# A New Generalized Class of Exponential Factor-Type Estimators for Population Distribution Function Using Two Auxiliary Variables

Authors: Sohaib Ahmad; Muhammad Aamir; Sardar Hussain; Javid Shabbir; Erum Zahid; Khanyaluck Subkrajang; Anuwat Jirawattanapanit
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this article, we propose a generalized class of exponential factor type estimators for estimation of the finite population distribution function (PDF) using an auxiliary variable in the form of the mean and rank of the auxiliary variable exist. The expressions of the bias and mean square error of the estimators are computed up to the first order approximation. The proposed estimators provide minimum mean square error as compared to all other considered estimators. Three real data sets are used to check the performance of the proposed estimators. Moreover, simulation studies are also carried out to observe the performances of the proposed estimators. The proposed estimators confirmed their superiority numerically as well as theoretically by producing efficient results as compared to all other competing estimators.


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## Body

## 1. Introduction

In survey sampling, it is well-known fact that the suitable use of auxiliary information improves the precision of estimators by taking advantage of the correlation between the study variable and the auxiliary variable. Several estimators exist in the literature for estimating various population parameters including mean, median, and total, but little attention has been paid to estimate the distribution function (DF). Some important references to the population mean using the auxiliary information include [1–21]. In their work, several authors have suggested improved ratio, product, and regression type estimators for estimating the finite population mean.The problem of estimating a finite population (DF) arises when we are interested to find out the proportion of certain values that are less than or equal to the threshold value. There are situations where estimating the cumulative distribution function is assessed as essential. For example, for a nutritionist, it is interesting to know the proportion of the population that consumed 25 percent or more of the calorie intake from saturated fat. Similarly, a soil scientist may be interested in knowing the proportion of people living in a developing country below the poverty line. In certain situations, the need of a cumulative distribution function is much more important. Some important work in the field of distribution function (DF) includes [22], which suggested an estimator for estimating the DF that requires information both on the study variable and the auxiliary variable. On similar lines, [23] proposed ratio and difference-type estimators for estimating the DF using the auxiliary information. Ahmad and Abu-Dayyeh [24] estimated the DF using the information on multiple auxiliary variables. Rueda et al. [25] used a calibration approach for estimating the DF. Singh et al. [26] considered the problem of estimating the DF and quantiles with the use of auxiliary information at the estimation stage, [27] considered a generalized class of estimators for estimating the DF in the presence of nonresponse, [28] suggested finite population distribution function estimation with dual use of auxiliary information under simple and stratified random sampling. Moreover, two new estimators were proposed for estimating the DF in simple and stratified sampling using the auxiliary variable and rank of the auxiliary variable.The rest of the article is composed as follows: in Section2, some notations and symbols are given. In Section 3, the existing estimators for estimating the DF are given. In Section 4, we define two new generalized exponential factor type estimators. Section 6 discusses the numerical study of the proposed class of estimators. We also conduct a simulation study for the support of our proposed generalized family of estimators in Section 7. Section 8 gives the concluding remarks.


## 2. Notations and Symbols

A finite populationΩ=1,2,…,N of N distinct and identified units is considered. To estimate the finite population distribution function (DF), a sample of size n units is drawn from a population using simple random sampling without replacement (SRSWOR). Let Yi, Xi and Rx be the values of the study variable Y, the auxiliary variable X and Rx rank of the auxiliary variable X, respectively. Let IY≤ty and IX≤tx be the indicator variables based on Y and X, respectively.LetF^ty=∑i=1nIYi≤ty/n and F^tx=∑i=1nIXi≤tx/n be the sampled distribution functions corresponding to the population distribution function F^ty=∑i=1nIXi≤ty/n and Ftx=∑i=1nIXi≤tx/n, respectively. Let Y¯^, X¯^, and R¯^x be the sample means corresponding to population means Y¯, X¯, and R¯x, respectively. Let δ12=∑i=1NIYi≤ty−Fty2/N−1, δ22=∑i=1NIYi≤tx−Ftx2/N−1, δ32=∑i=1NXi−X¯2/N−1, δ42=∑i=1NRx−R¯x2/N−1 be the population variances of IY≤ty, IX≤tx, X and Rx, respectively. Let C1=δ1/Fty, C2=δ2/Ftx, C3=δ3/X¯, C4=δ4/R¯x be the coefficients of variations of IY≤ty,IX≤tx, X, and Rx, respectively. Let δ12=∑i=1NIYi≤ty−FtyIXi≤tx−Ftx/N−1, δ13=∑i=1NIYi≤ty−FtyIXi≤tx−Ftx/N−1,(1)δ23=∑i=1NIYi≤tx−FtxXi≤X¯xN−1,(2)δ14=∑i=1NIYi≤ty−FtyRxi−R¯xN−1.δ24=∑i=1NIYi≤tx−FtyRxi−R¯x/N−1 be the population covariances between IY≤ty, IX≤tx, IY≤ty, IX≤tx and Rx, respectively. Let ρ12=δ12/δ1δ2, ρ13=δ13/δ1δ3, ρ23=δ23/δ1δ3, ρ14=δ14/δ1δ4, ρ24=δ24/δ2δ4, be the correlation coefficients between IY≤ty and IX≤tx, IY≤ty and X, IX≤tx and X, IY≤ty and Rx, IX≤tx and Rx, respectively. Let ρ1.232=ρ122+ρ132−2ρ12ρ13ρ23/1−ρ232, ρ1.242=ρ122+ρ142−2ρ12ρ14ρ24/1−ρ242 be the population coefficients of multiple determination of Y≤ty on IX≤tx and X, IY≤ty on IX≤tx and Rx, respectively.To obtain the biases and mean squared errors (MSEs) of the adapted and proposed estimators ofFty, we consider the following relative error terms. Let(3)ξ0=F^ty−FtyFty,ξ1=F^tx−FtxFtx,ξ2=X¯^−X¯X¯,ξ3=R¯^x−R¯xR¯x.Such thatEξi=0 for i=0,1,2,3. Vrstu=Eξ0rξ1sξ2tξ3u(4)Eξ02=λC12,Eξ12=λC22,Eξ22=λC32,Eξ32=λC42,Eξ0ξ1=λρ12C1C2.(5)Eξ0ξ2=λρ13C1C3,Eξ0ξ3=λρ14C1C4,Eξ1ξ2=λρ23C2C3,Eξ1ξ3=λρ24C2C4,whereλ=N−nnN.


## 3. Existing Estimators

In this section, we briefly review some existing estimators ofFty.(1)
The conventional unbiased mean per unit estimator ofFty is(6)F^ty=1n∑i=1nYi≤ty.The variance ofF^ty is given by the following equation:(7)VarF^ty=λF2tyC12.(2)
The traditional ratio estimator ofFty is(8)F^tyR=F^tyFtxF^tx.The bias and MSE ofF^tyR, to the first order of approximation, respectively, are given by the following equation:(9)BiasF^tyR≅λFtyC22−ρ12C1C2,(10)MSEF^tyR≅λF2tyC12+C22−2ρ12C1C2.The ratio estimatorF^tyR performs better than F^ty, in terms of MSE, if ρ12>C2/2C1.(3)
Reference [29] suggested the usual product estimator of Fty:(11)F^tyP=F^tyF^txFtx.The bias and MSE ofF^tyP, to the first order of approximation are given by the following equation:(12)BiasF^tyP=λFtyρ12C1C2,(13)MSEF^tyP≅λF2tyC12+C22+2ρ12C1C2.The product estimatorF^tyP is better than F^ty, in terms of MSE, if ρ12<−C2/2C1.(4)
The conventional difference estimator ofFty is(14)F^tyReg=F^ty+mFtx−F^tx,wherem is an unknown constant. The minimum variance of F^tyReg at the optimum value of mopt=ρ12δ1/δ2, is given by the following equation:(15)VarminF^tyReg=λF2tyC121−ρ122.(5)
Reference [4] suggested an improved difference-type estimator of Fty, given by the following equation:(16)F^tyR,D=m1F^ty+m2Ftx−F^tx,wherem1 and m2 are unknown constants.The optimum values ofm1 and m2 are(17)m1opt=11+λC121−ρ122,m2opt=Ftyρ12C1FtxC21+λC121−ρ122.The bias and minimum MSE ofF^tyR,D, to the first order of approximation, respectively, are given by the following equation:(18)BiasF^tyR,D=Ftym1−1,(19)MSEminF^tyR,D≅λF2tyC121−ρ1221+λC121−ρ122.(6)
Reference [30] suggested the exponential ratio-type and product-type estimators are given by the following equation:(20)F^tyBT,R=F^tyexpFtx−F^txF^tx+Ftx,(21)F^tyBT,P=F^tyexpF^tx−FtxF^tx+Ftx.The biases and MSEs ofF^tyBT,R and F^tyBT,P, to the first order of approximation, respectively, are given by the following equation:(22)BiasF^tyBT,R≅λFty3C228−ρ12C1C22,(23)MSEF^tyBT,R≅λF2ty44C12+C22−4ρ12C1C2,(24)BiasF^tyBT,P≅λFtyρ12C1C22−C228,(25)MSEF^tyBT,P≅λF2ty44C12+C22+4ρ12C1C2.(7)
Reference [14] suggested a generalized class of ratio-type exponential estimators as follows:(26)F^tyGK=m3F^ty+m4Ftx−F^txexpFtx−F^txFtx+F^tx,wherem3 and m4 are unknown constants.The optimum values ofm3 and m4, determined by minimizing (26) are given by the following equation:(27)m3opt=8−λ2C2281+λC121−ρ122,(28)m4opt=Ftyλ3C23+8ρ12C1−λ2ρ12C1C22−4C21−λC121−ρ1228FtxC21+λC121−ρ122.The bias ofF^yGK, are given by the following equation:(29)BiasF^tyGK≅Fty−m3Fty+38m32FtyλC12+12m4FtxλC22,(30)−12m3Ftyλρ12C1C2.The minimum MSE ofF^yGK at the optimum values of m3 and m4 is given by the following equation:(31)MSEF^tyGKmin=VarminF^Regty−λ2F2tyC22+8C121−ρ1222641+λC121−ρ122,


## 4. Proposed Estimator

Using the appropriate auxiliary information during the estimation stage or at the design stage improves an estimator’s efficiency. The sample distribution function of the auxiliary variable has already been employed to increase the efficiency and quality of estimators. The study of [20] suggested using the rank of the auxiliary variable as an additional auxiliary variable to improve the precision of a population distribution function estimator. In this article, we used two auxiliary variables to estimate the finite distribution function; we need additional auxiliary information on the sample mean and sample distribution function of the auxiliary variable, as well as the sample distribution function of the study variable. In literature, auxiliary information using the distribution function has been rarely attempted, therefore we are motivated towards it. The principal advantage of our proposed generalized class of estimator is that it is more flexible, and efficient than the existing estimators.


### 4.1. First Proposed Estimator

On the lines of [31], we suggest a generalized class of exponential factor type estimators which contains many stable and efficient estimators. By combining the idea of [30, 31], the first estimator is given by the following equation:(32)F^tyProp1k1,k2=F^tyexpS11−M11S11+M11expS22−M22S22+M22,where(33)S11=A1+T1Ftx+fB1F^tx,S22=A2+T2X¯+fB2X¯^,(34)M11=A1+fB1Ftx+T1F^tx,M22=A2+fB2X¯+T2X¯,^(35)Ai=Ki−1Ki−2,Bi=Ki−1Ki−4,(36)Ti=Ki−2Ki−3Ki−4,f=nN,Substituting different values ofKi (i = 1, 2, 3, 4) in Equation (32), we can generate many more types of estimators from our general proposed class of estimators, given in Table 1.Table 1
Family members of the proposed class of estimatorsF^tyProp1k1,k2.
S.NoK1K2EstimatorsI11F^tyProp11,1=F^tyexpFtx−F^tx/Ftx+F^txexpX¯−X¯^/X¯+X¯^II12F^tyProp11,1=F^tyexpFtx−F^tx/Ftx+F^txexpX¯−X¯^/X¯+X¯^III13F^tyProp11,3=F^tyexpFtx−F^tx/Ftx+F^txexpnX¯−X¯^/2NX¯−nX¯^+X¯IV14F^tyProp11,4=F^tyexpFtx−F^tx/Ftx+F^txV21F^tyProp12,1=F^tyexpF^tx−Ftx/F^tx+FtxexpX¯−X¯^/X¯+X¯^VI22F^tyProp12,2=F^tyexpF^tx−Ftx/F^tx+FtxexpX¯−X¯^/X¯+X¯^VII23F^tyProp12,3=F^tyexpF^tx−Ftx/F^tx+FtxexpnX¯−X¯^/2NX¯−nX¯^+X¯VIII24F^tyProp12,4=F^tyexpF^tx−Ftx/F^tx+FtxIX31F^tyProp13,1=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxexpX¯−X¯^X¯+X¯^X32F^tyProp13,2=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxexpX¯^−X¯/X¯^+X¯XI33F^tyProp13,3=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxexpnX¯−X¯^/2NX¯−nX¯^+X¯XII34F^tyProp13,4=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxXIII41F^tyProp14,1=F^tyexpX¯−X¯^/X¯+X¯^XIV42F^tyProp14,2=F^tyexpX¯^−X¯/X¯^+X¯XV43F^tyProp14,3=F^tyexpnX¯−X¯^/2NX¯−nX¯^+X¯XVI44F^tyProp14,4=F^tyBy solvingF^tyProp1k1,k2 given in (32) up-to first order of approximation, we have the following equation:(37)F^typrop1k1,k2=Fty1+ξ01+12ϑ1ξ1−14ϑ1γ1ξ12+18ϑ12ξ121+12ϑ2ξ2−14ϑ2γ2ξ22+18ϑ22ξ22,where(38)ϑ1=fB1−T1A1+fB1+T1,γ1=fB1−T1A1+fB1+T1,ϑ2=fB2−T2A2+fB2+T2,γ2=fB2−T2A2+fB2+T2.Or(39)F^tyProp1k1,k2−Fty=Ftyξ0+12ϑ1ξ1+12ϑ2ξ2+12ϑ1ξ0ξ1+12ϑ2ξ0ξ2−14ϑ1γ1ξ12−14ϑ2γ2ξ22+18ϑ12ξ12+18ϑ22ξ22.Using (39), the bias and MSE of F^tyProp1k1,k2 are given by the following equation:(40)BiasF^tyProp1k1,k2≅Ftyλ12ϑ1ρ12C1C2+12ϑ2ρ13C1C3+18C22ϑ12−2ϑ1γ1+18C32ϑ22−2ϑ2γ2,(41)MSEF^tyProp1k1,k2≅F2tyλC12+14ϑ12C22+14ϑ22C32+ϑ1ρ12C1C2+ϑ2ρ13C1C3+12ϑ1ϑ2ρ23C2C3.Differentiate (40) with respect to ϑ1 and ϑ2, we get the optimum values of ϑ1 and ϑ2 i.e.,(42)ϑ1opt=2C1ρ12−ρ23ρ13C2ρ232−1,ϑ2opt=2C1ρ12−ρ23ρ12C2ρ232−1.Substituting the optimum values ofϑ1opt and ϑ2opt in Equation ≠ 20, we get minimum MSE of F^tyProp1k1,k2 and is given by the following equation:(43)MSEF^tyProp1k1,k2≅F2tyλC121−ρ1.232,where(44)ρ1.232=ρ122+ρ132−2ρ12ρ13ρ231−ρ232.is the coefficient of multiple determination ofFty on Ftx and X.


### 4.2. Second Proposed Estimator

To increase the efficiency of the estimators both at the design stage as well as at the estimation stage, we utilize the auxiliary information. When there exists a correlation between the study variable and the auxiliary variable, then the rank of the auxiliary variable is also correlated with the study variable. The rank of the auxiliary variable can be treated as a new auxiliary variable, and this information may also be used to increase the precision of the estimators. Based on the idea of rank, we propose a second new class of factor type estimators of the finite population distribution function. The estimator is given by the following equation:(45)F^tyProp2k1,k2=F^tyexpP11−Q11P11+Q11expP22−Q22P22+Q22,where(46)P11=A1+T1Ftx+fB1F^tx,P22=A2+T2R¯^x+fB2R¯^x,(47)Q11=A1+fB1Ftx+T1F^tx,Q22=A2+fB2R¯x+T2R¯^x,(48)Ai=Ki−1Ki−2,Bi=Ki−1Ki−4,(49)Ti=Ki−2Ki−3Ki−4,f=nN.Substituting different values ofKi (i = 1, 2, 3, 4) in Equation (45), we can generate many more different types of estimators from our general proposed class of estimators, given in Table 2.Table 2
Family members of the proposed class of estimatorsF^tyProp2k1,k2.
S.NoK1K2EstimatorsI11F^tyProp21,1=F^tyexpFtx−F^tx/Ftx+F^txexpR¯x−R¯^x/R¯x+R¯^xII12F^tyProp21,2=F^tyexpFtx−F^tx/Ftx+F^txexpR¯x−R¯^x/R¯x+R¯^xIII13F^tyProp21,1=F^tyexpFtx−F^tx/Ftx+F^txexpnR¯x−R¯^x/2NR¯x−nR¯^x+R¯xIV14F^tyProp21,4=F^tyexpFtx−F^tx/Ftx+F^txV21F^tyProp22,1=F^tyexpF^tx−F^tx/F^tx+F^txexpR¯x−R¯^x/R¯x+R¯^xVI22F^tyProp22,2=F^tyexpF^tx−F^tx/F^tx+F^txexpR¯x−R¯^x/R¯x+R¯^xVII23F^tyProp22,3=F^tyexpF^tx−F^tx/F^tx+F^txexpnR¯x−R¯^x/2NR¯x−nR¯^x+R¯xVIII24F^tyProp22,4=F^tyexpF^tx−Ftx/F^tx+FtxIX31F^tyProp23,1=F^tyexpF^tx−Ftx/F^tx+FtxexpR¯x−R¯^x/R¯x+R¯^xX32F^tyProp23,2=F^tyexpF^tx−Ftx/F^tx+FtxexpR¯^x−R¯x/R¯^x+R¯xXI33F^tyProp23,3=F^tyexpF^tx−Ftx/F^tx+FtxexpnR¯x−R¯^x/2NX¯−nR¯^x+R¯xXII34F^tyProp23,4=F^tyexpnF^tx−Ftx/2NF^tx+nFtxXIII41F^tyProp24,1=F^tyexpR¯x−R¯^x/R¯x+R¯^xXIV42F^tyProp24,2=F^tyexpR¯x−R¯^x/R¯x+R¯^xXV43F^tyProp24,3=F^tyexpR¯x−R¯^x/R¯x+R¯^xXVI44F^tyProp24,4=F^tySolvingF^tyProp2k1,k2 given in (45) in terms of errors, we have the following equation:(50)F^tyProp2k1,k2=Fty1+ξ01+12ϑ1ξ1−14ϑ1γ1ξ12+18ϑ12ξ121+12ϑ2ξ3−14ϑ2γ2ξ32+18ϑ22ξ32,where(51)ϑ1=fB1−T1A1+fB1+T1,γ1=fB1−T1A1+fB1+T1,ϑ2=fB1−T1A1+fB1+T1,γ2=fB1−T1A1+fB1+T1.With first order approximation, we have the following equation:(52)F^tyProp2k1,k2−Fty=Ftyξ0+12ϑ1ξ1+12ϑ2ξ3+12ϑ1ξ0ξ1+12ϑ2ξ0ξ3−14ϑ1γ1ξ12−14ϑ2γ2ξ32+18ϑ12ξ12+18ϑ22ξ32.Using (52) the bias and MSE of F^tyProp2k1,k2 are given by the following equation:(53)BiasF^tyProp2k1,k2≅Ftyλ12ϑ1ρ12C1C2+12ϑ2ρ14C1C4+18C22ϑ12−2ϑ1γ1+18C42ϑ22−2ϑ2γ2.(54)MSEF^tyProp2k1,k2≅F2tyλC12+14ϑ12C22+14φ22C42+ϑ1ρ12C1C2+ϑ2ρ14C1C4+12ϑ1ϑ2ρ24C2C4.Differentiate Equation (54) with respect to ϑ1 and ϑ2 for minimum MSE, we get the optimum values of ϑ1 and ϑ2,(55)ϑ1opt=2C1ρ14−ρ24ρ12C4ρ242−1,ϑ2opt=2C1ρ14−ρ24ρ12C4ρ242−1.Substituting the optimum values ofϑ1opt and ϑ2opt in Equation ≠ 25, we get minimum MSE of F^tyProp2k1,k2 is given by the following equation:(56)MSEF^tyProp2k1,k2min≅F2tyλC121−ρ1.242,where(57)ρ1.242=ρ122+ρ142−2ρ12ρ14ρ241−ρ242.is the coefficient of multiple determination ofFty on Ftx and Rx.


## 4.1. First Proposed Estimator

On the lines of [31], we suggest a generalized class of exponential factor type estimators which contains many stable and efficient estimators. By combining the idea of [30, 31], the first estimator is given by the following equation:(32)F^tyProp1k1,k2=F^tyexpS11−M11S11+M11expS22−M22S22+M22,where(33)S11=A1+T1Ftx+fB1F^tx,S22=A2+T2X¯+fB2X¯^,(34)M11=A1+fB1Ftx+T1F^tx,M22=A2+fB2X¯+T2X¯,^(35)Ai=Ki−1Ki−2,Bi=Ki−1Ki−4,(36)Ti=Ki−2Ki−3Ki−4,f=nN,Substituting different values ofKi (i = 1, 2, 3, 4) in Equation (32), we can generate many more types of estimators from our general proposed class of estimators, given in Table 1.Table 1
Family members of the proposed class of estimatorsF^tyProp1k1,k2.
S.NoK1K2EstimatorsI11F^tyProp11,1=F^tyexpFtx−F^tx/Ftx+F^txexpX¯−X¯^/X¯+X¯^II12F^tyProp11,1=F^tyexpFtx−F^tx/Ftx+F^txexpX¯−X¯^/X¯+X¯^III13F^tyProp11,3=F^tyexpFtx−F^tx/Ftx+F^txexpnX¯−X¯^/2NX¯−nX¯^+X¯IV14F^tyProp11,4=F^tyexpFtx−F^tx/Ftx+F^txV21F^tyProp12,1=F^tyexpF^tx−Ftx/F^tx+FtxexpX¯−X¯^/X¯+X¯^VI22F^tyProp12,2=F^tyexpF^tx−Ftx/F^tx+FtxexpX¯−X¯^/X¯+X¯^VII23F^tyProp12,3=F^tyexpF^tx−Ftx/F^tx+FtxexpnX¯−X¯^/2NX¯−nX¯^+X¯VIII24F^tyProp12,4=F^tyexpF^tx−Ftx/F^tx+FtxIX31F^tyProp13,1=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxexpX¯−X¯^X¯+X¯^X32F^tyProp13,2=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxexpX¯^−X¯/X¯^+X¯XI33F^tyProp13,3=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxexpnX¯−X¯^/2NX¯−nX¯^+X¯XII34F^tyProp13,4=F^tyexpnFtx−F^tx/2NFtx−nF^tx+FtxXIII41F^tyProp14,1=F^tyexpX¯−X¯^/X¯+X¯^XIV42F^tyProp14,2=F^tyexpX¯^−X¯/X¯^+X¯XV43F^tyProp14,3=F^tyexpnX¯−X¯^/2NX¯−nX¯^+X¯XVI44F^tyProp14,4=F^tyBy solvingF^tyProp1k1,k2 given in (32) up-to first order of approximation, we have the following equation:(37)F^typrop1k1,k2=Fty1+ξ01+12ϑ1ξ1−14ϑ1γ1ξ12+18ϑ12ξ121+12ϑ2ξ2−14ϑ2γ2ξ22+18ϑ22ξ22,where(38)ϑ1=fB1−T1A1+fB1+T1,γ1=fB1−T1A1+fB1+T1,ϑ2=fB2−T2A2+fB2+T2,γ2=fB2−T2A2+fB2+T2.Or(39)F^tyProp1k1,k2−Fty=Ftyξ0+12ϑ1ξ1+12ϑ2ξ2+12ϑ1ξ0ξ1+12ϑ2ξ0ξ2−14ϑ1γ1ξ12−14ϑ2γ2ξ22+18ϑ12ξ12+18ϑ22ξ22.Using (39), the bias and MSE of F^tyProp1k1,k2 are given by the following equation:(40)BiasF^tyProp1k1,k2≅Ftyλ12ϑ1ρ12C1C2+12ϑ2ρ13C1C3+18C22ϑ12−2ϑ1γ1+18C32ϑ22−2ϑ2γ2,(41)MSEF^tyProp1k1,k2≅F2tyλC12+14ϑ12C22+14ϑ22C32+ϑ1ρ12C1C2+ϑ2ρ13C1C3+12ϑ1ϑ2ρ23C2C3.Differentiate (40) with respect to ϑ1 and ϑ2, we get the optimum values of ϑ1 and ϑ2 i.e.,(42)ϑ1opt=2C1ρ12−ρ23ρ13C2ρ232−1,ϑ2opt=2C1ρ12−ρ23ρ12C2ρ232−1.Substituting the optimum values ofϑ1opt and ϑ2opt in Equation ≠ 20, we get minimum MSE of F^tyProp1k1,k2 and is given by the following equation:(43)MSEF^tyProp1k1,k2≅F2tyλC121−ρ1.232,where(44)ρ1.232=ρ122+ρ132−2ρ12ρ13ρ231−ρ232.is the coefficient of multiple determination ofFty on Ftx and X.


## 4.2. Second Proposed Estimator

To increase the efficiency of the estimators both at the design stage as well as at the estimation stage, we utilize the auxiliary information. When there exists a correlation between the study variable and the auxiliary variable, then the rank of the auxiliary variable is also correlated with the study variable. The rank of the auxiliary variable can be treated as a new auxiliary variable, and this information may also be used to increase the precision of the estimators. Based on the idea of rank, we propose a second new class of factor type estimators of the finite population distribution function. The estimator is given by the following equation:(45)F^tyProp2k1,k2=F^tyexpP11−Q11P11+Q11expP22−Q22P22+Q22,where(46)P11=A1+T1Ftx+fB1F^tx,P22=A2+T2R¯^x+fB2R¯^x,(47)Q11=A1+fB1Ftx+T1F^tx,Q22=A2+fB2R¯x+T2R¯^x,(48)Ai=Ki−1Ki−2,Bi=Ki−1Ki−4,(49)Ti=Ki−2Ki−3Ki−4,f=nN.Substituting different values ofKi (i = 1, 2, 3, 4) in Equation (45), we can generate many more different types of estimators from our general proposed class of estimators, given in Table 2.Table 2
Family members of the proposed class of estimatorsF^tyProp2k1,k2.
S.NoK1K2EstimatorsI11F^tyProp21,1=F^tyexpFtx−F^tx/Ftx+F^txexpR¯x−R¯^x/R¯x+R¯^xII12F^tyProp21,2=F^tyexpFtx−F^tx/Ftx+F^txexpR¯x−R¯^x/R¯x+R¯^xIII13F^tyProp21,1=F^tyexpFtx−F^tx/Ftx+F^txexpnR¯x−R¯^x/2NR¯x−nR¯^x+R¯xIV14F^tyProp21,4=F^tyexpFtx−F^tx/Ftx+F^txV21F^tyProp22,1=F^tyexpF^tx−F^tx/F^tx+F^txexpR¯x−R¯^x/R¯x+R¯^xVI22F^tyProp22,2=F^tyexpF^tx−F^tx/F^tx+F^txexpR¯x−R¯^x/R¯x+R¯^xVII23F^tyProp22,3=F^tyexpF^tx−F^tx/F^tx+F^txexpnR¯x−R¯^x/2NR¯x−nR¯^x+R¯xVIII24F^tyProp22,4=F^tyexpF^tx−Ftx/F^tx+FtxIX31F^tyProp23,1=F^tyexpF^tx−Ftx/F^tx+FtxexpR¯x−R¯^x/R¯x+R¯^xX32F^tyProp23,2=F^tyexpF^tx−Ftx/F^tx+FtxexpR¯^x−R¯x/R¯^x+R¯xXI33F^tyProp23,3=F^tyexpF^tx−Ftx/F^tx+FtxexpnR¯x−R¯^x/2NX¯−nR¯^x+R¯xXII34F^tyProp23,4=F^tyexpnF^tx−Ftx/2NF^tx+nFtxXIII41F^tyProp24,1=F^tyexpR¯x−R¯^x/R¯x+R¯^xXIV42F^tyProp24,2=F^tyexpR¯x−R¯^x/R¯x+R¯^xXV43F^tyProp24,3=F^tyexpR¯x−R¯^x/R¯x+R¯^xXVI44F^tyProp24,4=F^tySolvingF^tyProp2k1,k2 given in (45) in terms of errors, we have the following equation:(50)F^tyProp2k1,k2=Fty1+ξ01+12ϑ1ξ1−14ϑ1γ1ξ12+18ϑ12ξ121+12ϑ2ξ3−14ϑ2γ2ξ32+18ϑ22ξ32,where(51)ϑ1=fB1−T1A1+fB1+T1,γ1=fB1−T1A1+fB1+T1,ϑ2=fB1−T1A1+fB1+T1,γ2=fB1−T1A1+fB1+T1.With first order approximation, we have the following equation:(52)F^tyProp2k1,k2−Fty=Ftyξ0+12ϑ1ξ1+12ϑ2ξ3+12ϑ1ξ0ξ1+12ϑ2ξ0ξ3−14ϑ1γ1ξ12−14ϑ2γ2ξ32+18ϑ12ξ12+18ϑ22ξ32.Using (52) the bias and MSE of F^tyProp2k1,k2 are given by the following equation:(53)BiasF^tyProp2k1,k2≅Ftyλ12ϑ1ρ12C1C2+12ϑ2ρ14C1C4+18C22ϑ12−2ϑ1γ1+18C42ϑ22−2ϑ2γ2.(54)MSEF^tyProp2k1,k2≅F2tyλC12+14ϑ12C22+14φ22C42+ϑ1ρ12C1C2+ϑ2ρ14C1C4+12ϑ1ϑ2ρ24C2C4.Differentiate Equation (54) with respect to ϑ1 and ϑ2 for minimum MSE, we get the optimum values of ϑ1 and ϑ2,(55)ϑ1opt=2C1ρ14−ρ24ρ12C4ρ242−1,ϑ2opt=2C1ρ14−ρ24ρ12C4ρ242−1.Substituting the optimum values ofϑ1opt and ϑ2opt in Equation ≠ 25, we get minimum MSE of F^tyProp2k1,k2 is given by the following equation:(56)MSEF^tyProp2k1,k2min≅F2tyλC121−ρ1.242,where(57)ρ1.242=ρ122+ρ142−2ρ12ρ14ρ241−ρ242.is the coefficient of multiple determination ofFty on Ftx and Rx.


## 5. Theoretical Comparison

In this section, the adapted and proposed estimators of F(ty) are compared in terms of the minimum mean square error.(1)
From (7) and (43),MSEF^tyProp1k1,k2min<VarF^ty if.VarF^ty−MSEF^tyProp1k1,k2min>0 or if.λF2tyC12−F2tyλC121−ρ1.232>0,or ifρ1.232>0(2)
From (10) and (43),MSEF^tyProp1k1,k2min<MSEF^tyR if.MSEF^tyR−MSEF^tyProp1k1,k2min>0 or if.F2tyλC12+C22−2ρ12C1C2−F2tyλC121−ρ1.232>0,or ifC22−2ρ12C1C2+C12ρ1.232>0(3)
From (13) and (43),MSEF^tyProp1k1,k2min<MSEF^tyP if.MSEF^tyP−MSEF^tyProp1k1,k2min>0 or if.F2tyλC12+C22+2ρ12C1C2−F2tyλC121−ρ1.232>0,or ifC22+2ρ12C1C2+C12ρ1.232>0(4)
From (15) and (43),MSEF^tyProp1k1,k2min<MSEF^tyReg if.MSEF^tyReg−MSEF^tyProp1k1,k2min>0 or if.F2tyλC12+1/4C22−ρ12C1C2−F2tyλC121−ρ1.232>0,or if1/4C22−ρ12C1C2+C12ρ1.232>0(5)
From (10) and (43),MSEF^tyProp1k1,k2min<MSEF^tyR,D if.MSEF^tyR,D−MSEF^tyProp1k1,k2min>0 or if.λF2tyC121−ρ122/1+λC121−ρ122−F2tyλC121−ρ1.232>0,or ifλC12ρ122+ρ1.232−ρ122+ρ1.232>0(6)
From (23) and (43),MSEF^tyProp1k1,k2min<MSEF^tyBT,Rif.MSEF^tyBT,R−MSEF^tyProp1k1,k2min>0 or if.λF2tyC12+1/4C22−ρ12C12C22−F2tyλC121−ρ1.232>0,or if1/4C22+C12ρ1.232−ρ12>0(7)
From (25) and (43),MSEF^tyProp1k1,k2min<MSEF^tyBT,P if.MSEF^tyBT,P−MSEF^tyProp1k1,k2min>0 or if.F2tyλC12+1/4C22+ρ12C1C2−F2tyλC121−ρ1.232>0,or if1/4C22+ρ12C1C2+C12ρ1.232>0(8)
From (31) and (43),MSEF^tyProp1k1,k2min<MSEF^tyG,K if.MSEF^tyG,K−MSEF^tyProp1k1,k2min>0 or if.λF2ty64C121−ρ122−λ4C24−16λ2C12C221−ρ122/641+λC121−ρ122−F2tyλC121−ρ1.232>0,or ifλF2ty64C121−ρ122−λ4C24−16λ2C12C221−ρ122−F2tyλC121−ρ1.232>0.(9)
From (7) and (56),MSEF^tyProp2k1,k2min<VarF^ty if.VarF^ty−MSEF^tyProp2k1,k2min>0 or if.λF2tyC12−F2tyλC121−ρ1.242>0,or ifρ1.242>0.(10)
From (10) and (56),MSEF^tyProp2k1,k2min<MSEF^tyR if.MSEF^tyR−MSEF^tyProp2k1,k2min>0 or if.F2tyλC12+C22−2ρ12C1C2−F2tyλC121−ρ1.242>0,or ifC22−2ρ12C1C2+C12ρ1.242>0.(11)
From (13) and (56),MSEF^tyProp2k1,k2min<MSEF^tyP if.MSEF^tyP−MSEF^tyProp2k1,k2min>0 or if.F2tyλC12+C22+2ρ12C1C2−F2tyλC121−ρ1.242>0,or ifC22+2ρ12C1C2+C12ρ1.242>0(12)
From (15) and (56),MSEF^tyProp2k1,k2min<MSEF^tyReg if.MSEF^tyReg−MSEF^tyProp2k1,k2min>0 or if.F2tyλC12+1/4C22−ρ12C1C2−F2tyλC121−ρ1.242>0,or if1/4C22−ρ12C1C2+C12ρ1.242>0(13)
From (19) and (56),MSEF^tyProp1k1,k2min<MSEF^tyR,D if.MSEF^tyR,D−MSEF^tyProp2k1,k2min>0 or if.λF2tyC121−ρ122/1+λC121−ρ122−F2tyλC121−ρ1.242>0,or ifλC12ρ122+ρ1.242−ρ122+ρ1.242>0(14)
From (23) and (56),MSEF^tyProp1k1,k2min<MSEF^tyBT,R if.MSEF^tyBT,R−MSEF^tyProp1k1,k2min>0 or if.λF2tyC12+1/4C22−ρ12C12C22−F2tyλC121−ρ1.242>0,or if1/4C22+C12ρ1.242−ρ12>0(15)
From (25) and (56),MSEF^typrop1min<MSEF^tyBT,P if.MSEF^tyBT,P−MSEF^typrop1min>0 or if.F2tyλC12+1/4C22+ρ12C1C2−F2tyλC121−ρ1.242>0,or if1/4C22+ρ12C1C2+C12ρ1.242>0(16)
From (31) and (56),MSEF^tyProp2k1,k2min<MSEF^tyG,K if.MSEF^tyG,K−MSEF^tyProp2k1,k2min>0 or if. MSEF^tyProp2k1,k2minλF2ty64C121−ρ122−λ4C24−16λ2C12C221−ρ122/641+λC121−ρ122−F2tyλC121−ρ1.242>0,or ifλF2ty64C121−ρ122−λ4C24−16λ2C12C221−ρ122−F2tyλC121−ρ1.242>0


## 6. Numerical Study

In this section, we conduct a numerical study to investigate the performances of the adapted and proposed DF estimators. For this purpose, three populations are considered. The summary statistics of these populations are reported in Tables3–5. The percentage relative efficiency PRE of the estimator F^ity with respect to F^ty is given by the following equation::(58)PREF^tyi,F^ty=VarF^tyMSEF^itymin×100.Table 3
Summary statistics.
ParameterValueParametertx,tyvalueX¯,Y¯Q1x,Q1,yX˜,Y˜Q3x,Q3yN923Fty0.77030.25560.50160.7497n180C10.54631.70700.99730.5780λ0.00447Ftx0.76930.25030.50050.7508X¯11440.5C20.54801.73170.99950.5764C31.86453ρ120.89300.87110.84620.8930R¯x462.000ρ13−0.6640−0.2861−0.4416−0.6465C40.57703ρ23−0.6753−0.2838−0.4480−0.6563ρ14−0.7169−0.7424−0.8286−0.7402ρ24−0.7298−0.7503−0.8660−0.7492β22.63332.32951.00002.3449Table 4
Summary statistics.
ParameterValueParametertx,tyvalueX¯,Y¯Q1x,Q1yX˜,Y˜Q3x,Q3yN923Fty0.77030.25560.50160.7497n180C10.54631.70700.99730.5780λ0.00447Ftx0.72910.25240.50160.7508X¯333.165C20.60981.72180.99730.5764C31.32809ρ120.87270.19100.89170.9162R¯x462.000ρ13−0.7385−0.3656−0.5366−0.7221C40.57703ρ23−0.7120−0.1046−0.5419−0.7298ρ14−0.7223−0.7424−0.8486−0.7430ρ24−0.7697−0.7503−0.8660−0.7491β22.06341.29901.00002.3449Table 5
Summary statistics.
ParameterValueParametertx,tyvalueX¯,Y¯Q1x,Q1yX˜,Y˜Q3x,Q3yN69Fty0.72460.24640.50720.7536n10C10.62091.76180.99280.5759λ0.08550Ftx0.76810.24640.50720.7536X¯4954.44C20.55351.76180.99280.5759C31.42478ρ120.66070.76580.94200.7658R¯x35.0000ρ13−0.7129−0.3612−0.5709−0.7424C40.57321ρ23−0.7745−0.3650−0.5427−0.7584ρ14−0.7168−0.7109−0.8558−0.7008ρ24−0.7310−0.7464−0.8660−0.7464β22.61442.38571.00002.3857wherei=R,P,Reg,R,D,BT,R,BT,P,G,K,Prop1,Prop2.The PREs of the distribution function estimators, computed from three populations, are given in Tables6–9Table 6
PREs of distribution function estimators for population I, II, and III, whenFty and Ftx  =  Y¯,X¯.
EstimatorsPopulation 1Population 2Population 3Y¯,X¯Y¯,X¯Y¯,X¯F^ty100100100F^tyR465.57336.04162.16F^tyP25.4723.3133.34F^tyBT,R281.07296.44164.00F^tyBT,P46.5743.7555.94F^tyReg493.79419.59177.46F^tyRao493.93419.73180.76F^tyG,K493.99419.78181.17F^tyProp1k1,k2511.03475.05216.33F^tyProp2k1,k2517.03430.94224.16Table 7
PREs of distribution function estimators for population I, II, and III, whenFty and Ftx  =  Q1y, Q1x.
EstimatorsPopulation 1Population 2Population 3Q1y,Q1xQ1y,Q1xQ1y,Q1xF^y100100100F^yR382.08411.14213.53F^tyP18.9518.9219.22F^tyBT,R267.67273.36206.54F^tyBT,P46.7046.6049.60F^yReg414.63442.7241.84F^tyRao415.93444.01268.39F^tyG,K415.95444.03268.65F^tyProp1k1,k2417.48447.26246.42F^tyProp2k1,k2448.24475.38270.52Table 8
PREs of distribution function estimators for population I, II, and III, whenFty and Ftx  =  Y˜,X˜ or Q2y, Q2x.
EstimatorsPopulation 1Population 2Population 3Y˜,X˜Y˜,X˜Y˜,X˜F^ty100100100F^tyR324.29461.49862.32F^tyP27.0626.4525.79F^tyBT,R248.08279.06324.69F^tyBT,P47.6446.6945.62F^tyReg352.08487.23888.06F^tyRao352.53488.37896.49F^tyG,K352.58488.43898.79F^tyProp1k1,k2358.25497.75899.77F^tyProp2k1,k2404.37550.62941.66Table 9
PREs of distribution function estimators for population I, II, and III, whenFty and Ftx  =  Q3y, Q3x.
EstimatorsPopulation 1Population 2Population 3Q3y,Q3xQ3y,Q3xQ3y,Q3xF^ty100100100F^tyR468.75598.06213.53F^tyP26.0425.7327.78F^yBT,R279.25298.47206.54F^tyBT,P46.7546.2549.60F^tyReg493.83622.46241.84F^yRao493.98622.6244.68F^tyG,K494.64622.69245.26F^tyProp1k1,k2509.95647.15284.03F^tyProp2k1,k2523.64652.19266.09Population 1.
(Source: [32]) Y: number of teachers and X: number of students.Population 2.
(Source: [32]) Y: Number of teachers and X: number of Schools.Population 3.
(Source: [6]) Y: the estimated number of fish caught by marine recreational fishermen in the year 1995 and X: the estimated number of fish caught by marine recreational fishermen in the year 1994.
In Tables6–9 we use Y¯,X¯,Y˜,X˜,Q1y,Q1x,Q3y and Q3x for indicator functions of Y and X, respectively. Here,(59)Fty=IYi≤ty=1ifYi≤ty,0ifYi>ty.(60)Ftx=IXi≤tx=1ifXi≤tx,0ifXi>tx.
when we usedY¯ as indicator function of Y and X¯ as indicator function of X, we get PRE in Table 6, and when we used Q1y as indicator function of Y and Q1x as indicator function of X, we get PRE given in Table 7. And, similarly when we used Y˜ as indicator function of Y and X˜ as indicator function of X, we get PRE in Table 8, and when we used Q3y as indicator function of Y and Q3x as indicator function of X, we get PRE given in Table 9.
Here we take three data sets for numerical illustration, respectively.
In Tables6–9, we observe that the proposed class of estimators are more precise than the existing estimators in terms of PREs.


## 7. Simulation Study

A simulation study is conducted to obtain the efficiency of the suggested estimators under simple random sampling when the auxiliary variables and rank of the auxiliary variable are used. We have generated three populations of size 1,000 from a multivariate normal distribution with different covariance matrices. All populations have different correlations, i.e., Population I is negatively correlated, Population II is positively correlated, and Population III has a strong positive correlation betweenX and Y variables. Population averages and covariance matrices are given as follows.


### 7.1. Population I

(61)μ1=55,(62)∑1=49.69.664,(63)ρXY=0.613800.


### 7.2. Population II

(64)μ2=55,(65)∑2=2410,(66)ρXY=0.902645,


### 7.3. Population III

(67)μ3=55,(68)∑3=4−9.7−.765,(69)ρXY=−0628261.The Percentage Relative Efficiency (PRE) is calculated as follows:(70)PREF^ity,F^ty=VarF^tyMSEF^itymin×100.In this study, we consider the generated population for summarizing the simulation procedures. The simulation results of PRE are given in Tables10–13.Table 10
PREs of distribution function estimators using simulation for Populations I, II, and III, whenFty and Ftx  =  Y¯,X¯.
EstimatorsPopulation 1Population 2Population 3Q1y,Q1xQ1y,Q1xQ1y,Q1xF^y100100100F^yR22.6935619.9412532.53728F^tyP85.22727176.886839.89362F^tyBT,R119.5219187.734766.51885F^tyBT,P60.1202450.83023100.3345F^yReg120.6046206.0017106.8579F^tyRao123.3073208.7044109.5606F^tyG,K123.3148208.7171109.5673F^tyProp1k1,k2121.8722210.9814111.9523F^tyProp2k1,k2129.5595228.8673120.3476Table 11
PREs of distribution function estimators using simulation for populations I, II, and III, whenFty and Ftx  =  Q1y, Q1x.
EstimatorsPopulation 1Population 2Population 3Q1y,Q1xQ1y,Q1xQ1y,Q1xF^y100100100F^yR34.6859529.0184791.81331F^tyP84.38889183.573733.68077F^tyBT,R120.7592190.348558.09967F^tyBT,P58.7990550.79537124.1445F^yReg121.9505211.4561124.8422F^tyRao122.8161211.863125.2238F^tyG,K122.8186211.8638125.2243F^tyProp1k1,k2123.3559220.4552128.5401F^tyProp2k1,k2134.4569249.8853134.4446Table 12
PREs of distribution function estimators using simulation for populations I, II, and III, whenFty and Ftx  =  Y˜,X˜ or Q2y, Q2x.
EstimatorsPopulation 1Population 2Population 3Y˜,X˜Y˜,X˜Y˜,X˜F^ty100100100F^tyR35.2050628.9309990.53 863F^tyP86.2069183.823534.53039F^tyBT,R120.4819191.570958.89282F^tyBT,P59.8802450.55612124.6883F^tyReg121.4182212.7587125.1101F^tyRao122.3191213.6596125.6773F^tyG,K122.3215213.6639125.6783F^tyProp1k1,k2122.7554220.0364128.8903F^tyProp2k1,k2133.9751251.1061136.1882Table 13
PREs of distribution function estimators using simulation for populations I, II, and III, whenFty and Ftx  =  Q3y, Q3x.
EstimatorsPopulation 1Population 2Population 3Q3y,Q3xQ3y,Q3xQ3y,Q3xF^ty100100100F^tyR36.0466229.4737366.445174F^tyP77.47934148.809539.0625F^yBT,R111.6903170.648565.35948F^tyBT,P62.3182452.2466103.0928F^tyReg114.3888178.8576108.5069F^yRao114.6891179.1579108.8072F^tyG,K114.6899177.6082108.808F^tyProp1k1,k2115.6248177.6082109.7103F^tyProp2k1,k2121.7216201.0239128.5133In Tables10–13, it can be seen that the proposed estimators perform better than all existing estimators. The percent relative efficiency shows that the second proposed family of estimators with simple random sampling yields the best result when the variables Y and X have a positive correlation. Overall, we can conclude that the performance of the family of suggested estimators is better than all existing estimators.


## 7.1. Population I

(61)μ1=55,(62)∑1=49.69.664,(63)ρXY=0.613800.


## 7.2. Population II

(64)μ2=55,(65)∑2=2410,(66)ρXY=0.902645,


## 7.3. Population III

(67)μ3=55,(68)∑3=4−9.7−.765,(69)ρXY=−0628261.The Percentage Relative Efficiency (PRE) is calculated as follows:(70)PREF^ity,F^ty=VarF^tyMSEF^itymin×100.In this study, we consider the generated population for summarizing the simulation procedures. The simulation results of PRE are given in Tables10–13.Table 10
PREs of distribution function estimators using simulation for Populations I, II, and III, whenFty and Ftx  =  Y¯,X¯.
EstimatorsPopulation 1Population 2Population 3Q1y,Q1xQ1y,Q1xQ1y,Q1xF^y100100100F^yR22.6935619.9412532.53728F^tyP85.22727176.886839.89362F^tyBT,R119.5219187.734766.51885F^tyBT,P60.1202450.83023100.3345F^yReg120.6046206.0017106.8579F^tyRao123.3073208.7044109.5606F^tyG,K123.3148208.7171109.5673F^tyProp1k1,k2121.8722210.9814111.9523F^tyProp2k1,k2129.5595228.8673120.3476Table 11
PREs of distribution function estimators using simulation for populations I, II, and III, whenFty and Ftx  =  Q1y, Q1x.
EstimatorsPopulation 1Population 2Population 3Q1y,Q1xQ1y,Q1xQ1y,Q1xF^y100100100F^yR34.6859529.0184791.81331F^tyP84.38889183.573733.68077F^tyBT,R120.7592190.348558.09967F^tyBT,P58.7990550.79537124.1445F^yReg121.9505211.4561124.8422F^tyRao122.8161211.863125.2238F^tyG,K122.8186211.8638125.2243F^tyProp1k1,k2123.3559220.4552128.5401F^tyProp2k1,k2134.4569249.8853134.4446Table 12
PREs of distribution function estimators using simulation for populations I, II, and III, whenFty and Ftx  =  Y˜,X˜ or Q2y, Q2x.
EstimatorsPopulation 1Population 2Population 3Y˜,X˜Y˜,X˜Y˜,X˜F^ty100100100F^tyR35.2050628.9309990.53 863F^tyP86.2069183.823534.53039F^tyBT,R120.4819191.570958.89282F^tyBT,P59.8802450.55612124.6883F^tyReg121.4182212.7587125.1101F^tyRao122.3191213.6596125.6773F^tyG,K122.3215213.6639125.6783F^tyProp1k1,k2122.7554220.0364128.8903F^tyProp2k1,k2133.9751251.1061136.1882Table 13
PREs of distribution function estimators using simulation for populations I, II, and III, whenFty and Ftx  =  Q3y, Q3x.
EstimatorsPopulation 1Population 2Population 3Q3y,Q3xQ3y,Q3xQ3y,Q3xF^ty100100100F^tyR36.0466229.4737366.445174F^tyP77.47934148.809539.0625F^yBT,R111.6903170.648565.35948F^tyBT,P62.3182452.2466103.0928F^tyReg114.3888178.8576108.5069F^yRao114.6891179.1579108.8072F^tyG,K114.6899177.6082108.808F^tyProp1k1,k2115.6248177.6082109.7103F^tyProp2k1,k2121.7216201.0239128.5133In Tables10–13, it can be seen that the proposed estimators perform better than all existing estimators. The percent relative efficiency shows that the second proposed family of estimators with simple random sampling yields the best result when the variables Y and X have a positive correlation. Overall, we can conclude that the performance of the family of suggested estimators is better than all existing estimators.


## 8. Concluding Remarks

In this article, we proposed a generalized class of exponential factor type estimators, utilizing the supplementary information in the form of the mean and rank of the auxiliary variable for estimating the finite population distribution function. The expressions for biases and mean squared errors of the proposed generalized class of estimators are derived up to the first order of approximation. The proposed estimatorsF^tyProp1k1,k2 and F^tyProp2k1,k2 are compared to all existing estimators numerically and theoretically. Based on the simulation studies as well as on the real data sets, it is observed that the proposed class of estimators performed better than their existing counterparts and should be preferable over the existing estimators available in the literature.



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*Source: 2545517-2022-08-08.xml*