# A New Exponential Factor-Type Estimator for Population Distribution Function Using Dual Auxiliary Variables under Stratified Random Sampling

Authors: Anum Iftikhar; Shi Hongbo; Sardar Hussain; Sajid Khan; Badr Alnssyan; Javid Shabbir; Sohaib Ahmad; Muhammad Aamir
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we propose a generalized class of exponential-type estimators for estimating the finite population distribution function using dual auxiliary variables under stratified sampling. The biases and mean squared errors (MSEs) of the proposed class of estimators are derived up to the first order of approximation. The empirical and theoretical study of comparisons is discussed. Four populations are taken for the support of the theoretical findings. It is observed that the proposed class of estimators performs better as compared to all other considered estimators in stratified sampling.


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## Body

## 1. Introduction

In survey sampling, the auxiliary information is often used to increase the precision of an estimator of population parameter(s), such as population mean, median, distribution function, quantiles, and standard deviation, etc., exist in the literature, which need single or two auxiliary information.Our primary goal is to enhance the precision of the estimator; for this reason, we use stratified random sampling. If the population of interest is homogeneous, then simple random sampling performs good. But there is a situation when the population of interest is heterogeneous, in such situation, it is advisable to use the stratified random sampling instead of simple random sampling. In stratified random sampling, we split the whole aggregate into number of nonoverlapping groups or subgroups called strata. These groups are homogeneous entirely and sample is drawn independently from each stratum separately. To obtain the maximum benefit from stratification, the values of the Nh must be known. When the strata have been determined, a sample is drawn from each, and the drawings being made independently. In stratified sampling, every stratum is handled as separate population, and consequently samples are drawn independently from every stratum.In other words, if SRS is used in each stratum for the selection of the sample, then the corresponding sample is called a stratified random sample. For good stratification, it requires that each stratum should be internally homogeneous but should externally differ from one another. Stratification may often produce gains in the precision of estimates. In stratified random sampling, the given population is divided into several strata. Then, from each stratum, a simple random sample is selected depending upon the size of the stratum. Estimators are first drawn from each stratum and then combined into a precise estimate of the population parameter.A lot of work has been done on the estimation of the population mean. Some important references on the population mean estimation using auxiliary information include Diana [1], Kadilar and Cingi [2, 3], Shabbir and Gupta [4], Shabbir and Gupta [5], Shabbir and Gupta [5], Haq and Shabbir [6], Aladag and Cingi [7], Singh and Khalid [8], Malik and Singh [9], Muneer et al. [10], Shabbir and Gupta [11], Haq et al. [12], Kaur et al. [13], Ahmad and Shabbir [14], Singh and Khalid [15], Al-Marzouki et al. [16], Ahmad et al. [17], Ahmad et al. [18], and Ahmad et al. [19]. In these works, the authors have suggested improved ratio, product, and regression-type estimators for estimating the finite population mean. They introduced estimators which used the auxiliary information to estimate the population mean and total.In the literature of sampling, the authors have estimated the DF using information on one or more auxiliary variable. Chambers and Dunstan [20] suggested an estimator for estimating the DF that requires information both on the study and auxiliary variables. Similarly, Rao et al. [21] and Rao [22] suggested ratio and difference/regression estimators for estimating the DF under a general sampling design. Kuk [23] suggested a kernel method for estimating the DF using the auxiliary information. Ahmed and Abu-Dayyeh [24] estimated the DF using information on multiple auxiliary variables. A calibration approach was used by Rueda et al. [25] to devise an estimator 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 of a survey. Moreover, Yaqub and Shabbir [27], Hussain et al. [28], and Hussain et al. [29] considered a generalized class of estimators for estimating the DF in the presence of non-response, while Hussain et al. [30] proposed two new families of estimators using dual auxiliary information under simple and stratified random sampling. Furthermore, Ahmad et al. [31] suggested a new estimator of DF using auxiliary information.In this paper, we propose a new estimator for estimating the DF using information on the distribution function and mean of the auxiliary variable. The biases and mean squared errors (MSEs) of the existing and proposed estimators of the DF are derived under the first order of approximation. From theoretical and numerical comparisons, we can say that the proposed estimator is more precise than the existing adapted estimators when estimating the DF.The rest of the paper is organized as follows. In Section2, some notations are given. In Section 3, some existing estimators of the finite population mean for estimating the finite DF are studied. The proposed estimator is given in Section 4. In Sections 5 and 6, theoretical and numerical comparisons are made, respectively. In Section 7, interpretation of the results in tables is deliberated. Finally, conclusions are drawn in Section 8.


## 2. Notation

Consider a finite populationΩ=1,2,…,N of N distinct units, which is divided into L homogeneous strata, where the size of hth stratum is Nh, for h=1,2,…,L, such that ∑h=1LNh=N. Let Y and X be the study and auxiliary variables which take values yh and xh, respectively, where i=1,2,…,Nh and h=1,2,…,L; for estimating finite population distribution function, assume that a sample of size nh is drawn from the h th stratum using simple random sampling without replacement, such that ∑h=1Lnh=N, where n is the sample size.Y: the study variable.X: the auxiliary variable.Let(1)Ui=IYih≤tyh=1ifYih≤tyh,0ifYih>tyh,and(2)Vi=IXih≤tx=1ifXih≤txh,0ifXih>txh.IYih≤tyh: indicator variable based on Y,IXih≤txh: indicator variable based on X,U¯h=∑i=1NhUih/Nh=∑i=1NIYih≤tyh/N: the population distribution function of Y for the hth stratum,U¯st=U¯=∑h=1LWhU¯h: the population distribution function of Y,V¯h=∑i=1NhVih/Nh=∑i=1NIXih≤txh/N: the population distribution function of X for the hth stratum,V¯st=V¯=∑h=1LWhV¯h: the population distribution function of X,X¯h=∑i=1NhXih/Nh: the population mean of X for the hth stratum,X¯st=X¯=∑h=1LWhX¯h: the population mean of X,U¯^h=∑i=1nhUih/nh=∑i=1nIYih≤tyh/n: the sample distribution function of Y for the hth stratum,U¯^st=U¯^=∑h=1LWhU¯^h: the sample distribution function of Y,V¯^h=∑i=1nhVih/nh=∑i=1nIXih≤txh/n: the sample distribution function of X for the hth stratum,V¯^st=V¯^=∑h=1LWhV¯^h: the sample distribution function of X,X¯^h=∑i=1nhXih/nh: the sample mean of X for the hth stratum,X¯^st=X¯^=∑h=1LWhX¯^h: the sample mean of X,Suh2=∑i=1NhUih−U¯h2/Nh−1: the population variance of U for the hth stratum,Svh2=∑i=1NhVih−V¯h2/Nh−1: the population variance of V for the hth stratum,Sxh2=∑i=1NhXih−X¯h2/Nh−1: the population variance of X for the hth stratum,Cuh=Suh/U¯h: the population coefficient of variation of U for the hth stratum,Cvh=Svh/U¯h: the population coefficient of variation of V for the hth stratum,Cxh=Sxh/U¯h: the population coefficient of variation of X for the hth stratum,Suvh=∑i=1NhUih−U¯hVih−V¯h/Nh−1: the population covariance between U and V, for the hth stratum,Suxh=∑i=1NhUih−U¯hXih−Xh/Nh−1: the population covariance between U and X, for the hth stratum,Svxh=∑i=1NhVih−V¯hXih−X¯h/Nh−1: the population covariance between V and X, for the hth stratum,Ruxh=Suxh/SuhSvh: the population correlation coefficient between U and V for the hth stratum,Ruxh=Suxh/SuhSxh: the population correlation coefficient between U and X for the hth stratum,Rvxh=Svxh/SvhSxh: the population correlation coefficient between V and X for the hth stratum,Ruv=∑h=1LWh2λhRuvhSuhSvh/∑h=1LWh2λhSuh2∑h=1LWh2λhSvh2: the population correlation coefficient between U and V,Rxv=∑h=1LWh2λhRxvhSxhSvh/∑h=1LWh2λhSuh2∑h=1LWh2λhSvh2: the population correlation coefficient between U and X,Rvx=∑h=1LWh2λhRxvhSvhSxh/∑h=1LWh2λhSuh2∑h=1LWh2λhSvh2: the population correlation coefficient between V and X,where λh=1/nh−1/Nh,Ru.vx2=Ruv2+Rux2−2RuvRuxRvx/1−Rvx2: multiple correlation coefficient of U on V and X.In order to obtain the biases and mean squared errors (MSEs) of the adapted and proposed estimators ofFy, we consider the following relative error terms.Lete0=U¯^−U¯/U¯,  e1=V¯^−V¯/V¯,  and e2=X¯^−X¯/X¯  such that Eei=0 for i=0,1,2, where E· is the mathematical expectation of ·. Let Eeih=0, i=0,1,2,(3)Ee0h2=λhCuh2,Ee1h2=λhCvh2,Ee2h2=λhCX¯^h2,Ee0he1h=λhRuvhCuhCvh,Ee0he2h=λhRuxhCuhCxh,andEe1he2h=λhRvxhCvhCxh.


## 3. Existing Estimators

In this section, we briefly review some existing estimators ofU¯.(1)
The conventional unbiased mean per unit estimator ofU¯ is as follows: the reference of this estimator is not included because this is a conventional unbiased estimator under simple random sampling.(4)U¯^st=∑h=1LWhU¯^h.The variance ofU¯^st is(5)VarU¯^st=∑h=1LWh2λhU¯h2Cuh2.(2)
Cochran [32] suggested the traditional ratio estimator of U¯, which is given by(6)U¯^R,h=∑h=1LWhU¯^hV¯hV¯^h.The bias and MSE ofU¯^R,h, to first order of approximation, respectively, are(7)BiasU¯^R,h≅∑h=1LWhλhU¯hCvh2−RuvhCuhCvh,(8)MSEU¯^R,h≅∑h=1LWh2λhU¯h2Cuh2+Cvh2−2RuvhCuhCvh.The ratio estimatorU¯^R,h is better than U¯^st, in terms of MSE, if Ruvh>Cvh/2Cuh.(3)
Murthy [33] suggested the usual product estimator of U¯, which is given by(9)U¯^P,h=∑h=1LWhU¯^hV¯^hV¯h.The bias and MSE ofU¯^P,h, to first order of approximation, are given by(10)BiasU¯^P,h=∑h=1LWhλhU¯hRuvhCuhCvh,(11)MSEU¯^P,h≅∑h=1LWh2λhU¯h2Cuh2+Cvh2+2RuvhCuhCvh.The product estimatorU¯^P,h is better than U¯^st, in terms of MSE, if Ruvh<−Cvh/2Cuh.(4)
The conventional difference estimator ofU¯ is(12)U¯^Reg,h=∑h=1LWhU¯^h1+mV¯h−V¯^h,wherem is an unknown constant. U¯^Reg,h is an unbiased estimator of U¯. The simplified minimum variance of U¯^Reg,h at the optimum value of mopt=Ruvδ1/δ2 is(13)VarminU¯^Reg,h=∑h=1LWh2λhU¯h2Cuh2−Cuh2Ruvh2.(5)
Rao [37] suggested an improved difference-type estimator of U¯, which is given by(14)U¯^R.D,h=∑h=1LWhU¯^hm1+m2V¯h−V¯^h,wherem1 and m2 are unknown constants. The bias and MSE of U¯^R.D,h, to the first order of approximation, respectively, are(15)BiasF^tyR,D=∑h=1LWhλhU¯hm1−1,MSEU¯^R.D,h≅∑h=1LWh2λhU¯h2m1−12+Cuh2m12+V¯2Cvh2m22−∑h=1LWh2λhU¯h22V¯hRuvhCuhCvhm1m2.The optimum values ofm1 and m2 are(16)m1opt=11+λhCuh21−Ruvh2andm2opt=∑h=1LWhλhU¯hRuvCuhV¯hCvh1+Cuh21−Ruvh2,respectively. The simplified minimum MSE ofU¯^R.D,h at the optimum values of m1 and m2 is given by(17)MSEminU¯^R.D,h≅∑h=1LWh2λhU¯h2Cuh21−Ruvh21+Cuh21−Ruvh2.(6)
Bahl and Tuteja’s exponential ratio-type and product-type estimators [34] are given by(18)U¯^BT.R,h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯^h+V¯h,(19)U¯^BT.P,h=∑h=1LWhU¯^hexpV¯^h−V¯hV¯^h+V¯h.The biases and MSEs ofU¯^BT.R,h and U¯^BT.P,h, to first order of approximation, respectively, are(20)BiasU¯^BT.R,h≅∑h=1LWhλhU¯h3Cvh28−RuvhCuhCvh2,(21)MSEU¯^BT.R,h≅∑h=1LWh2λhU¯h2144Cuh2+Cvh2−4RuvhCuhCvh,(22)BiasU¯^BT.P,h≅∑h=1LWhλhU¯hRuvhCuhCvh2−Cvh28,(23)MSEU¯^BT.P,h≅∑h=1LWh2λhU¯h2144Cuh2+Cvh2+4RuvhCuhCvh.(7)
Grover and Kaur [35] suggested a generalized class of ratio-type exponential estimators, which is given by(24)U¯^G.K,h=∑h=1LWhU¯^hm3+m4V¯h−V¯^hexpV¯h−V¯^hV¯h+V¯^h,wherem3 and m4 are unknown constants. The bias and MSE of F^t−yGK, to the first order of approximation, respectively, are(25)BiasU¯^G.K,h≅∑h=1LWhλhU¯h1−m3+38m32C12+12m4V¯hC22−∑h=1LWhλhU¯h12m3RuvC1C2.The optimum values ofm3 and m4 determined by minimizing (24) are(26)m3opt=8−λh2Cvh281+λhCuh21−R122andm4opt=∑h=1LWhU¯hλh3C23+8RuvCuh−λh2RuvCuhCvh2−4Cvh1+λhCuh21−R1228V¯hCvh1+λhCuh21−R122,respectively. The minimum MSE of F^yGK at the optimum values of m3 and m4 is given by(27)MSEminU¯^G.K,h≅∑h=1LWh2λhU¯h264Cuh21−R122−λh4C24−16λh2Cuh2Cvh21−R122641+λhCuh21−R122.


## 4. Proposed Class of Estimators

The precision of an estimator surges by using the appropriate secondary information at the estimation stage. In previous studies, the sample distribution function of the auxiliary variable was used to expand the productivities of the prevailing distribution function estimators. In a recent study, Hussain et al. [30] recommended to use ranks of the auxiliary variable as an additional auxiliary variable to increase the precision of an estimator of the population distribution function. Similarly, we use additional auxiliary information on sample mean and sample distribution function of the auxiliary variable along with the sample distribution function of study variable to estimate the finite CDF.Using the above idea on the lines of Shukla et al. [36], we suggest a general class of exponential factor-type estimators which contains many stable and efficient estimators. By combining the idea of Bahl and Tuteja and Shukla et al. [34, 36], the first estimator is given by(28)U¯^proph=∑h=1LWhU¯^hexpSuh−M1hSuh+M1hexpSvh−M2hSvh+M2h,where(29)Suh=A1h+C1hU¯h+fhB1hU¯^h,Svh=A2h+C2hX¯h+fhB2hX¯^h,M1h=A1h+fhB1hU¯h+C1hU¯^h,M2h=A2h+fhB2hU¯h+C2hU¯^h,Aih=Kih−1Kih−2,Bih=Kih−1Kih−4,Cih=Kih−2Kih−3Kih−4.Substituting different values ofKih (i = 1,2,3,4) in (18), we can generate many more different types of estimators from our general proposed class of estimators, which are given in Table 1.Table 1
Some members of the proposed class of family of estimatorsU¯^proph.
S.NoK1hK2hEstimators111U¯^prop1h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯h−X¯^hX¯h+X¯^h212U¯^prop2h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯h−X¯^hX¯h+X¯^h313U¯^prop3h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpnX¯h−X¯^h2NX¯h−nX¯h−X¯^h414U¯^prop4h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^h521U¯^prop5h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯h−X¯^hX¯h+X¯^h622U¯^prop6h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯h−X¯^hX¯h+X¯^h723U¯^prop7h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯h−X¯^hX¯h+X¯^h824U¯^prop8h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^h931U¯^prop9h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯h−V¯^hexpX¯h−X¯^hX¯h+X¯^h1032U¯^prop10h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯h−V¯^hexpX¯h−X¯^hX¯h+X¯^h1133U¯^prop11h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯h−V¯^hexpX¯h−X¯^hX¯h+X¯^h1234U¯^prop12h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯h−V¯^h1341U¯^prop13h=∑h=1LWhU¯^hexpX¯h−X¯^hX¯h+X¯^h1442U¯^prop14h=∑h=1LWhU¯^hexpX¯^h−X¯hX¯^h+X¯h1543U¯^prop15h=∑h=1LWhU¯^hexpnX¯h−X¯^h2NX¯h−nX¯h−X¯^h1644U¯^prop16h=∑h=1LWhU¯^hSolvingU¯^proph given in (28) in terms of errors, we have(30)U¯^proph=∑h=1LWhU¯h1+e0h1+12θ1he1h−14θ1hΘ1he1h2+18θ1h2e1h2+⋯1+12θ2he2h−14θ2hΘ2he2h2+18θ2h2e2h2+⋯,where(31)θ1h=fhB1h−C1hA1h+fhB1h+C1h,Θ1h=fhB1h+C1hA1h+fhB1h+C1h,(32)θ2h=fhB2h−C2hA2h+fhB2h+C2h,Θ2h=fhB2h+C2hA2h+fhB2h+C2h.To first-order approximation, we have(33)U¯^proph−U¯h≅∑h=1LWhU¯he0h+12θ1he1h+12θ2he2h+12θ1he0he1h+12θ2he0he2h−14θ1hΘ1he1h2−14θ2hΘ2he2h2+18θ1h2e1h2+18θ2h2e2h2.Taking squaring and expectation of (33) to first order of approximation, we get the bias and MSE:(34)BiasU¯^proph≅∑i−1LWhU¯hλh12θ1hRuvhCuhCvh+12θ2hRuxhCuhCxh+18Cvh2θ1h2−2θ1hΘ1h+18Cxh2θ2h2−2θ2hΘ2h,(35)MSEU¯^proph≅∑i=1LWh2U¯h2λhCuh2+14θ1h2Cvh2+14θ2h2Cxh2+θ1hRuvhCuhCvh+θ2hRuxhCuhCxh+12θ1hθ2hRvxhCvhCxh.Differentiate (35) with respect to θ1h and θ2h, and we get the optimum values of θ1h and θ2h, i.e.,(36)θ1hopt=2CuhRuvh−RvxhRuxhCvhRvxh2−1,θ2hopt=2CuhRuxh−RvxhRuvhCxhRvxh2−1.Substituting the optimum values ofθ1hopt   and   θ2hopt in (35), we get minimum MSE of U¯^prop which is given by(37)MSEminU¯^proph≅∑i=1LWh2U¯h2λhCuh21−Ruvxh2,where(38)Ruvxh2=Ruvh2+Ruxh2−2RuvhRuxhRvxh1−Rvxh2is the multiple correlation coefficient of yh on V¯^h and X¯^h. Now by putting different values of Kih in (28), some members of the proposed class of estimators can be obtained as(1)
ForK1h=1 and K2h=1,(39)U¯^prop1h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯h−X¯^hX¯h+X¯^h.The bias and MSE ofU¯^prop1h are given by(40)BiasU¯^prop1h≅∑h=1LWhλhU¯h38Cvh2+38Cxh2−12RuvhCuhCvh−12RuxhCuhCxh,(41)MSEU¯^prop1h≅∑h=1LWh2λhU¯h2Cuh2+14Cvh2+14Cxh2−RuvhCuhCvh−RuxhCuhCxh+12RvxhCvhCxh.(2)
ForK1h=1 and K2h=2,(42)U¯^prop2h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpX¯^h−X¯hX¯^h+X¯h.The bias and MSE ofU¯^prop2h are given by(43)BiasU¯^prop2h≅∑h=1LWhλhU¯h−18Cvh2−18Cxh2−12RuvhCuhCvh,(44)MSEU¯^prop2h≅∑h=1LWh2λhU¯h2Cuh2+14Cvh2+14Cxh2−RuvhCuhCvh+RuxhCuhCxh−12RvxhCvhCxh.(3)
ForK1h=1 and K2h=3,(45)U¯^prop3h=∑h=1LWhU¯^hexpV¯h−V¯^hV¯h+V¯^hexpnX¯h−X¯^h2NX¯h−nX¯^h+X¯h.The bias and MSE ofU¯^prop3h are given by(46)BiasU¯^prop3h≅∑h=1LWhλhU¯h38Cvh2+18Cxh2−14fh1−fh2Cxh2−12RuvhCuhCvh−12fh1−fhRuxhCuhCxh,(47)MSEU¯^prop3h≅∑h=1LWh2λhU¯h2Cuh2+14Cvh2+14fh1−fh2Cxh2−RuvhCuhCvh−fh1−fhRuxhCuhCxh+12fh1−fhRvxhCvhCxh.(4)
ForK1h=2 and K2h=1,(48)U¯^prop5h=∑h=1LWhU¯^hexpV¯^h−V¯hV¯^h+V¯hexpX¯h−X¯^hX¯h+X¯^h.The bias and MSE ofU¯^prop5h are given by(49)BiasU¯^prop5h≅∑h=1LWhλhU¯h−18Cvh2+38Cvh2+12RuvhCuhCvh−12RuxhCuhCxh,(50)MSEU¯^prop5h≅∑h=1LWh2λhU¯h2Cuh2+14Cvh2+14Cxh2+RuvhCuhCvh−RuxhCuhCxh−12RvxhCvhCxh.(5)
ForK1h=2 and K2h=2,(51)U¯^prop6h=∑h=1LWhU¯^hexpV¯^h−V¯hV¯^h+V¯hexpX¯^h−X¯hX¯^h+X¯h.The bias and MSE ofU¯^prop6h are given by(52)BiasU¯^prop6h≅∑h=1LWhλhU¯h−18Cvh2−18Cvh2+12ρyhp1CuhCvh+12RuxhCuhCxh,(53)MSEU¯^prop6h≅∑h=1LWh2λhU¯h2Cuh2+14Cvh2+14Cxh2+RuvhCuhCvh+RuxhCuhCxh+12RvxhCvhCxh.(6)
ForK1h=2 and K2h=3,(54)U¯^prop7h=∑h=1LWhU¯^hexpV¯^h−V¯hV¯^h+V¯hexpnX¯h−X¯^h2NX¯h−nX¯^h+X¯h.The bias and MSE ofU¯^prop7h are given by(55)BiasU¯^prop7h≅∑h=1LWhλhU¯h−18Cvh2−18fh1−fhCxh2+12ρyhp1CuhCvh−12fh1−fhRuxhCuhCxh,(56)MSEU¯^prop7h≅∑h=1LWh2λhU¯h2Cuh2+14Cvh2+14fh1−fh2Cxh2+RuvhCuhCvh−fh1−fhRuxhCuhCxh−12fh1−fhRvxhCvhCxh.(7)
ForK1h=3 and K2h=1,(57)U¯^prop9h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯^h+V¯hexpX¯h−X¯^hX¯h+X¯^h.The bias and MSE ofU¯^prop9h are given by(58)BiasU¯^prop9h≅∑h=1LWhλhU¯h−18fh1−fh2Cvh2+38Cxh2−12fh1−fhRuvhCuhCvh+12fh1−fhRuxhCuhCxh,(59)MSEU¯^prop9h≅∑h=1LWh2λhU¯h2Cuh2+14fh1−fh2Cvh2+14Cxh2−fh1−fhRuvhCuhCvh−RuxhCuhCxh+12fh1−fhRvxhCvhCxh.(8)
ForK1h=3 and K2h=2,(60)U¯^prop10h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯^h+V¯hexpX¯^h−X¯hX¯^h+X¯h.The bias and MSE ofU¯^prop10h are given by(61)BiasU¯^prop10h≅∑h=1LWhλhU¯h−18fh1−fh2Cvh2−18Cxh2−12fh1−fhRuvhCuhCvh+12RuxhCuhCxh,(62)MSEU¯^prop10h≅∑h=1LWh2λhU¯h2Cuh2+14fh1−fh2Cvh2+14Cxh2−fh1−fhRuvhCuhCvh+RuxhCuhCxh−12fh1−fhRvxhCvhCxh.(9)
ForK1h=3 and K2h=3,(63)U¯^prop11h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯^h+V¯hexpnX¯h−X¯^h2NX¯h−nX¯^h+X¯h.The bias and MSE ofU¯^prop11h are given by(64)BiasU¯^prop11h≅∑h=1LWhλhU¯h−18fh1−fh2Cvh2−18fh1−fh2Cxh2−12fh1−fhRuvhCuhCvh−12fh1−fhRuxhCuhCxh,(65)MSEU¯^prop11h≅∑h=1LWh2λhU¯h2Cuh2+14fh1−fh2Cvh2+14fh1−fh2Cxh2−fh1−fhRuvhCuhCvh−fh1−fhRuxhCuhCxh+12fh1−fh2RvxhCvhCxh.(10)
ForK1h=3 and K2h=4,(66)U¯^prop12h=∑h=1LWhU¯^hexpnV¯h−V¯^h2NV¯h−nV¯^h+V¯h.The bias and MSE ofU¯^prop12h are given by(67)BiasU¯^prop12h≅∑h=1LWhλhU¯h−18fh1−fh2Cvh2−12fh1−fhRuvhCuhCvh,(68)MSEU¯^prop12h≅∑h=1LWh2λhU¯h2Cuh2+14fh1−fh2Cvh2−fh1−fhRuvhCuhCvh.(11)
ForK1h=4 and K2h=1,(69)U¯^prop13h=∑h=1LWhU¯^hexpX¯h−X¯^hX¯h+X¯^h.The bias and MSE ofU¯^prop13h are given by(70)BiasU¯^prop13h≅∑h=1LWhλhU¯h38Cxh2−12RuxhCuhCxh,(71)MSEU¯^prop13h≅∑h=1LWh2λhU¯h2Cuh2+14Cxh2−RuvhCuhCxh.(12)
ForK1h=4 and K2h=2,(72)U¯^prop14h=∑h=1LWhU¯^hexpX¯^h−X¯hX¯^h+X¯h.The bias and MSE ofU¯^prop14h are given by(73)BiasU¯^prop14h≅∑h=1LWhλhU¯h−18Cxh2+12ρyhp2CuhCxh,(74)MSEU¯^prop14h≅∑h=1LWh2λhU¯h2Cuh2+14Cxh2+RuvhCuhCxh.(13)
ForK1h=4 and K2h=3,(75)U¯^prop15h=∑h=1LWhU¯^hexpnX¯h−X¯^h2NX¯h−nX¯^h+X¯h.The bias and MSE ofU¯^prop15h are given by(76)BiasU¯^prop15h≅∑h=1LWhλhU¯h−18fh1−fh2Cxh2−12fh1−fhRuxhCuhCxh,(77)MSEU¯^prop15h≅∑h=1LWh2λhU¯h2Cuh2+14fh1−fh2Cxh2−fh1−fhRuxhCuhCxh.


## 5. Theoretical Comparison

(i)
From (5) and (37),(78)MSEminU¯^proph<VarU¯^stifVarU¯^st−MSEminU¯^proph>0.(ii)
From (8) and (37),(79)MSEminU¯^proph<MSEU¯^R,hifMSEU¯^R,h−MSEminU¯^proph>0.(iii)
From (11) and (37),(80)MSEminU¯^proph<MSEU¯^P,hifMSEU¯^P,h−MSEminU¯^proph>0.(iv)
From (13) and (37),(81)MSEminU¯^proph<MSEU¯^Reg,hifMSEU¯^Reg,h−MSEminU¯^proph>0.(v)
From (17) and (37),(82)MSEminU¯^proph<MSEminU¯^R.D,hifMSEminU¯^R.D,h−MSEminU¯^proph>0.(vi)
From (21) and (37),(83)MSEminU¯^proph<MSEU¯^BT.R,hifMSEU¯^BT.R,h−MSEminU¯^proph>0.(vii)
From (23) and (37),(84)MSEminU¯^proph<MSEU¯^BT.P,hifMSEU¯^BT.P,h−MSEminU¯^proph>0.(viii)
From (27) and (37),(85)MSEminU¯^proph<MSEminU¯^GK,hifMSEminU¯^GK,h−MSEminU¯^proph>0.


## 6. Empirical Study

In this portion, we conduct a numerical study to judge the performances of the existing and proposed DF estimators. For this purpose, two datasets are taken. The summary statistics of these datasets are reported in Tables2 and 3. The PRE of an estimator U¯^hi with respect to U¯^st is(86)PREU¯^hi,U¯^st=VarU¯^stMSEminU¯^hi×100,where i=Rh,Ph,…,GK,h.Table 2
Summary statistics for Population I.
hNhnhWhλhU¯hV¯hX¯hSuhSvhSxhRuvhRuxhRvxh1127310.13750.02440.35430.3779208050.48020.4868304870.9164−0.4602−0.48322117210.12670.03900.41880.487292120.49550.5019151810.8709−0.4147−0.46003103290.11150.02480.42720.4660143090.49700.5013275500.9244−0.3928−0.41984170380.18410.02040.57650.611894790.49560.4888182190.8805−0.5074−0.53965205220.22210.04060.61460.653755700.48790.476984980.8772−0.5579−0.59096201390.21770.02070.50250.3532129980.50120.4792230940.7145−0.4334−0.3554Table 3
Summary statistics for Population II.
hNhnhWhλhU¯hV¯hX¯hSuhSvhSxhRuvhRuxhRvxh1127310.13750.02440.35430.3700498.2760.48020.4847555.580.8983−0.5398−0.55802117210.12670.03910.41880.4700318.3330.49550.5013365.450.8666−0.5205−0.55893103290.11150.02480.42720.4272431.3590.49700.4970613.950.9603−0.4803−0.48234170380.18410.02040.57650.5882311.3240.49560.4936458.020.9277−0.5818−0.59345205220.22210.04060.61460.6146227.1950.48790.4879260.850.8764−0.6395−0.64576201390.21770.02070.50250.4527312.7060.50120.4990397.040.8450−0.5063−0.4873The PREs of DF estimators, computed from five populations, are given in Tables4 and 5.Table 4
MSEs of DF estimators using Populations I and II.
EstimatorMSE valueMSE valueU¯^st0.0012297090.001229709U¯^R,h0.0002877690.000360417U¯^P,h0.0046352300.004491550U¯^Reg,h0.0002707180.000338073U¯^R.D,h0.0002704270.000337619U¯^BT.R,h0.0004507910.000495994U¯^BT.P,h0.0026245220.002561561U¯^G.k.h0.0002840020.000350169U¯^proph0.0002651310.000332562Table 5
PREs of DF estimators using Populations I and II.
EstimatorPRE valuePRE valueU¯^st100.00100.00U¯^R,h427.32341.19U¯^P,h26.53027.380U¯^Reg,h454.24363.74U¯^R.D,h454.73364.23U¯^BT.R,h272.79247.93U¯^BT.P,h46.85048.010U¯^G.k.h432.99351.18U¯^proph463.81369.77Population I (Source: 22)Y: the number of teachers.X: the number of students in both primary and secondary schools in Turkey in 2007 for 923 districts in six regions.Population II (Source: 22)Y: the number of teachers.X: the number of classes in both primary and secondary schools in Turkey in 2007 for 923 districts in six regions.


## 7. Interpretation of Results

As mention above, we used two datasets for numerical illustration. The proposed estimator and the existing estimators were compared between each other with respect to their MSE and PRE values. The results of PREs are presented in Tables4 and 5. In Tables 2 and 3, we see the summary statistics about the populations. It is further noted that the proposed estimator is more precise than the existing distribution function estimators of Cochran [32], Murthy [33], Rao [37], and Grover and Kaur [38], in terms of MSEs and PREs.


## 8. Conclusion

In this paper, we proposed an improved class of estimators of finite population DF by utilizing real-life datasets on dual auxiliary variables in stratified random sampling (StRS) scheme. Bias and MSE expressions of a proposed class of estimatorsU¯^proph are acquired up to first order of approximation. Based on the theoretical and numerical results, the proposed class of estimators performs better than the existing estimators considered under stratified random sampling. From these findings, we suggest the utilization of the proposed estimators for efficient estimation of population distribution function in the presence of the auxiliary information under stratified random sampling.



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*Source: 7673945-2022-05-10.xml*