# Generalized Difference-Cum Exponential Class of Estimators for Estimating Population Parameters of the Sensitive Variable

Authors: Gajendra K. Vishwakarma; Subhash Kumar Yadav; Tarushree Bari
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

We propose a generalized class of estimators to estimate general population parameters of susceptible research variables using additional auxiliary information in SRSWOR. The population constants such as the coefficient of variation, the population mean, the standard deviation, and the population mean square are defined using a conventional estimator. The expression for the mean square errors of the proposed class is derived up to the first order of approximation. To compare the effectiveness of the proposed class of estimators, an empirical study is conducted utilizing real and simulated data sets. Theoretical and empirical research demonstrates that the suggested generalized class of estimators outperforms other current estimators.


---

## Body

## 1. Introduction

The collection of data on sensitive topics such as induced abortions, drug misuse, and family income through personal interviews and surveys is a severe problem. Some questions, for example, are delicate: (a) On your 2009 tax return, how much did you underreport your income? (c) Have you undergone any abortions? (b) Have you molested any children? (e) Do you use illicit substances? One approach to get individuals to answer honestly is to use randomised response techniques. Horvitzet al. [1] and Greenberg et al. [2] have expanded Warner’s [3] model to include numeric responses to the sensitive question rather than simple “yes” or “no” responses. The respondent chooses one of two questions using a randomization device: one is the sensitive question, and the other is unrelated. Direct true responses on the study variable might become difficult to obtain in survey sampling, especially when the variable is sensitive. When the study variable is sensitive in nature, many survey statisticians have estimated the population mean of sensitive variables, such as Eichhron and Hayre [4], Gupta et al. [5], Saha [6], and Diana and Perri [7]. Different ratio, regression, and exponential estimators for estimating population parameters of sensitive variables based on scrambled responses were reported by Sousa et al. [8], Koyuncu et al. [9], Kalucha et al. [10], and Gupta et al. [11, 12]. Using a generalized quantitative optional randomised response model, Noor-Ul-Amin et al. [13] developed estimators based on generalized ratio and regression types, in which the nature of the auxiliary variable is nonsensitive. In the presence of measurement error, Khalil et al. [14] and Zahid and Shabbir [15] developed improved mean estimators of a sensitive research variable. Sanaullah et al. [16] introduced a new family of difference-cum-exponential-type estimators of the finite population mean of the susceptible research variable by using a single nonsensitive secondary information. In the case of nonresponse using RRT under two-phase sampling, Sanaullah et al. [17] proposed a generalized family of estimators for the finite population mean of a susceptible research variable. Quantitative scrambled randomised response models are considerably improved by Saleem et al. [18] by assessing the performance of the population mean estimator. Waseem et al. [19] presented a generalized exponential type estimator for the mean of a sensitive variable using information from a nonsensitive auxiliary variable. Singh et al. [20] provided some alternative additive randomised response models by offering a blank card option for predicting the population mean of a quantitative sensitive variable. Vishwakarma and Singh [21] developed the estimation procedure to estimate the effect of measurement errors under additive scramble response for sensitive variables.There have been a few extreme values in numerous populations, and estimating the unknown population parameters without considering this knowledge is extremely delicate. Otherwise, the results will be understated or exaggerated, respectively. In order to resolve this problem, it is necessary to incorporate this knowledge into the estimation of population parameters. Several researchers, including Isaki [22], Bahl and Tuteja [23], Upadhyaya and Singh [24], Kadilar and Cingi [25], Dubey and Sharma [26], Singh and Vishwakarm [27], Shabbir and Gupta [28], and Yadav et al. [29], have proposed some broader classes of estimators for estimating finite population variance. Gupta et al. [30] and Daraz and Khan [31] presented several variance estimators for sensitive variables that make use of additional information in SRSWOR.Motivated by Adichwal et al. [32], we suggest the general difference-cum exponential class of estimators for estimating general population parameters of the sensitive research variable.LetY represent the research variable, which contains sensitive elements that may not be accessible precisely as a result of the respondent’s response. Let X be a nonsensitive secondary variable with a positive correlation to Y. Let S be an uncorrelated scrambled variate with a predetermined distribution. The variable S is used to scramble the sensitive variable Y. Each respondent is given the task of selecting a random number from the S distribution, say s, and adding it to the actual value of Y to assess Z. Let Z stand for the reported scrambled answer to Y, which was first proposed by Warner [3] and then developed by Pollock and Bek [33].The general population parameter used in the study are discussed as follows:(1)μrs=1N∑i=1Nyi−Y¯rxi−X¯s.δrs=μrs/μ20r/2μ02s/2 where r and s are nonnegative integers.(2)μ20=Sz2,μ02=SX2,μ11=Szx,Cz2=Sz2Z¯2,Cx2=Sx2X¯2,ρzx=SzxSz=μ11μ20μ02.The error terms used in the study are defined as follows:(3)e0=z¯Z¯−1,e1=sz2Sz2−1,e2=x¯X¯−1,e3=sx2Sx2−1.Under the condition that(4)Eei=0;i=0,1,2,3.,Ee02=n−1Cz2,Ee12=n−1δ40−1,Ee22=n−1Cx2,Ee32=n−1δ04−1,Ee0e1=n−1δ30Cz,Ee0e2=n−1ρzxCzCx,Ee0e3=n−1δ12Cz,Ee1e2=n−1δ21Cx,Ee1e3=n−1δ22−1,Ee2e3=n−1δ03Cx.


## 2. Conventional Estimators

The population parameter of a sensitive research variable under discussion can be stated in its most general form as(5)za,b=Z¯aSzb,where a and b are suitably chosen scalars that take different values in order to get different parameters of a population. The parameters obtained after substituting a different combination of the scalars are given by(i)
For (a = 1, b = 0), z1,0 reduces to Z.¯(ii)
For (a = 0, b = 1), z0,1 reduces to Sz.(iii)
For (a = -1, b = 1), z−1,1 reduces to Cz.(iv)
For (a = 0, b = 2), z0,2 reduces to SZ2.For estimating the general population parameters discussed previously, the general traditional estimatorz^a,b is given by(6)z^a,b=z¯aszb.On expressing (6) in terms of errors, we have(7)z^a,b=Z¯aSzb1+e0a1+e1b/2=za,b1+ae0+b2e1+aa−12e02+ab2e0e1+bb−28e12z^a,b−za,b=za,bae0+b2e1+aa−12e02+ab2e0e1+bb−28e12.Squaring (7) and excluding terms with powers of three or more, we get(8)z^a,b−za,b2=za,b2a2e02+abe0e1+b24e12.We obtained the MSE of the general estimator by taking expectations on both the side of the abovementioned equation, as(9)MSEz^a,b=za,b2na2Cz2+abδ30+b24δ40−1.or(10)MSEz^a,b=za,b2nF1a,b,where, F1a,b=a2Cz2+abδ30+b2/4δ40−1.


## 3. Proposed Generalized Class of Estimators

Auxiliary information can be analyzed in a range of categories, including variables and attributes. It has been widely used by many researchers to develop numerous estimators for estimating population parameters in various sampling systems. Motivated by Adichwal et al. [32], we propose the general difference-cum exponential class of estimators for estimating population parameters of the sensitive research variable, given by(11)zd=z^a,b+wX¯−x¯expk1X¯−x¯X¯+α−1x¯expk2Sx2−sx2Sx2+β−1sx2.where k1 and k2 are considered as scalar constant which takes the values 0,−1,1, and w, α, and β are suitably chosen constants. Expressing (11) in terms of the errors, we get(12)zd=za,b1+e0a1+e1b/2−wX¯e2exp−k1e2α1+α−1αe2−1exp−k2e3β1+β−1βe3−1=za,b1+ae0+b2e1+aa−12e02+ab2e0e1+bb−28e12+…1−k2βe3−k1αe2+k22β−1+k22β2e32+k12α−1+k12α2e22+k1αk2βe2e3+…−wX¯e21−k2βe3−k1αe2+k22β−1+k22β2e32+k12α−1+k12α2e22+k1αk2βe2e3+….Multiplying and neglecting the terms of higher-order containing errors in (12) which are greater than order two, we get(13)zd=za,b[1+ae0+b2e1−k1αe2−k2βe3+aa−12e02+ab2e0e1+bb−18e12−−k1αae0e2+b2e1e2−k2βae0e3+b2e2e3+k12α−1+k12α2e22+k1αk2βe2e3−wX¯e2+k2βwX¯e2e3+k1αwX¯e22.Equivalently,(14)zd−za,b=za,bae0+b2e1−k1αe2−k2βe3+aa−12e02+ab2e0e1+bb−18e12−k1αae0e2+b2e1e2−k2βae0e3+b2e2e3+k12α−1+k12α2e22+k1αk2βe2e3−wX¯e2+k2βwX¯e2e3+k1αwX¯e22.Squaring (14) on both the sides, and neglecting error terms of power greater than two, we get(15)zd−za,b2=za,bae0+za,bb2e1−za,bk1α+wX¯e2−za,bk2βe32=za,bae0+b2e1−k1αe2−k2βe2−wX¯e22.Equivalently,(16)zd−za,b2=za,b2a2e02+b24e12+k1α2e22+k2β2e32+2ab2e0e1−2ak1αe0e2−2ak2βe0e3−2b2k1αe1e2−2b2k2βe1e3+2k1αk2βe2e3−2za,bawX¯e0e2−2za,bb2wX¯e1e2+2za,bk1αwX¯e22+2za,bk2βwX¯e2e3+w2X¯e22.Taking expectations on both sides of the abovementioned equation, we get MSE ofzd, given as(17)MSEzd=MSEz^a,b+za,b2nk12α2Cx2+k2β2δ04−1−2k12αaρzxCz+b2δ21Cx−2k2βaδ12Cz+b2δ22−1+2k1βk2βδ03Cx+2za,bnwX¯k1αCx2+k2βδ03Cx−2za,bnwX¯aρzxCz+b2δ21Cx+w2X¯2nCx2.or(18)MSEzd=MSEz^a,b+za,b2nP2Cx2+Q2δ04−1−2PF2a,bCx−2QF3a,b++2PQδ03Cx+2za,bnwX¯PCx2+Qδ03Cx−2za,bnwX¯F2a,bCx+w2X¯2nCx2,where(19)P=k1α,Q=k2β,F2a,b=aρzxCz+b2δ21,F3a,b=aρzxCz+b2δ22−1.Differentiating (17) partially w.r.t P and Q, and equating to zero, we get(20)za,bCxza,bδ03za,bδ03Cxza,bδ04−1PQ=za,baρzxCz+b2δ21−wX¯Cxza,baρzxCz+b2δ21−kX¯Cxδ03.After substituting the optimal values of P andQ, we get the minimum MSEzd as(21)MSEzd=MSEz^a,b−za,b2nF3a,b2−2F2a,bF3a,bδ03+δ04−1F2a,b2δ04−δ032−1=MSEz^a,b−za,b2nF2a,b2−za,b2nF2a,bδ03−F3a,b2δ04−δ032−1=MSEz^a,b−za,b2nF3a,b2δ04−1−za,b2nδ04−1F2a,b−δ03F3a,b2δ04−δ032−1δ04−1.Using known population meanX¯ and substituting k2=0, in the general difference-cum exponential class of estimators (11), we get the following general estimator for estimating population parameters of the sensitive research variable as(22)zd1=z^a,b+wX¯−x¯expk1X¯−x¯X¯+α−1x¯,where k1 takes real values (-1, 0, 1) and w and α are suitably chosen constants. Following the abovementioned procedure, the MSE of zd1 is given by(23)MSEzd1=MSEz^a,b+za,b2nk1α2Cx2−2k1αF2a,bCx.The minimum value of MSEzd1 is obtained, after minimizing the above equation for k1/αopt=F2a,b/Cx, as(24)MSEzd1min=MSEz^a,b−za,b2nF2a,b2.Table1 illustrates the existing estimators derived from the family of zd1 estimators by selecting appropriate values for a, b, w, k1, α and applying them appropriately.Table 1
Several members of thezd1 family of estimator.
S.No.Estimatorsabwk1α1zd11=z^a,bexpX¯−x¯/X¯+α−1x¯ab01α2zd12=z¯expX¯−x¯/X¯+α−1x¯1001α3zd13=z¯expX¯−x¯/X¯+x¯100124zd14=z¯+wX¯−x¯10w0αUsing known population varianceSx2 and substituting k1=0 in the general difference-cum exponential class of estimators (11), we get another general estimator for estimating population parameters of the sensitive research variable as(25)zd2=z^a,bexpk2Sx2−sx2Sx2+β−1sx2,where k2 takes real values (-1, 0, 1) and w and β are suitably chosen constants. Following the abovementioned procedure, the MSE of zd2 is given by(26)MSEzd2=MSEz^a,b+za,b2nk2β2δ04−1−2k2βF3a,b.The minimum value of MSEzd1 is obtained, after minimizing the above equation for k2/βopt=F3a,b/δ04−1, as(27)MSEzd2min=MSEz^a,b−za,b2nF2a,b2δ04−1.Table2 demonstrates the estimators derived from the family of zd2 estimators by selecting appropriate values for a, b, k2, α and applying them appropriately.Table 2
Several members of thezd2 family of estimator.
S.No.Estimatorsabk2β1zd11=z^a,bexpSx2−sx2/Sx2+β−1sx2ab1β2zd22=sz2expSx2−sx2/Sx2+β−1sx2021β3zd23=sz2expSx2−sx2/Sx2+sx20212


### 3.1. Efficiency Comparision ofzd

(28)MSEz^a,b−MSEzd=za,b2nF3a,b2−2F2a,bF3a,bδ03+δ04−1F2a,b2δ04−δ032−1≥0,MSEzd1−MSEzd=za,b2nF2a,bδ03−F3a,b2δ04−δ032−1≥0,MSEzd2−MSEzd=za,b2nδ04−1F2a,b−δ03F3a,b2δ04−δ032−1δ04−1≥0,provided thatδ04−δ032−1≥0 , the proposed class of estimator zd outperforms other competing estimators.


## 3.1. Efficiency Comparision ofzd

(28)MSEz^a,b−MSEzd=za,b2nF3a,b2−2F2a,bF3a,bδ03+δ04−1F2a,b2δ04−δ032−1≥0,MSEzd1−MSEzd=za,b2nF2a,bδ03−F3a,b2δ04−δ032−1≥0,MSEzd2−MSEzd=za,b2nδ04−1F2a,b−δ03F3a,b2δ04−δ032−1δ04−1≥0,provided thatδ04−δ032−1≥0 , the proposed class of estimator zd outperforms other competing estimators.


## 4. Estimtion of Population Mean of the Research Variable

Using known population parameters and substitutingdifferentvaluesofa,b,w,k1,k2,α,β as (1,0 w,k1,k2,,α,β) in the general difference-cum exponential class of estimators (equation 4), we obtain a class of estimators for estimating the population mean of the sensitive research variable as(29)z¯d′=z¯+wX¯−x¯expk1X¯−x¯X¯+α−1x¯expk2Sx2−sx2Sx2+β−1sx2.The expression of MSEz¯d′ upto O(n-1) is given by(30)MSEz¯d′=MSEz¯+Z¯2nP2Cx2+Q2δ04−1−2PρzxCxCz−2Qδ12Cz+2PQδ03Cx+2nZ¯X¯PCx2+Qδ03Cx−2nwZ¯X¯ρzx.where(31)P=k1α,Q=k2β.MSEz¯d′ is minimized for the optimal values of P and Q, given by(32)P=δ04−1ρzx−δ03δ12Czδ04−δ032−1Cx−wX¯Z¯,Q=δ12−δ03ρzxCzδ04−δ032−1.On substituting the optimal values ofP and Q in (30), the least value of MSEz¯′ takes the form, given by(33)MSEz¯′=Varz¯0−Z¯2δ122−2ρzxδ03δ12+δ04−1ρzx2Cz2nδ04−δ032−1.Table3 shows the members of the proposed generalized class of estimators z¯d′, and their values of different constant used in MSEs.Table 3
Several members of thez¯d′ family of estimator.
EstimatorsabcdMSEzd12=z¯expX¯−x¯/X¯+α−1x¯010αVarz¯−Z2¯/nρzx2Cz2zd13=z¯expX¯−x¯/X¯+x¯ (Gupta et al. [12]0102Varz¯−Z2¯/nCx2/4−ρzxCzCxzd14=z¯+wX¯−x¯ (Difference estimator)100αVarz¯−Z2¯/nρzx2Cz2


## 5. Relative Efficiency

In the current section, we analyze the theoretical efficiency of the proposed class of estimators and the criteria is discussed under which the estimator performs better than other existing estimators.(i)
Varz¯0−MSEz¯d′=Z¯2δ122−2ρzxδ03δ12+δ04−1ρzx2Cz2/nδ04−δ032−1≥0(ii)
MSEzd12−MSEz¯d′=Z¯2δ122−2ρzxδ03δ12+δ04−1ρzx2Cz2/nδ04−δ032−1−Z¯2/nρzx2Cz2≥0(iii)
MSEzd13−MSEz¯d′=Z¯2δ122−2ρzxδ03δ12+δ04−1ρzx2Cz2/nδ04−δ032−1−Z¯2/nCx2/4−ρzxCzCx≥0(iv)
MSEzd14−MSEz¯d′=Z¯2δ122−2ρzxδ03δ12+δ04−1ρzx2Cz2/nδ04−δ032−1−Z¯2/nρzx2Cz2≥0


## 6. Empirical Study

In a view to analysing the accuracy of the theoretical results, we use an empirical investigation to assess the effectiveness of the suggested class of estimators. The real population included in the study is Cochran ([34], page no. 325), where X denotes the number of rooms per block and Y denotes the number of person per block. The descriptive statistics of the real data population are given as(34)n=10,Y¯=101.1,X¯=58.8,Rzx=0.6507,Sx=7.9414,Sy=15.4449.PREs for the proposed families of estimators in comparison to RRT usual mean estimator are determined using the following equation:(35)PRE⋅,z¯0=MSEz¯0MSE⋅×100.


## 7. Simulation Study

A simulation model is constructed to examine the dynamics of the suggested class of estimators, incorporating the additive scrambling model given by Pollock and Bek [33]. The mean square errors of the proposed class of estimators and the competing estimators are compared using six artificial populations, namely, normal, beta, gamma, Poisson, uniform, and log-normal with differing population parameters. Following the procedure adopted by Singh and Horn [35], we generated study and ancillary variables as Y=10+1−ρxy2Y∗+ρxySy/SxX∗ and X=5+X∗, where X∗ and Y∗ are independent variates of pertaining family of distributions. The sensitive variable Z is derived by scrambling the variable Y with S where S follows the corresponding parent distribution with predefined parameters. With various selections of correlation coefficient, i.e., 0.6, 0.7, 0.8, and 0.9, samples of different sizes, i.e., 40, 60, 80, and 100, are collected from each bivariate distribution.


## 8. Results and Discussion

The findings of the study for real data set are presented in Table4 which describes the MSE and PRE of suggested class of estimators z¯d′ along with the competing estimators, i.e., RRT sample mean (z¯0), (zd13), Gupta et al. [12]’s estimator (zd13) and difference estimator (zd14). Table 5–10 summarise Table 6 PRE Table 7 of known Table 8 and proposed Table 9 estimators for various sample sizes, i.e., (40, 60, 80, and 100) and correlation coefficients, i.e., (0.6, 0.7, 0.8, and 0.9). Under simulation investigation, the proposed estimate z¯d′ has the lowest MSE for all available regression coefficient values, indicating that it is more efficient than the existing estimator in the literature reveal.Table 4
MSE and PRE of recommended and competing estimators for empirical data.
S. No.EstimatorsMSEPRE1z¯023.91043100.002zd1213.75848173.783zd1333.0129172.434zd1413.75848173.795z¯d′11.72704203.89Table 5
PRE of proposed and competing estimators for Normal population.
X∗∼N50,4Y∗∼N40,3S∼N0,1Estimatorsρ=0.6ρ=0.7ρ=0.8ρ=0.9N = 40z¯0100.00100.00100.00100.00zd12157.39188.02253.31437.25zd1373.9267.3862.5057.17zd14157.39189.02253.31437.25z¯d′160.83194.10257.55454.78z¯0100.00100.00100.00100.00zd12146.11177.43234.58361.75N = 60zd1377.5769.8563.6058.94zd14146.11177.43234.58361.75z¯d′164.15199.46263.62404.02z¯0100.00100.00100.00100.00zd12122.61143.31180.72257.52n=80zd1394.9580.5669.7062.19zd14122.61143.31180.72257.52z¯d′126.90148.24186.71265.23z¯0100.00100.00100.00100.00zd12150.99183.45242.34374.37n=100zd1375.6768.8163.1458.79zd14150.99183.45242.34374.37z¯d′154.06187.08246.87380.30Table 6
PRE of proposed and competing estimators for beta population.
X∗∼B15,3Y∗∼B3,5S∼B2,3Estimatorsρ=0.6ρ=0.7ρ=0.8ρ=0.9N = 40z¯0100.00100.00100.00100.00zd12100.87103.30108.50119.73zd1394.9292.6388.8386.00zd14100.87103.30108.50119.73z¯d′101.57104.15109.63121.49z¯0100.00100.00100.00100.00zd12118.84125.00132.26140.43n=60zd1386.5984.1481.8879.87zd14118.84125.00132.26140.43z¯d′122.48128.72135.99144.00z¯0100.00100.00100.00100.00zd12117.23123.74132.22143.66n=80zd1387.7085.1682.6680.12zd14117.23123.74132.22143.66z¯d′117.95124.56133.20144.89z¯0100.00100.00100.00100.00zd12107.58112.60120.22132.72n=100zd1393.5789.9786.1882.04zd14107.58112.60120.22132.72z¯d′108.87113.91121.53139.18Table 7
PRE of proposed and competing estimators for gamma population.
X∗∼G6,4Y∗∼G5,3S∼G3,10Estimatorsρ=0.6ρ=0.7ρ=0.8ρ=0.9N = 40z¯0100.00100.00100.00100.00zd12153.74198.05293.20593.14zd1376.4867.2760.3055.20zd14153.74198.05293.20593.14z¯d′154.63199.27295.21598.23z¯0100.00100.00100.00100.00n=60zd12133.09159.74211.36341.37zd1384.5173.9565.5358.92zd14133.09159.74211.36341.37z¯d′133.13159.76211.36341.55z¯0100.00100.00100.00100.00n=80zd12155.92192.21262.85455.35zd1373.6767.7262.5958.12zd14155.92192.21262.85455.35z¯d′165.20204.14280.32490.15z¯0100.00100.00100.00100.00n=100zd12141.60177.35254.68510.62zd1378.5869.6962.4956.81zd14141.60177.35254.68510.62z¯d′141.64177.42254.84511.29Table 8
PRE of proposed and competing estimators for Poisson population.
X∗∼P50Y∗∼P30S∼P3Estimatorsρ=0.6ρ=0.7ρ=0.8ρ=0.9N = 40z¯0100.00100.00100.00100.00zd12236.95278.38336.43422.00zd1363.9261.7859.8858.23zd14236.95278.38336.43422.00z¯d′238.92281.14340.68429.40z¯0100.00100.00100.00100.00n=60zd12237.36277.96334.13415.18zd1363.7861.6759.8158.23zd14237.36277.96334.13415.18z¯d′239.34280.72338.31422.30z¯0100.00100.00100.00100.00n=80zd12169.08200.04251.23355.45zd1371.0566.9663.1359.49zd14169.08200.04251.23355.45z¯d′176.18209.29264.69379.88z¯0100.00100.00100.00100.00n=100zd12132.77160.55217.90387.73zd1382.0172.9564.9057.93zd14132.77160.55217.90387.73z¯d′133.13160.98218.51388.85Table 9
PRE of proposed and competing estimators for uniform population.
X∗∼U50,100Y∗∼U40,80S∼U5,10Estimatorsρ=0.6ρ=0.7ρ=0.8ρ=0.9N = 40z¯0100.00100.00100.00100.00zd12175.69244.64438.103120.25zd1372.1263.2256.4751.70zd14175.69244.64438.103120.25z¯d′181.03253.28459.543959.40z¯0100.00100.00100.00100.00zd12167.76221.54355.871266.64n = 60zd1372.1264.6258.3753.33zd14167.76221.54355.871266.64z¯d′172.42228.50370.251398.22z¯0100.00100.00100.00100.00zd12176.59235.10386.241753.14n=80zd1370.1863.5157.7852.92zd14176.59235.10386.241753.14z¯d′177.26236.06388.071773.45z¯0100.00100.00100.00100.00zd12174.51230.18362.451063.53n=100zd1370.6463.8858.2953.84zd14174.51230.18362.451063.53z¯d′175.54231.61364.961075.18Table 10
PRE of proposed and competing estimators for uniform population.
X∗∼LN10,2Y∗∼LN10,3S∼LN5,3Estimatorsρ=0.6ρ=0.7ρ=0.8ρ=0.9N = 40z¯0100.00100.00100.00100.00zd12174.89220.82313.07591.36zd1371.3866.8362.6658.95zd14174.89220.82313.07591.36z¯d′152.22225.03319.08602.90z¯0100.00100.00100.00100.00zd12165.25202.36267.22406.01N = 60zd1372.3567.7863.8160.60zd14165.25202.36267.22406.01z¯d′166.85204.63270.96414.09z¯0100.00100.00100.00100.00zd12159.06204.38305.39698.66n=80zd1372.9866.7661.3456.89zd14159.06204.38305.39698.66z¯d′160.46206.21308.22706.10z¯0100.00100.00100.00100.00zd12149.76187.63266.47510.00n=100zd1376.5070.6165.1060.20zd14149.76187.63266.47510.00z¯d′150.18188.16267.22512.43


## 9. Conclusion

(i)
A new difference-cum exponential ratio type estimator in SRWOR is proposed in this study, which takes advantage of the additional information available to provide a more accurate general parameter estimate. Adopting the estimatorzd provided in Section 3, it is feasible to estimate a variety of population parameters. Tables 1 and 2 illustrate specific applications of the class of estimators for estimating the mean and variance of the sensitive variable using various choices of arbitrary constants.(ii)
In Section4, the specific situations of the generalized estimator are presented in order to estimate the mean of the sensitive research variable, and the relative efficiency of the class of estimators is discussed.(iii)
We tested the effectiveness of the suggested estimator on real and simulated data sets for the case of population mean estimation solely. The findings of the numerical illustration are demonstrated in Table4 which indicates the supremacy of the recommended class of estimator over other estimators discussed under Section 4.(iv)
We can infer from the results of the simulation study conducted on six artificial populations that for all the different values of correlation coefficient, the PRE of the recommended class of estimators is greater than that of the other estimators. Hence, the suggested class of estimators is more efficient than other estimators.(v)
The suggested general class of estimator can be used to estimate various population parameters of the sensitive research variable. Also, the suggested approach can be used in different sampling techniques.



---

*Source: 7890489-2022-05-02.xml*