sy22≅Sy4Ec11+ε0-πkε1-32πkε1ε0+158πk2ε12+c21+ε0+12πkε1+12πkε1ε0-18πk2ε12-12,

35 EτP2-Sy2≅Sy4Eψ1+ψ2-1+ψ1+ψ2ε0-ψ1+ψ22ε1+ψ1+3ψ28ε12-ψ1+ψ22ε0ε12,

or36 EτP3-Sy2≅Sy4Eψ3+ψ4-1+ψ3+ψ4ε0-πkψ3+ψ42ε1+ψ3+3ψ48πk2ε12-πkψ3+ψ42ε0ε12,

or37 MSEτP1≅Sy4c1+c2-12+c1+c22V40+123c1-c22+2c1+c2-1158c1-18c2πk2V04-2c1+c2-13c1-c2πkV22,

38 MSEτP2≅Sy4ψ1+ψ2-12+ψ1+ψ22V40+ψ1+ψ222+2ψ1+ψ2-1ψ1+3ψ28V04-2ψ1+ψ222ψ1+ψ2-1V22

and39 MSEτP3≅Sy4ψ3+ψ4-12+ψ3+ψ42V40+ψ3+ψ422+2ψ3+ψ4-1ψ3+3ψ48πk2V04-2ψ3+ψ422ψ3+ψ4-1πkV22.

The optimum value of c1 and c2 is obtain by differentiating (37) w.r.t c1andc2 respectively and equating to zero, as following∂∂c1MSEτP1= 0⇒c1OPT=-18A1,kV40+V04,k+A2,k+16V22,kA3,k,

∂∂c2MSEτP1= 0⇒c2OPT=18A1,kV40+21V04,k-3A2,k+16V22,kA3,k.

where,A1,k=16V04,k-V22,k,A2,k=83V04,kV22,k-2V22,k2+V04,k,andA3,k=V04,k2-4V04,kV22,k-16V04,kV22,k+16V22,k2-4V04,k.

putting the optimum value of c1andc2 in (37), we get40 MSEτP1min≅Sy4A3,k11616V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40.

whereV04,k=πk2V04andV22,k=πkV22

Similarly, using calculus rule, the optimum value of ψ1,ψ2,ψ3andψ4 can also be obtain by differentiating MSE of τP2andτP3 w.r.to ψ1,ψ2,ψ3andψ4 and equating to zero. Hence, we obtain after simplificationψ1opt=-19V044+40V042V402-60V042V22-32V402V22+32V222-16V042+32V22D,

ψ2opt=86V044+5V042V402-15V042V22-4V402V22+8λV222-4V042+4V22D,

ψ3opt=-19V04,k4+40V04,k2V402-60V04,k2V22,k-32V402V22,k+32V22,k2-16V04,k2+32V22,kDk

andψ4opt=86V04,k4+5V04,k2V402-15V04,k2V22,k-4V402V22,k+8λV22,k2-4V04,k2+4V22,kDk

whereD=33V044-16V042V402-80V042V22+64V222-16V042Dk=33V04,k4-16V04,k2V402-80V04,k2V22,k+64V22,k2-16V04,k2,k=1,2,…,6.

The MSE is given by41 MSEtP2=Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222

42 MSEτP3=Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2

Special Cases of τP1andτP3 in response to the transformation introduced

For k = 1, τP1andτP3 takes the following formτP1,1=sy2c1Sx2-ρyxsx2-ρyx+c2sx2-ρyxSx2-ρyxexpSx2-sx2Sx2+sx2-2ρyx

τP3,1=sy2ψ3Sx2-ρyxsx2-ρyx+ψ4expSx2-sx2Sx2+sx2-2ρyx.

where γ1=1andγ2=ρyx

The bias and MSEs are given by43 BiasτP1,1=Syh2c1+c2-1+32c1-12c2V22,1+158c1-18c2V04,1,

44 BiasτP3,1≅Sy2ψ3+ψ4-1+ψ3+3ψ48V04,1-ψ1+ψ22V22,1

45 MSEτP1,1min≅Sy4A3,111616V22,1V22,14V40-3V04,1+4+3V04,12-8V40V04,1+9V04,13+64V04,1V04,1-1V40

and46 MSEτP3,1=Sy4D1V04,16+25V402-10V22,12V04,14+V04,128V22,1-40V22,1V402+16V22,12V402-16V04,12V402+16V22,12.

V04,1=π12V04andV22,1=π1V22, π1=Sx2Sx2-ρyx.2. For k = 2, γ1=1andγ2=Cx, τP1andτP3 will take the following formτP1,2=sy2c1Sx2-Cxsx2-Cx+c2sx2-CxSx2-CxexpSx2-sx2Sx2+sx2-2Cx.

τP3,2=sy2ψ3Sx2-Cxsx2-Cx+ψ4expSx2-sx2Sx2+sx2-2Cx.

The bias and MSEs are given by47 BiasτP1,2=Syh2c1+c2-1+32c1-12c2V22,2+158c1-18c2V04,2,

48 BiasτP3,2≅Sy2ψ3+ψ4-1+ψ3+3ψ48V04,2-ψ1+ψ22V22,2.

49 MSEτP1,2min≅Sy4A3,211616V22,2V22,24V40-3V04,2+4+3V04,22-8V40V04,2+9V04,23+64V04,2V04,2-1V40

and50 MSEτP3,2=Sy4D2V04,26+25V402-10V22,22V04,24+V04,228V22,2-40V22,2V402+16V22,22V402-16V04,22V402+16V22,22.

where V04,2=π22V04andV22,2=π2V22,π2=Sx2Sx2-Cx.3. For k = 3, γ1=ρyxandγ2=Cx,τP1andτP3 will take the following formτP1,3=sy2c1ρyxSx2-Cxρyxsx2-Cx+c2ρyxsx2-CxρyxSx2-CxexpρyxSx2-sx2ρyxSx2+sx2-2Cx

τP3,3=sy2ψ3ρyxSx2-Cxρyxsx2-Cx+ψ4expρyxSx2-sx2ρyxSx2+sx2-2Cx.

The bias along with the mean square error (MSE) is given by51 BiasτP1,3=Sy2c1+c2-1+32c1-12c2V22,3+158c1-18c2V04,3,

52 BiasτP3,3≅Sy2ψ3+ψ4-1+ψ3+3ψ48V04,3-ψ1+ψ22V22,3

53 MSEτP1,3min≅Sy4A3,311616V22,3V22,34V40-3V04,3+4+3V04,32-8V40V04,3+9V04,33+64V04,3V04,3-1V40

and54 MSEτP3,3=Sy4D3V04,36+25V402-10V22,32V04,34+V04,328V22,3-40V22,3V402+16V22,32V402-16V04,32V402+16V22,32.

where V04,3=π32V04andV22,3=π3V22,π3=ρyxSx2ρyxSx2-Cx.4. For k = 4, γ1=Cxandγ2=ρyx. the estimators τP1andτP3 will take the following formτP1,4=sy2c1CxSx2-ρyxCxsx2-ρyx+c2Cxsx2-ρyxCxSx2-ρyxexpCxSx2-sx2CxSx2+sx2-2ρyxτP3,4=sy2ψ3CxSx2-ρyxCxsx2-ρyx+ψ4expCxSx2-sx2CxSx2+sx2-2ρyx.

The bias and MSEs are obtained as55 BiasτP1,4=Sy2c1+c2-1+32c1-12c2V22,4+158c1-18c2V04,4,

56 BiasτP3,4≅Sy2ψ3+ψ4-1+ψ3+3ψ48V04,4-ψ1+ψ22V22,4

57 MSEτP1,4min≅Sy4A3,411616V22,4V22,44V40-3V04,4+4+3V04,42-8V40V04,4+9V04,43+64V04,4V04,4-1V40

and58 MSEτP3,4=Sy4D4V04,46+25V402-10V22,42V04,44+V04,428V22,4-40V22,4V402+16V22,42V402-16V04,42V402+16V22,42.

where V04,1=π12V04andV22,1=π1V22,π4=CxSx2CxSx2-ρyx.5. For k = 5, γ1=X¯andγ2=ρyx, the estimators τP1andτP3 will take the following formτP1,5=sy2c1X¯Sx2-ρyxX¯sx2-ρyx+c2X¯sx2-ρyxX¯Sx2-ρyxexpX¯Sx2-sx2X¯Sx2+sx2-2ρyxτP3,5=sy2ψ3X¯Sx2-ρyxX¯sx2-ρyx+ψ4expX¯Sx2-sx2X¯Sx2+sx2-2ρyx.

The bias and MSEs are given by59 BiasτP1,5=Sy2c1+c2-1+32c1-12c2V22,5+158c1-18c2V04,5,

60 BiasτP3,5≅Sy2ψ3+ψ4-1+ψ3+3ψ48V04,5-ψ1+ψ22V22,5

61 MSEτP1,5min≅Sy4A3,511616V22,5V22,54V40-3V04,5+4+3V04,52-8V40V04,5+9V04,53+64V04,5V04,5-1V40

and62 MSEτP3,5=Sy4D5V04,56+25V402-10V22,52V04,54+V04,528V22,5-40V22,5V402+16V22,52V402-16V04,52V402+16V22,52.

where V04,5=π52V04andV22,5=π5V22 and π5=X¯Sx2X¯Sx2-ρyx.6. For k = 6, γ1=ρyxandγ2=1

τP1,6=sy2c1ρyxSx2-1ρyxsx2-1+c2ρyxsx2-1ρyxSx2-1expρyxSx2-sx2ρyxSx2+sx2-2τP3,6=sy2ψ3ρyxSx2-1ρyxsx2-1+ψ4expρyxSx2-sx2ρyxSx2+sx2-2.

The bias and MSEs are given by63 BiasτP1,6=Sy2c1+c2-1+32c1-12c2V22,6+158c1-18c2V04,6,

64 BiasτP3,6≅Sy2ψ3+ψ4-1+ψ3+3ψ48V04,6-ψ1+ψ22V22,6

65 MSEτP1,6min≅Sy4A3,611616V22,6V22,64V40-3V04,6+4+3V04,62-8V40V04,6+9V04,63+64V04,6V04,6-1V40

and66 MSEτP3,6=Sy4D6V04,66+25V402-10V22,62V04,64+V04,628V22,6-40V22,6V402+16V22,62V402-16V04,62V402+16V22,62.

where V04,6=π62V04andV22,6=π6V22w and π6h=ρhSwxh2ρhSwxh2-1. where V04,6=π62V04andV22,6=π6V22, π6=Sx2Sx2-1.

Efficiency comparisons

This section aims to compare the MSEs of the newly developed estimators with the competing estimators discussed in the literature.

Condition 1. By using (37), (38), (39) and (4) we can writeMSEτPi-Varτ1<0,i=1,2,3.

For i=1Sy4A3,k11616V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-16A3,kV40<0

For i=2Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DV40<0.

For i=3Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkV40.

Condition 2. By using (37), (38), (39) and (6) we can write

MSEτPi-MSEτ2<0,i=1,2,3.

For i = 1 Sy4A3,k11616V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-16A3,kV40+V04-2V22<0.

For i = 2 Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DV40+V04-2V22<0.

For i = 3 Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkV40+V04-2V22<0.

Condition 3. By using (37), (38), (39) and (8) we can write.

MSEτPi-MSEτ3<0, i = 1,2,3.

For i = 1 Sy4A3,k11616V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-16A3,kV40+V044-V22<0.

For i = 2 Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DV40+V044-V22<0.

For i = 3 Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkV40+V044-V22<0.

Condition 4. By using (37), (38), (39) and (10) we can write.

MSEτPi-MSEτ4<0, i = 1,2,3.

For i = 1 Sy4A3,k11616V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-16A3,kV401-V222V40V04<0.

For i = 2 Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DV401-V222V40V04<0.

For i = 3 Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkV401-V222V40V04<0.

Condition 5. By using (37), (38), (39) and (12) we can write

MSEτPi-MSEτ5<0, i = 1,2,3.

For i = 1 Sy4A3,k11616V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-16A3,k1-14q21+θ2V04-1-18q21+3θ+4θ2V0421-14q2θ1+3θV042+V401-V222V04V40<0.

For i = 2 Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-D1-14q21+θ2V04-1-18q21+3θ+4θ2V0421-14q2θ1+3θV042+V401-V222V04V40<0.

For i = 3 Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-Dk1-14q21+θ2V04-1-18q21+3θ+4θ2V0421-14q2θ1+3θV042+V401-V222V04V40<0.

Condition 6. By using Eqs. (37), (38), (39) and (14) we can write

MSEτPi-MSEτ6<0, i = 1,2,3.

For i = 1. 16V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-A3,k16MSEτ4-V40-1+8V041-V222V40V0441+V40-V222V04<0.

For i = 2.V046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DMSEτ4-V40-1+8V041-V222V40V04641+V40-V222V04<0.

For i = 3 V04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkMSEτ4-V40-1+8V041-V222V40V04641+V40-V222V04<0..

Condition 7. By using (37), (38), (39) and (16) we can write

MSEτPi-MSEτ7<0, i = 1,2,3.

For i = 1. 16V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-A3,kA316V22V224V40-3V04+4+3V042-8V40V04+9V043+64V04V04-1V40<0.

For i = 2.V046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-D16A316V22V224V40-3V04+4+3V042-8V40V04+9V043+64V04V04-1V40<0.

For i = 3. V04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-Dk16A316V22V224V40-3V04+4+3V042-8V40V04+9V043+64V04V04-1V40<0..

Condition 8. By using (37), (38), (39) and (19), we can write

MSEτPi-MSEτ8<0, i = 1,2,3.

For i = 1. 16V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-A3,kMSEτ4Sy4-V40-V222V04+14V0421+V40-V222V04<0.

For i = 2.V046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DMSEτ4Sy4-V40-V222V04+14V0421+V40-V222V04<0.

For i = 3 V04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkMSEτ4Sy4-V40-V222V04+14V0421+V40-V222V04<0..

Condition 9. By using (37), (38), (39) and (20), we can write

MSEτPi-MSEτ9<0, i = 1,2,3.

For i = 1 Sy4A3,k16V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-A3,kV40+ΩΩV04+2V22-D1D52+D2D42-2D3D4D5D1D2-D32<0.

For i = 2 Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-DV40+ΩΩV04+2V22-D1D52+D2D42-2D3D4D5D1D2-D32<0.

For i = 3 Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-DkV40+ΩΩV04+2V22-D1D52+D2D42-2D3D4D5D1D2-D32<0.

Condition 10. By using (37), (38), (39) and (24), we can write

MSEτPi-MSEτ10<0, i = 1,2,3.

For i = 1 Sy4A3,k16V22,kV22,k4V40-3V04,k+4+3V04,k2-8V40V04,k+9V04,k3+64V04,kV04,k-1V40-A3,k1-f30,22-2f20,2V40+V04-1f20,22V04-V402-1<0.

For i = 2 Sy4DV046+25V402-10V222V044+V0428V22-40V22V402+16V222V402-16V042V402+16V222-D1-f30,22-2f20,2V40+V04-1f20,22V04-V402-1<0.

For i = 3 Sy4DkV04,k6+25V402-10V22,k2V04,k4+V04,k28V22,k-40V22,kV402+16V22,k2V402-16V04,k2V402+16V22,k2-Dk1-f30,22-2f20,2V40+V04-1f20,22V04-V402-1<0.

The condition mentioned above always hold true for all type of real data where the correlation is positive between the main study variable and supplementary variable.

Empirical analysis

This section aims to investigate the performance of the proposed estimators against the competing estimators using data from some real-life situations. Table 1 consists of summary statistics of various datasets.Table 1 Summary statistics of various data sets.

Summary statistics	Population I:
Source26
Y: Amount (tons) of recyclable waste collection in Italy in 2003
X:Number of inhabitants in 2003	Population II: Source23
Y: The leaf area for the newly developed strain of wheat X: Weight of leaves	Population III: Source27
Y: weekly expenditure of food
X: size of persons	Population IV:
Source28
Y: Total amount of recyclable-waste collection in Italy (2003) X: Total amount of recyclable-waste collection in Italy (2002)	
Y¯	51.82	26.84	27.49091	62.62	
X¯	11.26	106.20	3.7272	556.55	
N	80	39	33	103	
n	20	14	9	40	
Sy	18.35	6.24	9.976094	91.35	
Sx	8.45	11.14	1.500198	610.16	
ϕ40	2.26	2.26	5.72	37.12	
ϕ04	2.86	2.99	2.380	17.87	
ϕ22	2.22	2.40	1.43	17.22	
ρyx	0.94	0.93	0.4237	0.72	

The percentage relative efficiency (PRE) of all estimators discussed in the literature against τ1 has been used as performance index. The formula of PRE is given below67 PRE=Varτ1MSEτiorMSEτPj,k×100

where, i=2,3,⋯,10. J = 1,2,3. and k = 1,2 ⋯,6. are denoted by existing estimators and proposed estimators.

It is obvious from the Table 2 that the newly transformed estimators always perform well than existing estimators for all real data sets. The transformation introduced results high gain in efficiency. The first proposed estimator given by (25) is more efficient than the parent estimator suggested by24, moreover it also outperforms all the competing estimators discussed in the literature. The second proposed estimator given by (26) is more efficient even than our first proposed estimators and all other competing estimators. Further, incorporating the transformed auxiliary variable in the second proposed estimator generates the third proposed estimators given by (27) which are more efficient than all other estimators.Table 2 PRE of proposed and competing estimators against the usual estimator τ1.

Estimator	Population no.	
1	2	3	4	
Conventional estimators	τ1	100.000	100.000	100.000	100.000	
τ2	185.2941	280.000	102.4253	175.7390	
τ3	249.5049	352.4475	101.1806	149.7570	
τ4	274.0411	458.0562	102.9216	175.9545	
τ5	195.2023	290.7025	124.2060	183.5658	
τ6	289.34199	473.2631	153.7084	257.5825	
τ7	336.50648	663.02559	159.1806	178.7570	
τ8	302.428041	584.1095	169.9411	196.2725	
τ9	236.64728	203.75595	102.7552	115.8661	
τ10	269.8926	430.5612	101.5689	171.9047	
Proposed estimators	τP11	339.83799	671.81178	171.4423	278.0721	
τP13	339.60901	664.96302	174.8473	279.5321	
τP16	341.674241	674.76219	169.2539	296.2645	
τP2	455.9280	1312.5474	218.2653	4059.0256	
τP32	456.2014	1421.83896	220.6593	4066.6241	
τP34	457.8603	1340.99191	216.2530	4080.5010	
τP35	278.8679	1314.1116	245.8746	4075.9032	

Simulation study

The simulation study of the suggested and existing estimators is conducted to assess the performance of both suggested and existing estimators. Three different populations of size 10,000 have been generated using positive correlation between main study and auxiliary variables. The intensity of correlation between the main study and the supplementary variable is high, moderate, and low in the first, the second and the third population respectively.

Population 1 μ=5.010.0 ∑=9.02.92.91.0ρ=0.9665.

Population 2 μ=5.010.0 ∑=10.03.03.01.5ρ=0.7559.

Population 3 μ=5.010.0 ∑=10.01.51.51.5ρ=0.3983.

We consider sample of sizes n = 50, 150, 300 are consider for each population, using simple random sampling without replacement approach. The steps below summarize the whole simulation procedure in R-Studio.

Step 1: Population is generated using Bivariate normal distribution with mean vector μ=c5.010.0 and ∑=9.02.92.91.0 and calculated all parameters of auxiliary variable and constant from the data.

Step 2: From each of the target population, samples of n =  50, 150, 300 have been drawn using SRSWOR method respectively, allow the loop to 100,000 times, select the sample, and calculated the estimates in each iteration.

Step 3: Utilizing samples generated in step 2, the 100,000 values of τi,i=1,2,..10. and τPjk,j=1,2,3andk=1,2,..,6., are obtained separately, using (4–20) and (23), (34) and (35), respectively.

Step 4: The estimates obtained from each iteration are store in a matrix and calculate the percent PRE averaging over all iteration of each estimates by using the formula (67), and report the results in Table 3.Table 3 Simulation results for the percent relative efficiencies of the newly developed estimators and conventional estimators with respect to the usual estimator considering different sample sizes and correlation coefficients.

Estimators	Population	
1	2	3	
n = 50	n = 150	n = 300	n = 50	n = 150	n = 300	n = 50	n = 150	n = 300	
τ1	100.00	100.000	100.00	100.00	100.00	100.00	100.00	100.00	100.00	
τ2	195.36	198.9839	201.47	101.55	102.46	100.37	104.84	102.95	103.30	
τ3	93.38	97.6381	101.63	110.53	104.32	102.56	103.45	101.36	102.49	
τ4	212.54	209.3710	218.63	115.63	108.37	111.35	105.25	108.54	106.26	
τ5	223.58	201.2898	219.35	123.92	119.73	113.27	110.47	107.16	108.57	
τ6	235.64	208.7391	202.63	120.46	123.18	115.27	115.65	107.16	110.75	
τ7	238.29	230.4728	224.38	139.47	130.36	128.38	112.59	110.47	108.73	
τ8	235.73	202.3729	223.24	130.28	121.46	119.37	110.45	108.38	106.39	
τ9	232.27	228.4732	214.73	137.47	128.56	126.18	107.38	106.75	106.95	
τ9	198.76	203.2417	215.76	119.78	108.45	110.56	104.24	107.96	104.22	
τP1,1	241.47	241.66	240.63	141.46	138.86	131.25	119.27	111.45	112.45	
τP1,4	242.45	240.25	238.58	140.48	132.45	131.57	115.26	110.47	110.46	
τP1,6	241.35	235.76	226.65	139.73	131.46	129.45	112.46	11.52	108.98	
τP2	260.85	262.37	275.46	185.36	189.83	190.89	135.85	136.86	141.86	
τP3,1	262.46	265.77	275.85	191.92	195.53	197.04	136.95	139.18	140.47	
τP3,3	265.66	266.75	276.85	195.84	195.95	198.90	140.43	142.65	145.96	
τP3,5	262.65	263.75	275.75	186.53	189.54	190.66	136.06	136.93	142.47	
τP3,6	271.46	270.01	276.66	197.59	197.49	198.45	140.94	142.95	147.88	

The simulation results indicate that the newly developed estimators are highly efficient as compared to the conventional estimators in all situations. It is obvious that for the high correlation ρ≥0.96, the estimators are tend to be more efficient than moderate ρ≥0.75 and low ρ≥0.39 correlation. Thus, the intensity of relationship between the study and supplementary variable plays a vital role in increasing the efficiency. The newly developed estimator retains its efficiency in all three cases. Further, the transformation also plays a remarkable role in enhancing efficiency. As it can be seen from the simulation in the Table 3, the first transformed proposed estimator is efficient than its parent estimator τ7 for different choices of γ1andγ2. Similarly, the third proposed estimator is also efficient not only from the conventional estimators but also from the second proposed estimator for different choices of γ1andγ2.

Conclusion

The empirical and simulation findings highlight the remarkable efficiency of the newly developed estimators in comparison to conventional counterparts across diverse scenarios. Particularly noteworthy is the enhanced efficiency of these estimators in situations with higher correlation, emphasizing the pivotal role of the strength of the relationship between the study and supplementary variable. This aligns with the expectation that a stronger correlation contributes to increased estimator efficiency.

Furthermore, the robustness of the newly developed estimator is evident across all correlation levels—high, moderate, and low. This consistent performance underscores the versatility and reliability of the proposed estimators, making them applicable in a broad spectrum of scenarios.

The introduced transformation in the estimators emerges as a key contributor to enhanced efficiency. As depicted in Table 3, simulation results affirm that the first transformed proposed estimator consistently outperforms its parent estimator across various choices. Similarly, the third proposed estimator not only demonstrates efficiency over conventional estimators but also surpasses the second proposed estimator for different choices. This emphasizes the significant role played by transformations in elevating the overall performance of the estimators.

Additionally, the proposed estimators exhibit flexibility and effectiveness in various sampling designs, such as stratified random sampling, non-response sampling, and adaptive cluster sampling. The extension of these estimators to non-conventional sampling designs, including adaptive cluster sampling and stratified adaptive cluster sampling, is also under consideration for variance estimation. These estimators display flexibility in exploring potential improvements in formulating estimates of population parameter utilizing two auxiliary variables.

In conclusion, the proposed estimators shows remarkable efficiency in finite population variance estimation under the simple random sampling scheme without replacement. The encouraging findings suggest their applicability in diverse survey scenarios, and future research avenues could further enhance their adaptability, extending their utility to more intricate sampling designs.

Author contributions

Hameed Ali: conceptualization, validation, investigation, data curation, methodology, writing, review & editing, formal analysis, visualization, original draft preparation. Syed Muhammad Asim: supervision, methodology, conceptualization, validation, review & editing, data curation. Muhammad Ijaz: supervision, validation, review & editing. Tolga Zaman: supervision, validation, review & editing. Soofia Iftikhar: supervision, validation, review & editing.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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