# A Nonconventional Auxiliary Information Based Robust Class of Exponential-type Difference Estimators

Authors: Muhammad Abid; Waqas Latif; Tahir Nawaz; Ronald Onyango; Muhammad Tahir
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study proposes a new class of improved exponential-type difference estimators of finite population mean by using supplementary information of known median along with suitable combinations of the conventional and non-conventional measures of the auxiliary variables under simple random sampling scheme. The expressions for the mean squared error and minimum mean squared error are derived up to first order of the approximation. Six real data sets were used to assess the performance of proposed class of estimators in comparison with existing estimators. The compariosn established that the suggested class of estimators are efficient than their existing counterparts considered in this study. To further support the findings of the numerical comparison, a simulation study was carried out which also proved the superiority of the proposed class of estimators of population mean. To gauge the performance of the propsoed class of estimators when some outliers are present in the data, a robustness study was carried out which showed that the proposed estimators considerably outperform their existing counterparts in terms of lower mean squared errors.


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## Body

## 1. Introduction

In survey sampling literature, several estimation methods such as the ratio, product, and regression are frequently used to take advantage of the supplementary information for estimation of unknown population parameters under different sampling schemes. The purpose behind utilization of auxiliary information is to improve the efficiency of the estimators of the parameters under consideration. In numerous statistical analyses, a primary goal is to estimate location parameter which is an important descriptive measure. When the observations are relatively homogeneous, the sample mean performs very well as measure of location. There are various situations in survey sampling when the distribution of the study variable is extremely skewed and sample mean becomes an inefficient measure of location. Moreover, if there are some extreme observations in the data which lies far from rest of the observations then the sample mean provides poor results. In these scenarios, an alternative measure of location known as median is usually considered as a preferred measure of location. Various authors have developed different estimators to estimate population parameters using auxiliary information under different sampling scheme. A bulk of literature is available based on utilization of supplementary variable. See, for example, Yan and Tain [1], Koyuncu [2], Subramani and Kumarapandiyan [3], Subramani and Prabavathy [4], John and Inyang [5], Singh and Pal [6], Singh et al. [7], Subramani [8], Abid et al. [9], Abbas et al. [10], Zaman [11], Zaman and Bulut [12], Yadav et al. [13]. Many researchers have used the known median of the study variable for the estimation of population mean. For example, Subramani and Prabavathy [14] and Yadav et al. [15] suggested new estimators using known median of the study variable, Irfan et al. [16] proposed some power–type ratio estimators, Shahzad et al. [17] suggested a family of exponential estimators utilizing known median of the study variable, Hafeez et al. [18] developed some new median based estimators utilizing supplementary information. In this study, a new class of exponential-type difference estimators is proposed to estimate finite population mean based on Known median of the study variable along with supplementary information of a single auxiliary variable under simple random sampling scheme.The notations used in this study are as follows:(i)
N, Population size(ii)
n, Sample size(iii)
f=n/N, Sampling fraction(iv)
abrY, Study Variables(v)
X Auxiliary variable(vi)
Y,¯X¯, Population means of study variable and auxiliary variable(vii)
y¯,x¯, Sample means of study variable and auxiliary variables(viii)
MY0.5MX0.5, Population median of study variable and auxiliary variable(ix)
M^Y0.5M^X0.5, sample median of study variable and auxiliary variable(x)
Cy,Cx, Coefficient of variation of the study variable and auxiliary variable(xi)
MSE (.), Mean square error of the estimator(xii)
Y¯^i,Existing estimators of Y¯,(xiii)
Y¯^pj, Proposed estimators of Y¯Subscript:.(i)
abri, For existing estimators(ii)
j, For existing estimatorsThe organization of the remaining article is as follows: Section2 presents some existing estimators that utilize auxiliary information based on known median of the study variable to estimate finite population mean. Section 3 describes the proposed new family of exponential-type difference estimators based on known median of the study variable coupled with information on an auxiliary variable. Moreover, the theoretical minimum MSE expression of the proposed family of estimators has been derived in this Section. The performance evaluation and comparison of the suggested class of estimators based on simulation, numerical and robustness studies are presented in Section 4. Finally, Section 5 concludes the paper with a summary and concluding remarks.


## 2. Some Existing Estimators of Population Mean

Following are some formulas and representations that are utilized in existing and proposed median based estimators for the estimation of finite population mean.Cy=Sy/Y¯, Sy2=1/N−1∑i−1NYi−Y¯2, Cyx=ρyxCyCx, f=n/N, g=1−f/n, Cx=Sx/X¯, R=Y¯/X, Rm=Y¯/MY0.5, SX2=1/N−1∑i−1NXi−X¯2, Covx,y=1/N−1∑i−1NYi−Y¯Xi−X¯, ρxy=Covx,y/SxSy, CyM^y0.5=SyM^y0.5/Y¯M^y0.5, CM^y0.5=SM^y0.5/My0.5,Traditionally, the sample mean,T1=y¯, is used to estimate the population mean of the study variable Y. The variance of traditional mean estimator of the study variable is(1)VT1=1−fnY¯2Cy2Subramani and Kumarapandiyan [3] proposed an estimator utilizing known median of auxiliary variable to estimate finite population mean as follow:(2)T2=y¯MX0.5+X¯MX0.5+x¯The MSE ofT2 is given bellow(3)MSET2≅gY¯2Cy2+θ22Cx2−2θ2Cyxwhere. θ2=X¯+MX0.5/X¯Subramani and Prabavathy [14] suggested two new estimators utilizing known median of both study and auxiliary variables and mean of the auxiliary variable for the estimation of population mean. Their proposed estimators are given as(4)T3=y¯MX0.5MY0.5+X¯MX0.5M^Y0.5+X¯,T4=y¯X¯MY0.5+MX0.5X¯M^Y0.5+MX0.5.The MSE ofT3andT4 as follows(5)MSETi≅gSy2+Rm2θi2SM^Y0.52−2RmθiCovy,¯M^Y0.5fori=3,4.where Rm=Y¯/MY0.5, θ3=MX0.5MY0.5/MX0.5MY0.5+X¯, θ4=X¯MY0.5/X¯MY0.5+MX0.5Subramani and Prabavathy [4] suggested median based estimators using quartiles and their functions for the estimation of population mean, which are given bellow(6)T5=y¯MY0.5+Q1M^Y0.5+Q1,T6=y¯MY0.5+Q3M^Y0.5+Q3,T7=y¯MY0.5+QrM^Y0.5+Qr,T8=y¯MY0.5+QdM^Y0.5+Qd,T9=y¯MY0.5+QaM^Y0.5+Qa,The mean square errors of the suggested estimatorsTi are(7)MSETi≅gSy2+Rm2θi2SM^Y0.52−2RmθiCovy,¯M^Y0.5fori=5,6,…,9.where Rm=Y¯/MY0.5, θ5=MY0.5/MY0.5+Q1, θ6=MY0.5/MY0.5+Q3, θ7=MY0.5/MY0.5+Qrθ8=MY0.5/MY0.5+Qdθ9=MY0.5/MY0.5+Q3Yadav et al. [15] proposed two estimators to estimate finite population mean using known median of both study and auxiliary variables, which are(8)T10=K10y¯MX0.5MY0.5+X¯MX0.5M^Y0.5+X¯,T11=K11y¯X¯MY0.5+MX0.5X¯M^Y0.5+MX0.5.The minimum MSE ofT10 and T11 are as follows(9)Kiopt=AiBifori=10,11,Ai=1+gθi2SM^Y0.52My0.52−θiCovy,¯M^Y0.5Y¯My0.5,Bi=1+gCy2+3θi2SM^Y0.52My0.52−4θiCovy,¯M^Y0.5Y¯My0.5.where θ10=MX0.5MY0.5/MX0.5MY0.5+X¯, θ11=X¯MY0.5/X¯MY0.5+MX0.5(10)MSEminTi≅Y¯21−Ai2Bifori=10,11.Subramani [8] suggested an estimator based on known median of study variable for the estimation of population mean, which is given below(11)T12=y¯MY0.5M^Y0.5.The MSE of the suggested estimatorT12 is(12)MSETi≅gSy2+Rm2SM^Y0.52−2RmCovy,¯M^Y0.5.where Rm=Y¯/MY0.5,Kumar et al. [19] proposed an estimator to estimate finite population mean using known median of study variables, which is defined as(13)T13=y¯expMY0.5−M^Y0.5MY0.5+1−aM^Y0.5.The MSE ofT13 is(14)MSET13≅gY¯2Cy2+CM^y0.52a2−2aCyM^y0.5.Which is optimum for.aopt=CM^y0.52/CyM^y0.5So, the minimum MSE ofT13 is(15)MSEminT13≅gY¯2Cy2−CyM^y0.52CM^y0.52.Irfan et al. [16] introduced a class of power–type ratio estimators using both known median of the study and auxiliary variables to estimate population mean, which are(16)T14=K1y¯MX0.5MY0.5+X¯MX0.5M^Y0.5+X¯MX0.5MY0.5/MX0.5MY0.5+X¯,T15=K2y¯X¯MY0.5+MX0.5X¯M^Y0.5+MX0.5X¯MY0.5/X¯MY0.5+MX0.5,T16=K3y¯MY0.5+1M^Y0.5+1MY0.5/MY0.5+1,T17=K4y¯MX0.5MY0.5+RmX¯MX0.5M^Y0.5+RmX¯MX0.5MY0.5/MX0.5MY0.5+RmX¯,T18=K5y¯X¯RmMY0.5+MX0.5X¯RmM^Y0.5+MX0.5X¯RmMY0.5/X¯RmMY0.5+MX0.5,T19=K6y¯MY0.5+RmM^Y0.5+RmMY0.5/MY0.5+Rm.The MSEs of the suggested estimatorsTi=14,15,…,19 are given as(17)MSETi≅Y¯2Ki−12+gKi2Cy2+2θi4−θi3CM^y0.52−4θi2Covy,¯M^Y0.5Y¯MY0.5−gKiY¯2Cy2+θi4−θi3CM^y0.52−2θi2Covy,¯M^Y0.5Y¯M0.5.The optimum value ofKi is(18)Kiopt=AiBifori=14,15,…,19.where(19)Ai=2+gθi4−θi3CM^y0.52−2θi2Covy,¯M^Y0.5Y¯My0.5,Bi=2+2gCy2+2θi4−θi3CM^y0.52−4θi2Covy,¯M^Y0.5Y¯My0.5.and(20)MSEminTi≅Y¯21−Ai22Bifori=14,15,…,19.Yadav et al. [20] suggested two estimators of population mean utilizing known median of the study variable(21)T20=y¯MY0.5M^Y0.5a,T21=y¯MY0.5MY0.5+aM^Y0.5−MY0.5.The MSE of suggested estimatorsT20andT21 are(22)MSETi=gY¯2Cy2+ai2CM^y0.52−2aiCyM^y0.5fori=20,21.Which is optimum for.aiopt=CyM^y0.5/CM^y0.52So, the minimum MSE ofTi is(23)MSEminTi≅gY¯2Cy2−CyM^y0.52CM^y0.52fori=20,21.Kumar et al. [21] suggested an estimator of population mean using known median of the study variable(24)T22=y¯a2−MY0.5M^Y0.5+1−a2−MY0.5M^Y0.5.The MSE of proposed estimatorT22 is(25)MSET22≅gY¯2Cy2+a2CM^y0.52−2aCyM^y0.5.Which is optimum for.aopt=−CyM^y0.5/CM^y0.52So, the minimum MSE ofT22 is(26)MSEminT22≅gY¯2Cy2−CyM^y0.52CM^y0.52.Yadav et al. [13] suggested two estimators of population mean utilizing known median of the study variable(27)T23=y¯MY0.5M^Y0.5expaMY0.5−M^Y0.5MY0.5+M^Y0.5.The MSE ofT23 is(28)MSET23≅gY¯2Cy2+a2CM^y0.52−2aCyM^y0.5.Which is optimum for.aopt=−CyM^y0.5/CM^y0.52So, the minimum MSE ofT23 is(29)MSEminT23≅gY¯2Cy2−C2yM^y0.5CM^y0.52Hafeez et al. [18] introduced new estimators utilizing known median of the study variable for the estimation of finite population mean(30)T24=y¯expMY0.5−M^Y0.5MY0.5+M^Y0.5,T25=y¯+KMY0.5−M^Y0.5,T26=K1y¯+K2MY0.5−M^Y0.50.5,T27=K1y¯+K2MY0.5−M^Y0.5expMY0.5−M^Y0.5MY0.5+M^Y0.5,The MSE ofT24 estimator is(31)MSET24≅Y¯2Cy2+CM^y0.524−CyM^y0.5The minimum MSE ofT25 at Kopt=Covy¯,M^Y0.5/VM^y0.5. is given as(32)MSEminT25≅Sy21−ρyM^y0.52The minimum MSE ofT26 at optimum values(33)K1opt=11+Cy21−ρyM^y0.52,K2opt=Sy2ρyM^y0.521+Cy21−ρyM^y0.52.is(34)MSEminT26≅Y¯2Cy21−ρyM^y0.521+Cy21−ρyM^y0.52The minimum MSE ofT27 at optimum values(35)K1opt=1−1/8CM^y0.521+Cy21−ρyM^y0.52,K2opt=1−1/8CM^y0.521+Cy21−ρyM^y0.52CyM^y0.5CM^y0.52−1+1/2R2,MSEminT27≅Y¯2Cy21−ρyM^y0.521+Cy21−ρyM^y0.52−Y¯2CM^y0.524Cy21−ρyM^y0.52+CM^y0.524161+Cy21−ρyM^y0.52.


## 3. Proposed Class of Exponential-type Difference Estimator Based on Median

The proposed class of exponential-type difference estimators for the estimation of population mean using known median of the study variable is defined as follows(36)Tpj=λ1y¯+λ2My0.5−M^y0.5expaMy0.5−M^y0.5aMy0.5−M^y0.5+2bforj=1,2,…,6.where λ1 and λ2 are the constants at which mean squared error is minimized and a,b be the known characteristics of the parameters of the population. The mean squared error of proposed class Tpj can be obtained as follows:To study the properties of the suggested estimators based on known median of study variable under simple random sampling scheme the error terms are defined as follows:Let us defineeo=y¯−Y¯/Y¯ and e1=M^Y0.5−MY0.5/MY0.5 so that y¯=Y¯1+e0 and M^y0.5=MY0.51+e1. From these definitions of e0 and e1, We get Ee0=Ee1=0, while Ee02=1−f/nCy2, Ee12=1−f/nCMYo.52, and Eeoe1=1−f/nρyMYo.5CyCMY0.5 with. CyMY0.5=Covy,¯M^Y0.5/Y¯My0.5The proposed class of estimatorsTpj can be written in the terms of e0ande1 as(37)Tpj=λ1Y¯1+e0+λ2My0.5−My0.51+e1expaMy0.5−My0.51+e1aMy0.5−My0.51+e1+2b,Tpj=λ1Y¯1+e0−λ2My0.5e1exp−aMy0.5e12aMy0.5+b1+aMy0.5e1/2aMy0.5+b.Now,(38)Tpj−Y¯=Y¯λ1−1+Y¯e0−Y¯θe02+38Yθ2e12−Y¯θe0e12−λ2My0.5e1+λ2My0.5θe122,where θ=aMy0.5/aMy0.5+b.By squaring both sides and ignoring theei terms which are greater than two, the mean squared error is given as(39)MSETpj≅Y¯2+Y¯2λ12ΔA+λ22ΔB+λ1Y¯2ΔC−λ2Y¯ΔD+λ1λ2Y¯ΔE.where(40)ΔA=1+gCy2+14θ2CMy0.52+38θ2CMy0.52−2θCyMy0.5,ΔB=gMy0.52CMy0.52ΔC=gθCyMy0.5−38θ2CMy0.52−2,ΔD=gθMy0.5CMy0.52ΔE=g2θMy0.5CMy0.52−2My0.5CyMy0.5.TheMSETpj is minimized for the values(41)λ1∗=ΔDΔE+2ΔBΔCΔE2−4ΔAΔB,λ2∗=−Y¯ΔCΔE+2ΔAΔDΔE2−4ΔAΔB.Hence the minimum MSE of classTpj is(42)MSEminTpj≅Y¯21+ΔAΔD2+ΔBΔC2+ΔCΔDΔEΔE2−4ΔAΔB.


## 4. Simulation Study

A simulation study is also conducted to evaluate the performance of existing and suggested estimators. For this purpose, the statistical programming R language is utilized. The following procedure is adopted to compute MSE of proposed estimators and their competitors.(i)
Generate a random sample of sizen=3 and 5 from bivariate normal distribution.(ii)
Compute the MSE of existing and suggested estimators by using generated random sample with the help of the formulas defined in Section2 and Section 3.(iii)
Repeat step (i) and (ii) 30000 times to obtain MSEs(iv)
Average these MSEs to obtain the values of the MSE of existing and proposed estimators.The results of the simulation study in the terms of MSEs are Table1presented in Table 2. The key findings from these results are summarized as:(i)
The proposed estimators are more efficient as compared to other traditional mean and other existing estimators based on median.(ii)
As value ofn increases, the value of MSE decreases, and vice versa.(iii)
Among all the proposed estimators, the estimatorsTP21 have smaller MSE for each choice of n.Table 1
Proposed class of median based estimators.
EstimatorsabTP1MX0.5X¯TP2X¯MX0.5TP311TP4Y¯RmMX0.5TP5MX0.5Y¯RmTP61RmTP71TMTP81HLTP91DMTP10RmTMTP11RmHLTP12RmDMTP13ρyxTMTP14ρyxHLTP15ρyxDMTP16CxTMTP17CxHLTP18CxDMTP19CM^Y0.5TMTP20CM^Y0.5HLTP21CM^Y0.5DMTable 2
MSEs of existing and proposed estimators with simulated data.
ExistingEstimatorsn = 3n = 5ProposedEstimatorsn = 3n = 5T187.501351.1371TP169.934641.3588T281.010647.3438TP269.935741.3592T373.187442.7718TP369.935241.3590T473.258642.8135TP469.935741.3592T5227.8969227.8969TP569.934641.3588T6226.0171226.0171TP669.935241.3590T7226.6818226.6818TP769.853141.3295T8227.6978227.6978TP869.852241.3292T9226.8375226.8375TP969.851941.3291T10205.2060205.206TP1069.853141.3295T11205.2845205.2845TP1169.852341.3292T12228.9906228.9906TP1269.851941.3291T1371.974542.0629TP1369.852741.3294T1470.453141.7661TP1469.851941.3291T1570.553441.8309TP1569.851641.3290T1670.506041.8004TP1669.853841.3298T1770.552241.8302TP1769.852841.3294T1870.454641.7671TP1869.852441.3292T1970.507441.8012TP1969.850141.3284T2071.974442.0630TP2069.850041.3284T2171.974442.0630TP2169.849941.3283T2271.974442.0630T2371.974442.0630T24230.0941230.0941T25224.3360224.3360T26204.9097204.9097T27203.0698203.0698


### 4.1. Numerical Study

For assessing the performance of the suggested class of estimators over existing estimators, we have performed a numerical study utilizing the datasets from the previous studies of the Subramani and Prabavathy [4] and Subramani [8]. The descriptive statistics of these datasets are given in Table 3.Table 3
Datasets for numerical study.
CharacteristicsPop-1Pop-2Pop-3Pop-4Pop-5Pop-6N343434348080n335533Y¯856.4118856.4118856.4118856.41185182.6375182.637X¯208.8824199.4412208.8824199.4412285.1251126.463MY0.5767.5767.5767.5767.551055105MX0.5150142.5150142.5148757.5Q194.2599.2594.2599.2586.50517.50Q3254.75278.00254.75278.00445.251693.75Qr160.50178.75160.50178.75358.751176.25Qd80.2589.37580.2589.375179.375588.125Qa174.50188.625174.50188.625265.8751105.625R4.09994.29414.09994.294118.17674.6008Rm1.11581.11581.11581.11581.01521.0152Sy2163356.41163356.4191690.37191690.3713365953.463365953.46Sx26884.44556857.85553864.17263849.248073061.2791715671.523SMY0.52101518.77101518.7759396.283659396.283617302151730215Covy,¯M^Y0.590236.29490236.29448074.954248074.954211983751198375ρyx0.44910.44530.44910.44530.9150.941Covy,¯x¯15061.40114905.0498453.81878366.0597453752.2411460496.58Cy20.2227260.2227260.1250140.1250140.1253160.125316Cx20.1577850.1724080.088560.0967710.8987040.564001CMY0.520.1723410.1723410.1723410.1008330.0663910.066391CyMY0.50.1372840.1372840.1372840.073140.0452950.045295Cyx0.0841940.087260.0472570.0489810.30710.25017TM162.2589.375162.2589.375206.937931.562HL190.00320.00190.00320.00249.0001040.500DM234.82206.944234.82206.944276.1891150.700For comparison of the proposed and existing estimators, the PREs of the suggested estimatorsTpj with respect to the existing estimators Ti is calculated as(43)PRETi,Tpj=MSETiMSETpj∗100.The MSEs and PREs for suggested and existing estimators are given in Table4–6. The key findings from the results are summarized as:(1)
The performance of existing estimatorT18 and T13 is comparatively better with small MSE and higher PRE among other competitor estimators of the study (cf. Table 4).(2)
The suggested family of estimators is more efficient as compared to traditional and existing estimators of the mean. The proposed estimatorTP18 is most efficient (cf. Table 5).(3)
The suggested estimators are superior in performance with large value of PRE than the existing estimators of the study. The smallest gained value of PRE of a proposed estimator is also higher than largest achieved value of the existing estimator. The proposed class of estimators got highest PRE values of202.80 and 175.73 percent against usual estimator of population mean for the sample size n=3 and 5, respectively (cf. Table 6).(4)
The MSE tends to decrease as sample sizen increases, and vice versa (cf. Table 5).(5)
The PRE of the suggested class of estimators in comparison to the existing estimator with least MSE among all existing estimators increases asn increases (cf. Table 6).Table 4
Mean square error of the existing estimators.
Estimatorsn = 3n = 5Population.1Population.2Population.3Population.4T149647.5449647.5415641.3015641.30T239715.6940030.5212512.3012611.49T326832.1226831.989942.4969942.433T426845.7326845.809948.5389948.570T584266.7184147.8954798.7654675.13T683413.7083642.2052826.6652784.47T783266.0183175.3253543.5053322.41T884643.7584390.4355174.3054924.52T983190.4483153.4653369.8153221.37T1073465.0673465.0250891.4450891.29T1173468.7473468.7650906.4250906.50T1288379.0788379.0758356.9258356.92T1325270.6825270.689003.4479003.447T1425270.8425270.649692.5929692.479T1525290.0425290.159703.3869703.444T1625281.9925281.999698.8639698.863T1725292.1825292.279704.5859704.638T1825266.2825266.069690.0249689.899T1925278.6925278.699697.0089697.008T2025270.6825270.689003.4479003.447T2125270.6825270.689003.4479003.447T2225270.6825270.689003.4479003.447T2325270.6825270.689003.4479003.447T2494267.1794267.1756534.8956534.89T2583148.6983148.6952778.8352778.83T2674682.1474682.1449235.7949235.79T2771158.7171158.7147885.9547885.95Table 5
Mean squared error of the proposed estimators.
Estimatorsn = 3n = 5Population.1Population.2Population.3Population.4TP124567.50024567.5008911.9068911.906TP224567.73024567.7408911.9368911.936TP324567.64024567.6408911.9238911.923TP424567.76024567.7608911.9398911.939TP524567.44024567.4408911.8998911.899TP624567.60024567.6008911.9188911.918TP724524.79024524.1308906.4868906.401TP824519.45024520.5608905.8068905.948TP924513.57024516.4008905.0588905.418TP1024528.27024527.6508906.9288906.848TP1124523.20024524.2608906.2838906.418TP1224517.58024520.2908905.5688905.913TP1324494.49024493.3008902.6298902.478TP1424487.96024488.9408901.7988901.922TP1524481.30024484.0908900.9498901.305TP1624489.41024490.4108900.4798900.602TP1724482.91024486.0508899.6928900.068TP1824476.37024481.2408898.9258899.493TP1924491.24024490.4008900.8128900.708TP2024484.72024486.0408900.0118900.171TP2124478.13024481.2308899.2268899.592Table 6
PRE’s of existing and proposed estimators with respect to usual estimator of mean.
ExistingEstimatorsPop-1Pop-3ProposedEstimatorsPop-1Pop-3n = 3n = 5n = 3n = 5T1100.00100.00TP1202.08175.51T2125.01125.01TP2202.08175.51T3185.03157.32TP3202.09175.51T4184.94157.22TP4202.08175.51T558.9228.54TP5202.09175.51T659.5229.61TP6202.09175.51T759.6329.21TP7202.44175.62T858.6528.35TP8202.48175.63T959.6829.31TP9202.53175.65T1067.5830.73TP10202.41175.61T1167.5830.73TP11202.45175.62T1256.1826.80TP12202.50175.64T13196.46173.73TP13202.69175.69T14196.46161.37TP14202.74175.71T15196.31161.19TP15202.80175.73T16196.37161.26TP16202.73175.73T17196.30161.17TP17202.78175.75T18196.50161.41TP18202.84175.77T19196.40161.30TP19202.72175.73T20196.46173.72TP20202.77175.74T21196.46173.72TP21202.82175.76T22196.46173.72T23196.46173.72T2452.6727.67T2559.7129.64T2666.4831.76T2769.7732.66The graphical comparison of suggested estimators with the usual and other competing estimators is also presented in this study by using datasets 1 and 3 with sample size 3 and 5. Figures1 and 2, shows that the proposed estimators have smaller MSEs as compared to traditional mean and other existing estimators considered in this study, which indicate the dominancy of proposed estimators over its competitors.Figure 1
Mean squared error of the proposed and existing estimators for population-1 withn = 3.Figure 2
Mean squared error of the proposed and existing estimators of population-3 withn = 5.


### 4.2. Robustness Study of the Proposed Estimators

The non-conventional measures utilized in the study such as tri-mean, decile mean, and Hodges-Lehmann are robust in the existence of outliers. Therefore, these non-conventional measures attain precise results than other conventional measures in the presence of extreme values in the data. In the present section, we assess the performance of our suggested estimator in case of outliers. For this purpose, we utilized both population 5 and 6 which have some extreme values. In Figures3 and 4, the scatter diagrams are clearly indicating the existence of outliers in the data. We obtained MSEs of existing and suggested estimator with both dataset, which are given in Table 7 to compare the performance. It is observed that proposed estimator have smaller values of MSE as compared to the MSE of the usual mean estimator and other competing estimators considered in this study. This comparison shows that the suggested estimators are more precise in the presence of extreme observations in the data. The estimator TP21 is most efficient among all existing and suggested estimators with least MSE values with both datasets.Figure 3
Scatter diagram between study and auxiliary variables of Population-5.Figure 4
Scatter diagram between study and auxiliary variables of Population-6.Table 7
MSEs of existing and proposed estimators in the presence of outliers.
ExistingEstimatorsn = 3ProposedEstimatorsn = 3Population-5Population-6Population-5Population-6T110799101079910TP1791577.3791577.6T2952162.7239478.2TP2791578.3791578.2T3871244.7871275.9TP3791578.0791578.0T4871344.8871333.9TP4791578.3791578.2T526976082626793TP5791577.3791577.6T626365572544339TP6791578.0791578.0T726492722566306TP7791436.9791048.1T826795812617952TP8791410.0791001.1T926642642570760TP9791393.0790955.9T1023649262364965TP10791438.9791054.3T1123650522365038TP11791412.4791007.6T1227159952715995TP12791395.6790962.8T13813613.9813613.9TP13791424.5791022.6T14831680.9831731.2TP14791395.5790974.0T15831842.0831824.4TP15791377.1790927.6T16831786.9831786.9TP16791429.6790920.8T17831842.9831825.6TP17791401.4790867.1T18831677.6831728.6TP18791383.6790816.3T19831785.2831785.2TP19791106.9790346.5T20813613.9813613.9TP20791032.8790287.7T21813613.9813613.9TP21790987.9790236.0T22813613.9813613.9T23813613.9813613.9T2425951642595164T2525359392535939T2623171662317166T2722770162277016


## 4.1. Numerical Study

For assessing the performance of the suggested class of estimators over existing estimators, we have performed a numerical study utilizing the datasets from the previous studies of the Subramani and Prabavathy [4] and Subramani [8]. The descriptive statistics of these datasets are given in Table 3.Table 3
Datasets for numerical study.
CharacteristicsPop-1Pop-2Pop-3Pop-4Pop-5Pop-6N343434348080n335533Y¯856.4118856.4118856.4118856.41185182.6375182.637X¯208.8824199.4412208.8824199.4412285.1251126.463MY0.5767.5767.5767.5767.551055105MX0.5150142.5150142.5148757.5Q194.2599.2594.2599.2586.50517.50Q3254.75278.00254.75278.00445.251693.75Qr160.50178.75160.50178.75358.751176.25Qd80.2589.37580.2589.375179.375588.125Qa174.50188.625174.50188.625265.8751105.625R4.09994.29414.09994.294118.17674.6008Rm1.11581.11581.11581.11581.01521.0152Sy2163356.41163356.4191690.37191690.3713365953.463365953.46Sx26884.44556857.85553864.17263849.248073061.2791715671.523SMY0.52101518.77101518.7759396.283659396.283617302151730215Covy,¯M^Y0.590236.29490236.29448074.954248074.954211983751198375ρyx0.44910.44530.44910.44530.9150.941Covy,¯x¯15061.40114905.0498453.81878366.0597453752.2411460496.58Cy20.2227260.2227260.1250140.1250140.1253160.125316Cx20.1577850.1724080.088560.0967710.8987040.564001CMY0.520.1723410.1723410.1723410.1008330.0663910.066391CyMY0.50.1372840.1372840.1372840.073140.0452950.045295Cyx0.0841940.087260.0472570.0489810.30710.25017TM162.2589.375162.2589.375206.937931.562HL190.00320.00190.00320.00249.0001040.500DM234.82206.944234.82206.944276.1891150.700For comparison of the proposed and existing estimators, the PREs of the suggested estimatorsTpj with respect to the existing estimators Ti is calculated as(43)PRETi,Tpj=MSETiMSETpj∗100.The MSEs and PREs for suggested and existing estimators are given in Table4–6. The key findings from the results are summarized as:(1)
The performance of existing estimatorT18 and T13 is comparatively better with small MSE and higher PRE among other competitor estimators of the study (cf. Table 4).(2)
The suggested family of estimators is more efficient as compared to traditional and existing estimators of the mean. The proposed estimatorTP18 is most efficient (cf. Table 5).(3)
The suggested estimators are superior in performance with large value of PRE than the existing estimators of the study. The smallest gained value of PRE of a proposed estimator is also higher than largest achieved value of the existing estimator. The proposed class of estimators got highest PRE values of202.80 and 175.73 percent against usual estimator of population mean for the sample size n=3 and 5, respectively (cf. Table 6).(4)
The MSE tends to decrease as sample sizen increases, and vice versa (cf. Table 5).(5)
The PRE of the suggested class of estimators in comparison to the existing estimator with least MSE among all existing estimators increases asn increases (cf. Table 6).Table 4
Mean square error of the existing estimators.
Estimatorsn = 3n = 5Population.1Population.2Population.3Population.4T149647.5449647.5415641.3015641.30T239715.6940030.5212512.3012611.49T326832.1226831.989942.4969942.433T426845.7326845.809948.5389948.570T584266.7184147.8954798.7654675.13T683413.7083642.2052826.6652784.47T783266.0183175.3253543.5053322.41T884643.7584390.4355174.3054924.52T983190.4483153.4653369.8153221.37T1073465.0673465.0250891.4450891.29T1173468.7473468.7650906.4250906.50T1288379.0788379.0758356.9258356.92T1325270.6825270.689003.4479003.447T1425270.8425270.649692.5929692.479T1525290.0425290.159703.3869703.444T1625281.9925281.999698.8639698.863T1725292.1825292.279704.5859704.638T1825266.2825266.069690.0249689.899T1925278.6925278.699697.0089697.008T2025270.6825270.689003.4479003.447T2125270.6825270.689003.4479003.447T2225270.6825270.689003.4479003.447T2325270.6825270.689003.4479003.447T2494267.1794267.1756534.8956534.89T2583148.6983148.6952778.8352778.83T2674682.1474682.1449235.7949235.79T2771158.7171158.7147885.9547885.95Table 5
Mean squared error of the proposed estimators.
Estimatorsn = 3n = 5Population.1Population.2Population.3Population.4TP124567.50024567.5008911.9068911.906TP224567.73024567.7408911.9368911.936TP324567.64024567.6408911.9238911.923TP424567.76024567.7608911.9398911.939TP524567.44024567.4408911.8998911.899TP624567.60024567.6008911.9188911.918TP724524.79024524.1308906.4868906.401TP824519.45024520.5608905.8068905.948TP924513.57024516.4008905.0588905.418TP1024528.27024527.6508906.9288906.848TP1124523.20024524.2608906.2838906.418TP1224517.58024520.2908905.5688905.913TP1324494.49024493.3008902.6298902.478TP1424487.96024488.9408901.7988901.922TP1524481.30024484.0908900.9498901.305TP1624489.41024490.4108900.4798900.602TP1724482.91024486.0508899.6928900.068TP1824476.37024481.2408898.9258899.493TP1924491.24024490.4008900.8128900.708TP2024484.72024486.0408900.0118900.171TP2124478.13024481.2308899.2268899.592Table 6
PRE’s of existing and proposed estimators with respect to usual estimator of mean.
ExistingEstimatorsPop-1Pop-3ProposedEstimatorsPop-1Pop-3n = 3n = 5n = 3n = 5T1100.00100.00TP1202.08175.51T2125.01125.01TP2202.08175.51T3185.03157.32TP3202.09175.51T4184.94157.22TP4202.08175.51T558.9228.54TP5202.09175.51T659.5229.61TP6202.09175.51T759.6329.21TP7202.44175.62T858.6528.35TP8202.48175.63T959.6829.31TP9202.53175.65T1067.5830.73TP10202.41175.61T1167.5830.73TP11202.45175.62T1256.1826.80TP12202.50175.64T13196.46173.73TP13202.69175.69T14196.46161.37TP14202.74175.71T15196.31161.19TP15202.80175.73T16196.37161.26TP16202.73175.73T17196.30161.17TP17202.78175.75T18196.50161.41TP18202.84175.77T19196.40161.30TP19202.72175.73T20196.46173.72TP20202.77175.74T21196.46173.72TP21202.82175.76T22196.46173.72T23196.46173.72T2452.6727.67T2559.7129.64T2666.4831.76T2769.7732.66The graphical comparison of suggested estimators with the usual and other competing estimators is also presented in this study by using datasets 1 and 3 with sample size 3 and 5. Figures1 and 2, shows that the proposed estimators have smaller MSEs as compared to traditional mean and other existing estimators considered in this study, which indicate the dominancy of proposed estimators over its competitors.Figure 1
Mean squared error of the proposed and existing estimators for population-1 withn = 3.Figure 2
Mean squared error of the proposed and existing estimators of population-3 withn = 5.


## 4.2. Robustness Study of the Proposed Estimators

The non-conventional measures utilized in the study such as tri-mean, decile mean, and Hodges-Lehmann are robust in the existence of outliers. Therefore, these non-conventional measures attain precise results than other conventional measures in the presence of extreme values in the data. In the present section, we assess the performance of our suggested estimator in case of outliers. For this purpose, we utilized both population 5 and 6 which have some extreme values. In Figures3 and 4, the scatter diagrams are clearly indicating the existence of outliers in the data. We obtained MSEs of existing and suggested estimator with both dataset, which are given in Table 7 to compare the performance. It is observed that proposed estimator have smaller values of MSE as compared to the MSE of the usual mean estimator and other competing estimators considered in this study. This comparison shows that the suggested estimators are more precise in the presence of extreme observations in the data. The estimator TP21 is most efficient among all existing and suggested estimators with least MSE values with both datasets.Figure 3
Scatter diagram between study and auxiliary variables of Population-5.Figure 4
Scatter diagram between study and auxiliary variables of Population-6.Table 7
MSEs of existing and proposed estimators in the presence of outliers.
ExistingEstimatorsn = 3ProposedEstimatorsn = 3Population-5Population-6Population-5Population-6T110799101079910TP1791577.3791577.6T2952162.7239478.2TP2791578.3791578.2T3871244.7871275.9TP3791578.0791578.0T4871344.8871333.9TP4791578.3791578.2T526976082626793TP5791577.3791577.6T626365572544339TP6791578.0791578.0T726492722566306TP7791436.9791048.1T826795812617952TP8791410.0791001.1T926642642570760TP9791393.0790955.9T1023649262364965TP10791438.9791054.3T1123650522365038TP11791412.4791007.6T1227159952715995TP12791395.6790962.8T13813613.9813613.9TP13791424.5791022.6T14831680.9831731.2TP14791395.5790974.0T15831842.0831824.4TP15791377.1790927.6T16831786.9831786.9TP16791429.6790920.8T17831842.9831825.6TP17791401.4790867.1T18831677.6831728.6TP18791383.6790816.3T19831785.2831785.2TP19791106.9790346.5T20813613.9813613.9TP20791032.8790287.7T21813613.9813613.9TP21790987.9790236.0T22813613.9813613.9T23813613.9813613.9T2425951642595164T2525359392535939T2623171662317166T2722770162277016


## 5. Summary and Conclusion

A class of median based exponential-type difference estimators under SRS scheme have been proposed utilizing auxiliary information on non-conventional measures. After empirical and graphical study, it is found that suggested class of estimators are more efficient in term of MSE as compared to usual sample mean and other existing estimators considered in this study. The PREs also reveal the dominancy of suggested estimators over traditional and competing estimators. A simulation study based on generated simple random sample with samples of sizen=3, and 5 also showed that the proposed median based estimator of the study variable are more efficient with smaller MSE than usual mean and other existing estimators. The present study can also be extended to the other sampling schemes such as stratified random sampling, systematic sampling, and ranked set sampling to improve efficiency of the mean estimators.



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*Source: 9600982-2022-09-19.xml*