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

Authors: Showkat Ahmad Lone; Mir Subzar; Ankita Sharma
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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 populationZ=Z1,Z2,…,ZM of M units and let z,r be the study and auxiliary variables defined on Z taking values zi,ri, respectively, on Zii=1,2,…,M. It is desired to estimate the population variance Wz2 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.Z¯=M−1∑i=1Mzi: the population mean of the study variable zR¯=M−1∑i=1Mri: the population mean of the auxiliary variable rWz2=M−1−1∑i=1Mzi−Z¯2: the population mean square/variance of the study variable zWr2=M−1−1∑i=1Mri−R¯2: the population mean square/variance of the auxiliary variable rTM=Q1+2Q2+Q3/4: tri-mean of the auxiliary variable rHL=medianRj+Rk/2,1≤j≤k≤M: Hodges–Lehmann estimator of the auxiliary variable rMR=R1+RM/2: mid-range of the auxiliary variable rG=4/M−1∑i=1M2i−M−1/2MRi: Gini’s mean difference of the auxiliary variable rD=2π/MM−1∑i=1Mi−M+1/2Ri: Downton’s method of the auxiliary variable rSpw=π/M2∑i=1M2i−M−1Ri: probability weighted moments of the auxiliary variable r:(1)μ40∗=μ40−1,μ04∗=μ04−1,μ22∗=μ22−1,where μrs=λbc/λ02c/2λ20b/s,λrs=M−1∑i=1Mzi−Z¯bri−R¯c,andb,c 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 as(2)l=α1wz2+α2Wr2−wr2γWr2+φQ2γwr2+φQ2,where γWr2+φQ2>0,γwr2+φQ2>0, and γ,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 α1,α2 are constants such that the mean squared error (MSE) is minimum. Here, we note that 0≤φ≤1. For suitable values of α1,α2,γ,φ, the proposed class of estimators “l” reduces to some known existing estimators based on quartiles and their functions given in Table 1.Table 1
Some existing known estimators of the class of estimatorsl.
S. no.EstimatorsValues of constantsα1α2γQφ1l1=wz2, usual unbiased estimator100——2l2=wz2Wr2/wr2, Das and Tripathi [1] and Isaki [4] estimator101—03l3=wz2Wr2+φQ12/wr2+φQ12, Singh et al. [5] estimator101Q1δ4l4=wz2Wr2+φQ22/wr2+φQ22, Singh et al. [5] estimator101Q2δ5l5=wz2Wr2+φQ32/wr2+φQ32, Singh et al. [5] estimator101Q3δ6l6=wz2Wr2+φQr2/wr2+φQr2, Singh et al. [5] estimator101Qrδ7l7=wz2Wr2+φQd2/wr2+φQd2, Singh et al. [5] estimator101Qdδ8l8=wz2Wr2+φQa2/wr2+φQa2, Singh et al. [5] estimator101Qaδ9l9=wz2CrWr2+φQ12/Crwr2+φQ12, Solanki et al. [17] estimator10CrQ1δ10l10=wz2β2rWr2+φQ22/β2rsr2+φQ22, Solanki et al. [17] estimator10β2rQ2δ11l11=wz2R¯Wr2+φQ32/R¯wr2+φQ32, Solanki et al. [17] estimator10R¯Q3δ12l12=wz2β2rWr2+φQr2/β2rwr2+φQr2, Solanki et al. [17] estimator10β2rQrδ13l13=wz2ρWr2+φQd2/ρwr2+φQd2, Solanki et al. [17] estimator10ρQdδ14l14=wz2ρWr2+φQa2/ρwz2+φQa2, Solanki et al. [17] estimator10ρQaδFrom (2), we would like to remark the suitable values of α1,α2,γ,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.Table 2
Some new members of the class of estimatorsl.
S. no.EstimatorsValues of constantw1w2γQφ15l15=α1wz2+α2Wr2−wr2γWr2+φTM2/γwr2+φTM2α1α2CrTMφ16l16=α1wz2+α2Wr2−wr2γWr2+φMR2/γwr2+φMR2α1α2CrMRφ17l17=α1wz2+α2Wr2−wr2γWr2+φHL2/γwr2+φHL2α1α2CrHLφ18l18=α1wz2+α2Wr2−wr2γWr2+φG2/γwr2+φG2α1α2CrGφ19l19=α1wz2+α2Wr2−wr2γWr2+φD2/γwr2+φD2α1α2CrDφ20l20=α1wz2+α2Wr2−wr2γWr2+φSpw2/γwr2+φSpw2α1α2CrSpwφWhile obtaining the expression of bias and mean square error (MSE) for the suggested class of estimator “l”, we write (3)wz2=Wz21+ϑ0,wr2=Wr21+ϑ1,such that Eϑ0=Eϑ1=0, and up to the first order of approximation, while fpc term is ignored, we have(4)Eϑ02=μ40∗m,Eϑ12=μ04∗m,Eϑ0ϑ1=μ22∗m.Now, in terms ofϑ′s, the estimator “l”, given in (2), is expressed as(5)l=Wz2α11+ϑ0−α2bϑ11+θ∗ϑ1−1,where(6)b=Wr2Wz2,θ∗=γWr2γWr2+φQ2.We assume thatθ∗ϑ1<1, so that 1+θ∗ϑ1−1 is expendable. Expanding the right-hand side of (5) and multiplying out, we have(7)θ≅Wz2α11+ϑ0−θ∗ϑ1−θ∗ϑ0ϑ1+θ∗2ϑ12+α2bθ∗ϑ12−ϑ1orθ−Wz2≅Wz2α11+ϑ0−θ∗ϑ1−θ∗ϑ0ϑ1+θ∗2ϑ12+α2bθ∗ϑ12−ϑ1−1.Taking expectation of both sides of (7), we get the bias of the estimators l to the first degree of approximation as(8)Bl=Wz2α11+θ∗mθ∗μ04∗−μ22∗+α2bθ∗mμ04∗−1.Squaring both sides of (7), neglecting terms of ϑ′s having the power greater than two, we have(9)θ−Wz22≅Wz41+α121+2ϑ0−2θ∗ϑ1+ϑ02+3θ∗2ϑ12−4θ∗ϑ0ϑ1+α22b2ϑ12+2α1α2b2θ∗ϑ12−ϑ0ϑ1−ϑ1−2α11+ϑ0−θ∗ϑ1−θ∗ϑ0ϑ1+θ∗2ϑ1−2α2bθ∗ϑ12−ϑ1.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(10)MSEl=Wz41+α12d1+α22d2+2α1α2−2α1d4−2α2d5,where(11)d1=1+1mμ40∗+3θ∗2μ04∗−4θ∗μ22∗,d2=b2μ04∗m,d3=bm2θ∗μ04∗−μ22∗,d4=1+θ∗mθ∗μ04∗−μ22∗,d5=bθ∗mμ04∗.Differentiating MSE (l) with respect to α1 and α2 and equating them to zero, we have(12)d1d3d3d2α1α2=d4d5.Simplifying (12), we get the optimum values of α1 and α2 as(13)α1=d2d4−d3d5d1d2−d32=α10say,α2=d1d5−d3d4d1d2−d32=α20say.Inserting (13) in (10), we get the resulting minimum MSE of “l” given by(14)min⋅MSEl=Wz41−d2d42−2d3d4d5+d1d52d1d2−d32.Thus we establish the following theorem.Theorem 1.
To the first degree of approximation(15)MSEl≥Wz41−d2d42−2d3d4d5+d1d52d1d2−d32,with equality holding if(16)α1=α10,α2=α20,where αi0′si=1,2 are given in (13).Special Case 1:.
forα1=1 in (2), we get an alternative class of estimators for the population variance Wz2 as(17)t∗=wz2+α2Wr2−wr2γWr2+φQ2γwr2+φQ2.
Puttingα1=0 in (8) and (10), we get the bias and MSE of l∗ to the first degree of approximation, given by(18)Bl∗=Wz2mθ∗θ∗+bμ04∗−μ22∗,MSEl∗=Wz41+d1−2d4+α22d2−2α2d5−d3.
TheMSEl∗ is minimum when(19)α20∗=d5−d3d2.
Thus, the resulting minimumMSE of l∗ is given by(20)min⋅MSEl∗=Wz41+d1−2d4−d5−d32d2=Wz4mμ40∗−μ22∗2μ04∗.
This equals to the minimum MSE of the difference estimatorld envisaged by [1](21)ld=wz2+α2Wr2−wr2.Special Case 2:.
forα2=0 in (2), we get another class of estimators for population variance Wz2 as(22)l1∗=α1wz2γWr2+φQ2γwr2+φQ2.
Puttingα2=0 in (8) and (10), we get the bias and MSE of l1∗, respectively, as(23)Bl∗=Wz2α11+θ∗mθ∗μ04∗−μ22∗−1,MSEl∗=Wz41+α12d1−2α1d4.
TheMSEl∗ is minimum for(24)α10∗=d4d1.
Thus, the resulting minimum MSE ofl∗ is given by(25)min⋅MSEl∗=Wz41−d42d1.
Thus, we state the following corollary.Corollary 1.
To the first degree of approximation(26)MSEl1∗≥Wz41−d42d1,with equality holding if(27)α1=α10∗=d4d1.Special Case 3:.
if we setα1,α2=1,0 in (2), the class of estimators l reduces to the estimator(28)l1∗∗=wz2γWr2+φQ2γwr2+φQ2,which includes Solanki et al. [17] estimators l9 to l14 .
Puttingα1,α2=1,0 in (8) and (10), we get the bias and MSE of l1∗∗ to the first degree of approximation (ignoring fpc term), respectively, as(29)Bl1∗∗=Wz2θ∗mθ∗μ04∗−μ22∗,(30)MSEl1∗∗=Wz41+d1−2d4,(31)=Wz4mμ40∗+θ∗2μ04∗−2θ∗μ22∗.Special Case 4:.
if we setα1,α2,γ=α1,0,1in (2), the class of estimators l reduces to the class of estimators of Wz2 as(32)l2∗=α1wz2Wr2+φQ2wr2+φQ2.
Insertingα1,α2,γ=α1,0,1 in (8) and (10), we get the bias and MSE of l2∗ to the first degree of approximation (ignoring fpc term), respectively, as(33)Bl2∗=Wz2α11+θmθμ04∗−μ22∗−1,(34)MSEl2∗=Wz41+α12d1∗−α1d4∗,where(35)d1∗=1+1mμ40∗+3θ2μ04∗−4θμ22∗,d4∗=1+θmθμ04∗−μ22∗,θ=Wr2Wr2+ϕQ2.
TheMSEl2∗ at (34) is minimized for(36)α10∗∗=d4∗d1∗.
Substitution of (36) in (34) yields the minimum MSE of l2∗ and is given by(37)min⋅MSEl2∗=Wz41−d4∗2d1∗.
Thus, we established the following corollary.Corollary 2.
To the first degree of approximation(38)MSEl2∗≥Wz41−d4∗2d1∗,with equality holdings if(39)α1=α10∗∗=d4∗d1∗.Special Case 5:.
forα1,α2,γ=1,0,1 in (2), the class of estimators “l” reduces to the class of estimators of Wz2 as(40)l2∗∗=wz2Wr2+φQ2wr2+φQ2,which includes Singh et al. [5] estimators l3 to l8.
Insertingα1,α2,γ=1,0,1 in (8) and (10), we get the bias and MSE of l2∗∗ to the first degree of approximation (ignoring fpc term), respectively, given by(41)l2∗∗=Wz2θ∗mθ∗μ04∗−μ22∗,(42)MSEl2∗∗=Wz41+d1∗−2d4∗=Wz4mμ40∗+θ2μ04∗−2θμ22∗.


## 4. Efficiency Comparison

To the first degree of approximation, the MSE ofwz2 and l2 are, respectively, given by(43)MSEwz2=Wz4mμ40∗,(44)MSEl2=Wz4mμ40∗+μ04∗+2μ22∗.From (20), (43), and (44), we have(45)MSEwz2−min⋅MSEl∗orld=Wz4μ22∗2mμ04∗>0,(46)MSEl2−min⋅MSEl2∗=Wz4μ04∗−μ22∗2mμ04∗≥0.It follows from (45) and (46) that the proposed class of estimators l∗ or difference estimator ld is better than the usual unbiased estimator wz2 and the ratio type estimator l2 due to [1, 4].From (37) and (42), we have(47)MSEl2∗∗−min⋅MSEl2∗=Wz4d1∗−d4∗2d1∗≥0.This follows that the proposed class of estimatorsl2∗ is more efficient than the proposed class of estimators l2∗∗ according to Singh et al. [5]. Thus, the estimator l2∗ is an improvement over the Singh et al. [5] estimators l2∗∗.From (25) and (30), we have(48)MSEl1∗∗−min⋅MSEl1∗=Wz4d1−d42d1≥0.Expression (48) clearly indicates that the proposed class of estimators l1∗ is better than class of estimator l1∗∗ recently proposed by Solanki et al. [17]. Thus, the estimators l1∗ are an improvement over the estimator l1∗∗ due to Solanki et al. [17].Furthermore, from (14) and (20), we have(49)min⋅MSEl∗−min⋅MSEl=Wz4d2d1−d4+d3d5−d32d2d1d2−d32>0.It follows from (49) that the proposed class of estimators “l” is better than the class of estimators l∗ and the difference estimator ld.Next, from (14) and (25), we have(50)min⋅MSEl1∗−min⋅MSEl=Wz4d1d5−d3d42d1d1d2−d32≥0.Thus, from (48) and (50), we have the following inequality:(51)min⋅MSEl≤min⋅MSEl1∗≤min⋅MSEl1∗∗.This clearly shows that the proposed family of estimatorsl is better than families of estimators l1∗ and l1∗∗.Finally, from the above theoretical comparisons we conclude that the proposed family of estimatorsl is better than the usual unbiased estimator l1=wz2, [4] ratio type estimator l2, and the estimators (l3 to l8) and (l9 to l14) recently proposed by Singh et al. [5] and Solanki et al. [17], respectively, and for other families of estimators (l1∗, l2∗).


## 5. Empirical Study

The performances of the proposed estimatorslkk=15,16,…,20, which are members of the suggested class of estimators, l , are evaluated against the usual unbiased estimators wz2 and the estimators lii=2,3,…,14 proposed by [1, 4, 5, 10] for the population data set (source: [18]) summarized in Table 3.Table 3
The parameters of population data set.
M80Cz0.3542Q15.1500TM9.318m20Wr8.4563Q210.300MR10.405Z¯51.8264Cr0.7507Q316.975HL17.955R¯11.2646μ042.8664Qr11.825G9.0408ρ0.9413μ402.2667Qd5.9125D8.0138Wz18.3569μ222.2209Qa11.0625Spw7.9136We have computed the percent relative efficiencies (PREs) of the suggested estimatorslkk=15,16,…,20 with respect to the usual unbiased estimator wz2 in the range of φ∈−1.0,1.0 by using the following formula:(52)PRElk,wz2=MSEwz2MSElk×100=μ40∗m1−d1d42−2d3d4d5+d1d32/d1d2−d32×100,and the findings are summarized in Table 4.Table 4
PRE of estimatorslkk=15,16,…,20 with respect to l1=wz2.
ϕPRE (wz2)l15l16l17l18l19l20−1.00257.0391259.3591291.8213258.2640256.4163253.0021−0.75255.2481257.2658275.3213256.0011253.9286252.8256−0.50254.2359256.9329273.3492255.3651253.6165252.4152−0.25253.9139256.1231273.1029254.9606253.0259252.00810.00253.1029255.8569273.0051254.0436252.8169251.56210.25252.9320255.0091272.9321253.8186252.0411251.05610.50253.1001254.8563272.3649253.5646251.9103250.39610.75254.2569254.6561272.1536253.3963251.3851250.00111.00254.2031254.5341271.9806252.8646251.0091249.5651For the purpose of comparison of the proposed estimatorsl15 to l20 with that of the usual unbiased estimator l1=wz2, [4] estimator l2, [5] estimators (l3 to l8), and [17] estimators (l9 to l14), in Table 5, we give the PREs of the estimators l1, l2 and the PREs of the estimators l3 to l14 with respect to usual unbiased estimator wz2 along with the value of δfor which the PREs of the estimators l3 to l14 are maximum as given in [17].Table 5
PREs of the estimatorsl1=wz2, l2 and lii=3,…,14.
EstimatorValue ofφPRE (wz2)l1—100.00l2—183.23l31.00209.67l40.50230.60l50.30230.28l60.70230.57l71.00200.87l80.60230.80l91.00270.38l101.00270.61l111.00263.50l120.80270.58l131.00270.61l140.30270.58It is observed from Tables4 and 5 that all the estimators lkk=15,16,…,20 which are the members of the proposed class of estimators l performed better than the usual unbiased estimator l1=wz2, ratio type estimators l2 due to [4, 5] estimators (l3 to l8) and [17] estimators (l9 to l14), for φ∈−1.0,1.0.


## 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 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.



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