# On Estimation of Distribution Function Using Dual Auxiliary Information under Nonresponse Using Simple Random Sampling

Authors: Saddam Hussain; Mi Zichuan; Sardar Hussain; Anum Iftikhar; Muhammad Asif; Sohail Akhtar; Sohaib Ahmad
Journal: Journal of Probability and Statistics
 (2020)

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## Abstract

In this paper, we proposed two new families of estimators using the supplementary information on the auxiliary variable and exponential function for the population distribution functions in case of nonresponse under simple random sampling. The estimations are done in two nonresponse scenarios. These are nonresponse on study variable and nonresponse on both study and auxiliary variables. As we have highlighted above that two new families of estimators are proposed, in the first family, the mean was used, while in the second family, ranks were used as auxiliary variables. Expression of biases and mean squared error of the proposed and existing estimators are obtained up to the first order of approximation. The performances of the proposed and existing estimators are compared theoretically. On these theoretical comparisons, we demonstrate that the proposed families of estimators are better in performance than the existing estimators available in the literature, under the obtained conditions. Furthermore, these theoretical findings are braced numerically by an empirical study offering the proposed relative efficiencies of the proposed families of estimators.


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## Body

## 1. Introduction

It is a well-known phenomenon that the known auxiliary information in the study of sample survey gives us an efficient estimate of population parameters, i.e., the population mean and population distribution function, under some essential conditions. This information (auxiliary) may be used for drawing a random sample using SRSWR or SRSWOR. Also, simple random sampling can be improved using the following sampling methods.Stratification, systematic, nonresponse sampling, and probability proportional sampling schemes are used for estimating the population parameter. Auxiliary information gives us some sort of techniques by means of the ratio, product, regression, and other methods. In a practical situation, one of the important issues in surveys is that it suffers from nonresponse. Nonresponse is a common problem which may crawl with sampling survey. Nonresponse has many ways of occurrence. Examples are linguistic problems, illness, nonresponse, nonacceptance, process of return address misguided, and capture by another person. Research has labelled that various types of nonresponse may have different effects on estimators. A lot of work has been done on the estimation of population mean under nonresponse to control the nonresponse bias and to increase the efficiency of the estimators by different authors. The problem of nonresponse in sample surveys is more common and more prevalent in mail surveys than in special interview surveys. Hansen and Hurwitz [1] assumed that a part of sample of earlier nonrespondents to be recommunicated with a more expensive system; they attempted the first effort by mail questionnaire and performed the second attempt by a personal interview. However, Hansen and Hurwitz [1] have not used any kind of supplementary information to increase the efficiency of the estimator. For the first time, the author of [2] used the auxiliary information for estimating the population mean. Cochran [3] used the auxiliary information for estimating the population mean under nonresponse. Then, work on nonresponse extended by many authors (cf., [4–7]) recommends various types of estimators for estimation of population mean and distribution function using the secondary information under nonresponse. Okafor and Lee [8] presented ratio and regression estimation with partial sampling of the nonrespondents for estimating the population mean. Furthermore, the authors of [9, 10] proposed estimators for estimating population mean using multiauxiliary information in different directions and Zhao et al. [11] used the idea of robust estimation of the distribution function and quantiles with nonignorance missing data.Also, for estimating population mean under the two-phase sampling strategy in the presence of nonresponse, the authors of [12–15] have made significant contributions. Diana and Perri [16] suggested a class of estimators in two-phase sampling with subsampling of nonrespondents in estimating the finite population mean. In this paper, we introduce the use of sample distribution functions of the study variable and auxiliary variable along with the mean of the auxiliary variable and also the ranks of the auxiliary variable for estimating the population distribution function.Extensive literature has been published on estimation of population mean under nonresponse; however, no effort has been dedicated to the development of efficient methods for population cumulative distribution function. In survey sampling, the statisticians are often interested in proportion size of the study variable, i.e., proportion of units in population with values less than or equal to a specified value ofy; for instance, we may be interested to know the proportion of the population in which 31% or more people are educated.Motivated byF^R,Dy, F^Sy, and average of F^BT,Ry and F^BT,Py, two new families of estimators are proposed for estimating distribution function in the presence of nonresponse. By numerical results, we will show that the proposed family of estimators is more precise than the existing estimators.We planned the paper as follows: In Section2, some notations are introduced. In Section 3, the existing estimators are reviewed briefly. Two new families of estimators are introduced in Section 4, respectively. The existing and proposed estimators are compared (theoretically and numerically) in Sections 5 and 6. In Section 7, the concluding remarks of the paper are discussed.


## 2. Notations

Consider a finite populationΩ=V1,V2,…,VN of N distinct units, which is partitioned into respondents Ω1=V1,V2,…,VN1 and nonrespondents Ω2=VN1+1,VN1+2,…,VN groups with sizes N1 and N2, respectively, for estimating the CDF, where N=N1+N2. A sample of size n has been drawn from this population by simple random sampling (SRSWOR), out of which n1 units respond and n2=n−n1 do not respond. It is assumed that the sample size n1 is drawn from the response group of Ω1 and n2 is drawn from the nonresponse group of Ω2. Moreover, a sample of size r=n2/kk>1 is drawn by simple random sampling (SRSWOR) from n2, and this time response is obtained from all r units. Let Y and X be the study and auxiliary variables, respectively. Let Z be used for the ranks of the X and IY≤y and IX≤x be the indicator variables based on Y and X. Furthermore, Fy=∑i=1NIYi≤y/N and F^y=∑i=1nIYi≤y/n and Fx=∑i=1NIXi≤x/N and F^x=∑i=1nIXi≤x/n are the population and sample distribution functions of Y and X, respectively. Similarly, let X¯=∑i=1NXi/N and X¯^=∑i=1nXi/n and Z¯=∑i=1NZi/N and Z¯^=∑i=1nZi/n be the population and sample means of X and Z,respectively.Furthermore,F2y=∑i=N1+1NIYi≤y/N2 and F2x=∑i=N1+1NIXi≤x/N2 are the population distribution functions of IY≤y and IX≤x for the nonresponse group and X¯2=∑i=N1+1NXi/N2 and Z¯2=∑i=N1+1NZi/N2 are the population means of X and Z for the nonresponse group, respectively.Here, (x=X¯ and Θ2x) and (y=Y¯ and Θ2y), where X¯ and Y¯ are the population means of XY. Similarly, Θ2x and Θ2y are the population second quartiles of XY, respectively.To obtain the bias and MSE of the proposed estimator, we consider the following error terms. Let(1)e1∗=F^H∗y−FyFy,e2∗=F^H∗x−FxFx,e3∗=X¯^H∗−X¯X¯,e4∗=Z¯^H∗−Z¯Z¯,e2=F^Hx−FxFx,e3=X¯^H−X¯X¯,e4=Z¯^H−Z¯Z¯.Here,F^H∗y, F^H∗x, and X¯^∗ and Z¯^∗ are the notations used for CDFs, mean, and mean of ranks when there are no responses on both study and auxiliary variables. And, F^Hx, X¯^, and Z¯^ are the notations used for CDF, mean, and mean of ranks when there are no responses on only auxiliary variable, shown in Table 1.Table 1
Estimators, variances, covariances, and correlation under nonresponse situations.
EstimatorSituation=IIIF^y=F^H∗yF^H∗yF^x=F^H∗xF^HxX¯^=X¯^H∗X¯^HZ¯^=Z¯^H∗Z¯^HVariance/covarianceΘrstu=ψrstuΨrstuΘ2000=ψ2000Ψ2000Θ0200=ψ0200Ψ0200Θ0020=ψ0020Ψ0020Θ0002=ψ0002Ψ0002Θ1100=ψ1100Ψ1100Θ1010=ψ1010Ψ1010Θ1001=ψ1001Ψ1001Θ0110=ψ0110Ψ0110Θ0101=ψ0101Ψ0101Coefficient of correlationϱ12=ρ122ρ12ϱ13=ρ132ρ13ϱ23=ρ232ρ23ϱ14=ρ142ρ14ϱ24=ρ242ρ24Coefficient of multiple determinationℜ1.232=Φ1.232Φ1.232ℜ1.242=Φ1.242Φ1.242LetEei∗=0 for i∗=1,2,3,4 and Eei=0 for i=2,3,4, where E⋅ is the mathematical expectation of ⋅. Let(2)Ψrstu=Ee1re2se3te4u,ψrstu=Ee1∗re2∗se3∗te4∗u,where r,s,t,u=1,2,3,4. Here,(3)Ee1∗2=λC12+λ2C122=ψ20002,Ee2∗2=λC22+λ2C222=ψ02002,Ee3∗2=λC32+λ2C322=ψ00202,Ee4∗2=λC42+λ2C422=ψ00022,Ee22=λC22=Ψ0200,Ee32=λC32=Ψ0020,Ee42=λC42=Ψ0002,Ee1∗e2∗=λρ12C1C2+λ2ρ122C12C22=ψ1100,Ee1∗e3∗=λρ13C1C3+λ2ρ132C12C32=ψ1010,Ee1∗e4∗=λρ14C1C4+λ2ρ142C12C42=ψ1001,Ee2∗e3∗=λρ23C2C3+λ2ρ232C22C32=ψ0110,Ee2∗e4∗=λρ24C2C4+λ2ρ242C22C42=ψ0101,Ee1∗e2=λρ12C1C2=Ψ1100,Ee1∗e3=λρ13C1C3=Ψ1010,Ee1∗e4=λρ14C1C4=Ψ1001,Ee2e3=λρ23C2C3=Ψ0110,Ee2e4=λρ24C2C4=Ψ0101.Here,(4)ϱ1.232=ψ11002ψ0020+ψ10102ψ0200−2ψ1010ψ1100ψ0110ψ2000ψ0200ψ0020−ψ01102.where it is the coefficient of multiple determination of I(Y ≤ y) on I(X ≤ x) and X with situation-I. Also,(5)ϱ1.232=Ψ11002Ψ0020+Ψ10102Ψ0200−2Ψ1010Ψ1100Ψ0110ψ2000Ψ0200Ψ0020−Ψ01102is the coefficient of multiple determination of I(Y ≤ y) on I(X ≤ x) and X with situation-II. And,(6)ϱ1.242=ψ11002ψ0002+ψ10012ψ0200−2ψ1001ψ1100ψ0101ψ2000ψ0200ψ0002−ψ01012is the coefficient of multiple determination of I(Y ≤ y) on I(X ≤ x) and Z with situation-I. Finally,(7)ϱ1.242=Ψ11002Ψ0002+Ψ10012Ψ0200−2Ψ1001Ψ1100Ψ0101ψ2000Ψ0200Ψ0002−Ψ01012is the coefficient of multiple determination of I(Y ≤ y) on I(X ≤ x) and Z with situation-II. Here, λ=1/n1/N, λ2=W2k−1/n, S12=∑i=1NIYi≤y−Fy2/N−1, S22=∑i=1NIXi≤x−Fx2/N−1, S32=∑i=1NXi−X¯2/N−1, and S42=∑i=1NZi−Z¯2/N−1 are the population variances of IY≤y, IX≤x, X, and Z for the response group, respectively.Similarly,S122=∑i=N1+1NIYi≤y−Fy22/N2−1, S222=∑i=N1+1NIXi≤x2−Fx22/N2−1, S322=∑i=N1+1NXi−X2¯2/N2−1, and S422=∑i=N1+1NZi−Z2¯2/N2−1 are the population variances of IY≤y, IX≤x, X, and Z for the nonresponse group, respectively.C1=S1/Fy, C2=S2/Fx, C3=S3/X¯, and C4=S4/Z¯ are the population coefficient of variations for the response group, and C12=S12/Fy2, C22=S22/Fx2, C32=S32/X¯2, and C42=S42/Z2¯ are the population coefficient of variations for the nonresponse group.S12=∑i=1NIYi≤y−FyIXi≤x−Fx/N−1, S13=∑i=1NIYi≤y−FyXi−X¯/N−1, S23=∑i=1NIXi≤x−FxXi−X¯/N−1, S14=∑i=1NIYi≤y−FyZi−Z¯/N−1, and S24=∑i=1NIXi≤x−FxZi−Z¯/N−1 are the population covariances for the response group.S122=∑i=N1+1NIYi≤y−Fy2IXi≤x−Fx2/N2−1, S132=∑i=N1+1NIYi≤y−Fy2Xi−X2¯/N2−1, S232=∑i=N1+1NIXi≤x−Fx2Xi−X2¯/N2−1, S142=∑i=N1+1NIYi≤y−Fy2Zi−Z¯/N2−1, and S242=∑i=N1+1NIXi≤x−Fx2Zi−Z2¯/N2−1 are the population covariances for the nonresponse group.Similarly,ρ12=S12/S1S2, ρ13=S13/S1S3, ρ23=S23/S2S3, ρ14=S14/S1S4, and ρ24=S24/S2S4 are the population correlation coefficients for the response group, respectively.ρ122=S122/S12S22, ρ132=S132/S12S32, ρ232=S232/S22S32, ρ142=S142/S12S42, and ρ242=S242/S22S42 are the population correlation coefficients for the nonresponse group.LetFy=W1F1y+W2F2y, where Wj=Nj/N and Fjy=∑i=1NjIYi≤y/Nj for j=1,2. Also, F^1y=∑i=1n1IYi≤y/n1 denote the sample distribution function of n1 responding units out of n units and F^2ry=∑i=1rIYi≤y/r denote the sample distribution function of r responding units out of nonresponse units.The existing Hansen and Hurwitz [1] unbiased estimator of Fy with its variance is(8)F^Hy=w1F^1y+w2F^2ry,VarF^Hy=λS12+λ2S122.Similarly, the unbiased estimators forF^Hx, X¯^H, and Z¯^H and their corresponding variances are(9)F^Hx=w1F^1x+w2F^2rx,X¯^H=w1X¯^1+w2X¯^2r,Z¯^H=w1Z¯^1+w2Z¯^2r,VarF^HxλS12+λ2S122,VarX¯^H=λS22+λ2S222,VarZ¯^H=λS32+λ2S322,respectively.In practice, we use three situations, occurring under nonresponse, but here, we use two situations which mostly occur, namely, nonresponse on both the study variable and the auxiliary variable (say situation-I) and nonresponse just on study variable only (say situation-II). For notational convenience, we follow the notations given in Table1.


## 3. Existing Estimators

In this section, some estimators of finite population mean exist for estimating the finite CDF under nonresponse; the biases and MSEs of these existing estimators are derived under the first order of approximation.(1)
Cochran’s [17] existing ratio estimator of Fy is(10)F^Ry=F^yFxF^x.The bias and MSE ofF^Ry, to the first order of approximation, are(11)biasF^Ry≅FyΘ0200−Θ1100,MSEF^Ry≅F2yΘ2000+Θ0200−2Θ1100.(2)
Murthy’s [18] existing product estimator of Fy is(12)F^Py=F^yF^xFx.The bias and MSE ofF^Py, to the first order of approximation, are(13)biasF^Py=FyΘ1100,MSEF^Py≅F2yΘ2000+Θ0200+2Θ1100.(3)
The existing regression estimator ofFy is(14)F^Regy=F^y+kFx−F^x,wherek is an unknown constant. Here, F^Regy is an unbiased estimator of F^y. The minimum variance of F^Regy at the optimum value kopt=FyΘ1100/FxΘ0200 is(15)VarminF^Regy=F2yΘ2000Θ0200−Θ11002Θ0200.Here, (15) may be written as(16)VarminF^Regy=F2yΘ20001−ϱ122.(4)
Rao’s [19] existing difference-type estimator of Fy is(17)F^R,Dy=k1F^y+k2Fx−F^x,wherek1 and k2 are unknown constants. The bias and MSE of F^R,Dy, to the first order of approximation, are(18)biasF^R,Dy=Fyk1−1,MSEF^R,Dy≅F2y−2k1F2y+k12F2y+k12F2yΘ2000−2k1k2FyFxΘ1100+k22F2xΘ0200.The optimum values ofk1 and k2, determined by minimizing (18), are(19)k1opt=Θ0200Θ0200Θ2000−Θ11002+Θ0200,k2opt=FyΘ1100FxΘ2000Θ0200−Θ11002+Θ0200.The minimum MSE ofF^R,Dy at the optimum values of k1 and k2 is(20)MSEminF^R,Dy=F2yΘ2000Θ0200−Θ11002Θ2000Θ0200−Θ11002+Θ0200.Here, (20) may be written as(21)MSEminF^R,Dy=F2yΘ20001−ϱ1221+Θ20001−ϱ122.(5)
Grover and Kaur’s [20] existing generalized class of ratio-type exponential estimator of Fy is(22)F^G,Ky=k3F^y+k4Fx−F^xexpaFx−F^xaFx+F^x+2b,wherek3 and k4 are unknown constants. The bias and MSE of F^G,Ky, to the first order of approximation, are(23)biasF^G,Ky≅Fyk3−1+38θ2k3Fy+12θk4FxΘ0200−12θFyΘ1100,MSEF^G,Ky≅k42F2xΘ0200+k32F2yΘ2000+2θk3k4FyFxΘ0200−2k3k4FyFxΘ1100+F2y−2k3F2y+θk32F2y+k3F2yΘ1100−θk4FyFxΘ0200−2θk32F2yΘ1100−34θ2k3F2yΘ0200+θ2k32F2yΘ0200.The optimum values ofk3 and k4, determined by minimizing (15), are(24)k3opt=Θ0200θ2Θ0200−88−Θ2000Θ0200+Θ11002−Θ0200,k4opt=Fyθ3Θ02002−θ2Θ0200Θ1100+4θΘ2000Θ0200−4θΘ11002−4θΘ0200+8Θ11008FxΘ2000Θ0200−Θ11002+Θ0200.The simplified minimum MSE ofF^G,Ky at the optimum values of k3 and k4 is(25)MSEminF^G,Ky≅F2y6464−16θ2Θ0200−Θ0200−8+θ2Θ02002Θ02001+Θ2000−Q11002.Here, (25) may be written as(26)MSEminF^G,Ky≅VarminF^Regy−F2yθ2Θ02002−8Θ11002+8Θ0200Θ2000264Θ020021+Θ20001−ϱ122,which shows that F^G,Ky is more precise than F^4y.


## 4. Proposed Estimators

On the lines ofF^R,Dy, F^Sy, and average of F^BT,Ry and F^BT,Py, the first proposed family of estimators for estimating Fy is given by(27)F^Pr1y=F^y2expFx−F^xF^x+Fx+expF^x−FxF^x+Fx+m5Fx−F^xFx+m6F^y+m7X¯−X¯^X¯expaFx−F^xaFx+F^x+2b,where m5, m6, and m7 are unknown constants and a≠0 and b are either two real numbers or functions of known population parameters of IX≤x, such as ρ12, β2 (coefficient of kurtosis), and C2.The estimatorF^Pr1y can also be written as(28)F^Pr1y=Fy1+ξ01+m6−m5ξ1−m7ξ2+18θ2Fyξ121−12θξ1+38θ2ξ12+⋯.Simplifying (28) and keeping terms only up to the second power of ξs, we can write(29)F^Pr1y−Fy≅m6Fy+Fyξ0+m6Fyξ0−12θFyξ1+12θ2Fyξ12−12θFyξ0ξ1−m5ξ1+12θm5ξ12−12θm6Fyξ1+38θ2m6Fyξ12−12θm6Fyξ0ξ1−m7ξ2+12θm7ξ1ξ2.The bias and MSE ofF^Pr1y, to the first order of approximation, respectively, are(30)biasF^Pr1y≅12θ2FyΘ0200−12θFyΘ1100+12m5θΘ0200+m6Fy+38m6θ2FyΘ0200−12m6θFyΘ1100+12m7θΘ0110,MSEF^Pr1y≅−θF2yΘ1100+32m6θ2F2yΘ0200+m62θ2F2yΘ0200+m5θFyΘ0200−2m6m7FyΘ1010+14θ2F2yΘ0200+2m6F2yΘ2000+m52Θ0200−2m62θF2yΘ1100+F2yΘ2000+m62F2y+m7θFyΘ0110+2m5m6θFyΘ0200−2m5m6FyΘ1100+2m5m7Θ0110−3m6θF2yΘ1100−2m5FyΘ1100−2m7FyΘ1010+m62F2yΘ2000+m72Θ0020+2m6m7θFyΘ0110.The optimum values ofm5, m6, and m7, determined by minimizing (29), are(31)m5opt=Fyθ3Θ02003/2ϱ232−1+Θ20001/2−4+θ2Θ0200ϱ12−ϱ23ϱ13+2θΘ02001/2ϱ232−1−1+Θ20001−ϱ1.2328Θ02001/2ϱ232−1−1+Θ20001−ℜ1.232,m6opt=4Θ2000−2ϱ12ϱ13ϱ23+ϱ122+ϱ132+ϱ232−1+θ2Θ0200ϱ232−1−4ϱ232−11+Θ20001−ℜ1.232,m7opt=FyΘ20001/2θ2Θ0200−ϱ13+ϱ12ϱ23+4ϱ13−ϱ12ϱ23−4Θ00201/2ϱ232−11+Θ20001−ℜ1.232.The simplified minimum MSE ofF^Pr1y at the optimum values of m5, m6, and m7 is(32)MSEminF^Pr1y≅F2y16Θ20001−ℜ1.232−θ4Θ02002−8θ2Θ0200Θ20001−ℜ1.232161+Θ20001−ℜ1.232,where ℜ1.232=Θ11002Θ0020+Θ10102Θ0200−2Θ1010Θ1100Θ0110/Θ2000Θ0200Θ0020−Θ01102.It can be seen thatF^Pr1y is more precise than F^Regy.On similar lines, the second proposed family of estimators for estimatingFy is given by(33)F^Pr2y=F^y2expFx−F^xF^x+Fx+expF^x−FxF^x+Fx+m8Fx−F^xFx+m9F^y+m10Z¯−Z¯^Z¯expaFx−F^xaFx+F^x+2b,where m8, m9, and m10 are unknown constants and a≠0 and b are either two real numbers or functions of known population parameters of IX≤x, such as ρ12, β2 (coefficient of kurtosis), and C2.The estimatorF^Pr2y can also be written as(34)F^Pr2y=m8Fy1+e1−m9e2−m10e41−12θe2+38θ2e22+⋯.Simplifying (34) and keeping terms only up to the second power of eis, we can write(35)F^Pr2y−Fy=−Fy+m8Fy+m8Fye1−12θm8Fye2−m9e2−m10e4+38θ2m8Fye12+12θm9e12−12θm8Fye1e2+12θm10e2e4.The bias and MSE ofF^9y, to the first order of approximation, are(36)biasF^Pr2y≅12θ2FyΘ0200−12θFyΘ1100+12m8θΘ0200+m9Fy+38m9θ2FyΘ0200−12m9θFyΘ1100+12m10θΘ0101,MSEF^Pr2y≅−θF2yΘ1100+32m9θ2F2yΘ0200+m92θ2F2yΘ0200+m8θFyΘ0200−2m9m10FyΘ1001+14θ2F2yΘ0200+2m9F2yΘ2000+m82Θ0200−2m92θF2yΘ1100+F2yΘ2000+m92F2y+m10θFyΘ0101+2m8m9θFyΘ0200−2m8m9FyΘ1100+2m8m10Θ0101−3m9θF2yΘ1100−2m8FyΘ1100−2m10FyΘ1001+m92F2yΘ2000+m102Θ0002+2m9m10θFyΘ0101.The optimum values ofk8, k9, and k10, determined by minimizing (36), are(37)m8opt=Fyθ3Θ02003/2ϱ242−1+Θ20001/2−4+θ2Θ0200ϱ12−ϱ24ϱ14+2θΘ02001/2ϱ242−1−1+Θ20001−ϱ1.2428Θ02001/2ϱ242−1−1+Θ20001−ℜ1.242,m9opt=4Θ2000−2ϱ12ϱ14ϱ24+ϱ122+ϱ142+ϱ242−1+θ2Θ0200ϱ242−1−4ϱ242−11+Θ20001−ℜ1.242,m10opt=FyΘ20001/2θ2Θ0200−ϱ14+ϱ12ϱ24+4ϱ14−ϱ12ϱ24−4Θ00201/2ϱ242−11+Θ20001−ℜ1.242.The simplified minimum MSE ofF^Pr2y at the optimum values of k8, k9, and k10 is(38)MSEminF^Pr2y≅F2y16Θ20001−ℜ1.242−θ4Θ02002−8θΘ0200Θ20001−ℜ1.242161+Θ20001−ℜ1.242,where ℜ1.242=Θ11002Θ0002+Θ10012Θ0200−2Θ1001Θ1100Θ0101/Θ2000Θ0200Θ0002−Θ01012.It can be seen thatF^Pr2y is more precise than F^Regy.In Table2, we put some members of the Grover and Kaur [20] and proposed families of estimators with selected choices of a and b.Table 2
Some members of the adapted and proposed distribution function estimators.
abF^G,KyF^8yF^9y1C2F^GK1yF^Pr11yF^Pr21y1β2F^GK2yF^Pr12yF^Pr22yβ2C2F^GK3yF^Pr13yF^Pr23yC2β2F^GK4yF^Pr14yF^Pr24y1ρ12F^GK5yF^Pr15yF^Pr25yC2ρ12F^GK6yF^Pr16yF^Pr26yρ12C2F^GK7yF^Pr17yF^Pr27yβ2ρ12F^GK8yF^Pr18yF^Pr28yρ12β2F^GK9yF^Pr19yF^Pr29y1NFxF^GK10yF^Pr110yF^Pr210y


## 5. Efficiency Comparisons

In this section, the adapted and proposed estimators ofFy are compared in terms of the minimum MSEs.(i)
From (8) and (32),(39)MSEminF^Pr1y<VarF^y,if VarF^y−MSEminF^Pr1y>0.(ii)
From (11) and (32)(40)MSEminF^Pr1y<MSEF^Ry,if MSEF^Ry−MSEminF^Pr1y>0.(iii)
From (13) and (32),(41)MSEminF^Pr1y<MSEF^Py,if MSEF^Py−MSEminF^Pr1y>0.(iv)
From (16) and (32),(42)MSEminF^Pr1y<VarminF^Regy,ifVarminF^Regy−MSEminF^Pr1y>0.(v)
From (21) and (32),(43)MSEminF^Pr1y<MSEminF^R,Dy,ifMSEminF^R,Dy−MSEminF^Pr1y>0.(vi)
From (26) and (32),(44)MSEminF^Pr1y<MSEminF^G,Ky,ifMSEminF^G,Ky−MSEminF^Pr1y>0.(vii)
From (8) and (38),(45)MSEminF^Pr2y<VarF^y,if VarF^y−MSEminF^Pr2y>0.(viii)
From (11) and (38),(46)MSEminF^Pr2y<MSEF^Ry,if MSEF^Ry−MSEminF^Pr2y>0.(ix)
From (13) and (38),(47)MSEminF^Pr2y<MSEF^Py,if MSEF^Py−MSEminF^Pr2y>0.(x)
From (16) and (38),(48)MSEminF^Pr2y<VarminF^Regy,ifVarminF^Regy−MSEminF^Pr2y>0.(xi)
From (21) and (38),(49)MSEminF^Pr2y<MSEminF^R,Dy,ifMSEminF^R,Dy−MSEminF^Pr2y>0.(xii)
From (26) and (38),(50)MSEminF^Pr2y<MSEminF^G,Ky,ifMSEminF^G,Ky−MSEminF^Pr2y>0.The proposed families of estimators are always more precise than the adapted estimators as conditions (i)–(xii) are always true.


## 6. Empirical Study

In this section, we conduct a numerical study to see the performance of the existing and proposed distribution function 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 an estimator F^iy with respect to F^Hy is where i=R,P,Reg,R,D,…,Pr2.Table 3
Summary statistics for population I.
ParameterValueParameterValueN30Z¯15.5000n8S39.23238λ0.09167S48.79557X¯67.2667ρ340.98899ParameterX¯, Y¯Θ2y, Θ2xFy—0.606300.16535Fx—0.769230.20513S1—0.508550.50855S2—0.508550.50742ρ12—−0.73333−0.80178ρ13—0.719750.71975ρ23—−0.83726−0.84109ρ14—0.736220.73622ρ24—−0.86728−0.86535NonresponseParameterValueParameterValueN28S329.38749W20.26667S422.44949λ20.03333ρ3420.91015ParameterX¯, Y¯Θ2y, Θ2xS12—0.517550.51755S22—0.517550.51755ρ122—−0.60000−0.60000ρ132—0.363870.36387ρ232—−0.81228−0.81228ρ142—0.281720.28172ρ242—−0.84515−0.84515Table 4
Summary statistics for population II.
ParameterValueParameterValueN50Z¯25.5000n15S321.3175λ0.04667S414.5756X¯78.2900ρ340.94677ParameterX¯, Y¯Θ2y, Θ2xFy—0.500000.16535Fx—0.500000.20513S1—0.505080.47121S2—0.505080.49856ρ12—−0.12000−0.14941ρ13—0.229250.28411ρ23—−0.78936−0.80938ρ14—0.184350.25257ρ24—−0.86630−0.85514NonresponseParameterValueParameterValueN212S3218.2593W20.24000S423.60555λ20.01600ρ3420.97952ParameterX¯, Y¯Θ2y, Θ2xS12—0.514930.38924S22—0.522230.51493ρ122—−0.16903−0.37796ρ132—0.256950.13750ρ232—−0.81370−0.84530ρ142—0.220340.12955ρ242—−0.86905−0.85689Table 5
Summary statistics for population III.
ParameterValueParameterValueN50Z¯25.5000n15S322.18052λ0.04667S414.57598X¯75.8720ρ340.95742ParameterX¯, Y¯Θ2y, Θ2xFy0.500000.66000Fx—0.500000.58000S1—0.505080.50508S2—0.505080.50508ρ12—−0.20000−0.18306ρ13—0.300940.34288ρ23—−0.79517−0.81844ρ14—0.257810.33356ρ24—−0.86628−0.85512NonresponseParameterValueParameterValueN212S3219.5392W20.24000S423.60555λ20.01600ρ3420.98710ParameterX¯, Y¯Θ2y, Θ2xS22—0.522230.52223ρ122—−0.50709−0.44721ρ132—0.358480.18467ρ232—−0.82900−0.82900ρ142—0.367240.19433ρ242—−0.86905−0.86905(51)PREF^iy,F^Hy=VarF^HyMSEminF^iy×100,The PREs of distribution function estimators, computed from three populations, are given in Tables6 and 7.Population I (source: [21]).Y: duration of sleep of persons with age more than 50 yearsX: the age of persons in years. The proportion of the non-response units in the given population is considered to be the last 25% unitsPopulation II (source: [22]).Y: the eggs produced in 1990 (millions)X: the price per dozen (cents) in 1990. The proportion of the non-response units in the given population is considered to be the last 25% unitsPopulation III (source: [22]).Y: the eggs produced in 1990 (millions)X: the price per dozen (cents) in 1991. The proportion of the non-response units in the given population is considered to be the last 25% unitsTable 6
PREs of distribution function estimators using populations I, II and III with situation-I and situation-II, whenx,y=Θ2x,Θ2y.
EstimatorsPopulation IPopulation IIPopulation IIIF^GKyF^Pr1yF^Pr2yF^GKyF^Pr1yF^Pr2yF^GKyF^Pr1yF^Pr2yF^GK1yF^Pr11yF^Pr21y208.21211.36216.86108.45113.58109.94115.27118.13115.63F^GK2yF^Pr12yF^Pr22y208.22211.39216.90108.46113.58109.95115.28118.13115.64F^GK3yF^Pr13yF^Pr23y208.21211.36216.86108.45113.58109.94115.27118.13115.63F^GK4yF^Pr14yF^Pr24y208.24211.43216.93108.46113.59109.95115.28118.14115.65F^GK5yF^Pr15yF^Pr25y270.64579.94610.90111.76121.61117.66121.61134.60131.64F^GK6yF^Pr16yF^Pr26y284.55855.14921.21111.73121.54117.60121.50134.27131.32F^GK7yF^Pr17yF^Pr27y209.72214.62220.21108.27113.18109.56115.09117.74115.26F^GK8yF^Pr18yF^Pr28y270.64579.94610.90111.76121.61117.66121.61134.60131.64F^GK9yF^Pr19yF^Pr29y209.85579.94220.51108.27113.18109.56115.09117.75115.26F^GK10yF^Pr110yF^Pr210y207.47209.83215.29108.26113.17109.55115.07117.70115.22F^y100.00100100F^Ry29.4743.9738.87F^Py164.9457.4669.33F^Regy194.40101.8108.61F^R,Dy207.46108.26115.07F^GK1yF^Pr11yF^Pr21y177.62185.80186.47107.67111.58109.81109.64114.27112.09F^GK2yF^Pr12yF^Pr22y177.63185.82186.49107.68111.58109.81109.64114.28112.09F^GK3yF^Pr13yF^Pr23y177.62185.80186.47107.67111.58109.81109.64114.27112.09F^GK4yF^Pr14yF^Pr24y177.65185.84186.52107.68111.59109.82109.65114.28112.09F^GK5yF^Pr15yF^Pr25y207.48295.08296.44109.98116.91115.04113.76124.47122.04F^GK6yF^Pr16yF^Pr26y212.84327.94329.57109.96116.87115.00113.69124.28121.86F^GK7yF^Pr17yF^Pr27y178.54187.79188.48107.54111.29109.53109.51114.00111.82F^GK8yF^Pr18yF^Pr28y207.48295.08296.44109.98116.91115.04113.76124.47122.04F^GK9yF^Pr19yF^Pr29y178.62187.97188.65107.54111.29109.53109.51114.00111.82F^GK10yF^Pr110yF^Pr210y177.17184.83185.50107.53111.28109.52109.50113.97111.79F^y100.00100.00100.00F^Ry35.8252.2449.21F^Py151.2864.0969.33F^Regy164.11101.07103.04F^R,Dy177.16107.53109.50Table 7
PREs of distribution function estimators using populations I, II, and III with situation-I and situation-II, whenx,y=X¯,Y¯,.
EstimatorsPopulation IPopulation IIPopulation IIIF^GKyF^Pr1yF^Pr2yF^GKyF^Pr1yF^Pr2yF^GKyF^Pr1yF^Pr2yF^GK1yF^Pr11yF^Pr21y239.81240.75243.79107.15110.21109.02109.41114.28114.10F^GK2yF^Pr12yF^Pr22y239.73240.59243.63107.09110.08108.90109.35114.16113.97F^GK3yF^Pr13yF^Pr23y239.83240.79243.84107.18110.26109.08109.44114.34114.16F^GK4yF^Pr14yF^Pr24y239.68240.48243.52107.07110.02108.85109.33114.09113.91F^GK5yF^Pr15yF^Pr25y287.13431.28439.07109.76116.82115.55112.69122.96122.76F^GK6yF^Pr16yF^Pr26y270.32338.77343.87110.19118.07116.77113.44125.30125.09F^GK7yF^Pr17yF^Pr27y243.94250.01253.20106.96109.80108.63109.22113.88113.70F^GK8yF^Pr18yF^Pr28y296.41502.56512.87109.55116.22114.96112.35121.92121.72F^GK9yF^Pr19yF^Pr29y242.79247.30250.44106.95109.79108.61109.21113.86113.67F^GK10yF^Pr110yF^Pr210y238.92238.93241.95106.94109.77108.59109.20113.83113.65F^y100.00100.00100.00F^Ry30.5531.0031.42F^Py208.9145.8650.75F^Regy225.86104.18106.19F^R,Dy238.91106.94109.20F^GK1yF^Pr11yF^Pr21y201.21203.59203.62104.76111.86110.04105.97116.90117.29F^GK2yF^Pr12yF^Pr22y201.16203.50203.53104.72111.77109.95105.93116.80117.19F^GK3yF^Pr13yF^Pr23y201.22203.62203.65104.77111.90110.08105.99116.94117.33F^GK4yF^Pr14yF^Pr24y201.13203.43203.46104.70111.73109.91105.91116.76117.15F^GK5yF^Pr15yF^Pr25y224.45274.00274.05106.51116.30114.38108.12122.64123.06F^GK6yF^Pr16yF^Pr26y217.00246.45246.49106.79117.08115.15108.58124.07124.49F^GK7yF^Pr17yF^Pr27y203.62208.84208.87104.62111.56109.75105.84116.60116.99F^GK8yF^Pr18yF^Pr28y228.30290.50290.56106.38115.92114.01107.90121.99122.41F^GK9yF^Pr19yF^Pr29y202.96207.35207.39104.61111.55109.74105.83116.58116.97F^GK10yF^Pr110yF^Pr210y200.67202.49202.52104.61111.54109.72105.82116.56116.95F^y100.00100.00100.00F^Ry36.6939.2540.01F^Py183.1551.3755.81F^Regy187.61101.84102.81F^R,Dy200.66104.61105.82From the numerical results, presented in Tables6 and 7, it is observed that the PREs of all families of estimators change with the choices of a and b. It is further noted that the proposed families of estimators are more precise than the existing distribution function estimators of Hansen and Hurwitz [1]; Cochran [17]; Murthy [18]; Rao [19]; and Grover and Kaur [20], in terms of PRE under both situations.


## 7. Concluding Remarks

In this paper, we have proposed two new families of estimators for estimating the finite population distribution function. The proposed estimators needed supplementary data on the sample mean and ranks of the auxiliary variable. The biases and mean squared error of the proposed families of estimators were derived using the first order of approximation. Based on theoretical as well as numerical comparative studies, it is concluded that the proposed families of estimators are more precise than their existing counterparts under situation-I and situation-II. So, we recommend using the sample mean and ranks of the auxiliary variable with the proposed families of estimators for estimating the finite population distribution function.



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