# A Simulation Study: Population Distribution Function Estimation Using Dual Auxiliary Information under Stratified Sampling Scheme

Authors: Sohaib Ahmad; Sardar Hussain; Erum Zahid; Anum Iftikhar; Saddam Hussain; Javid Shabbir; Muhammad Aamir
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In survey sampling, information on auxiliary variables related to the main variable is often available in many practical problems. Since the mid-twentieth century, researchers have taken a keen interest in the use of auxiliary information, due to its usefulness in estimation methods. In this article, our main objective is to discover the problem associated with estimation of the finite population distribution function, using the known auxiliary variable, which occurs as the sample distribution function and the rank of the auxiliary variable. A new family of the finite population distribution function estimators is proposed in the stratified sampling scheme. The mathematical equations for the bias and mean square error have been obtained for each proposed estimator, along with the eﬃciency conditions. Besides theoretical eﬃciency comparison, an empirical study has also been conducted to analyze the performance of estimators. A simulation study is also performed to observe the efficiency of the proposed estimators. The implementation of the proposed sampling scheme is illustrated by a practical example.


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## Body

## 1. Introduction

In the literature of survey sampling, in certain situations, the use of the auxiliary variable may increase the precision of estimators in estimating the population parameters of interest. Researchers have already found out how to obtain estimates for unknown population parameters such as mean, median, variance, and standard deviation that possess maximum statistical properties. For that purpose, a representative part of population is needed. (i) When population is homogeneous, then preferably one can utilize the idea of simple random sampling (SRS). (ii) On the other hand, when population is heterogeneous, then stratified random sampling is useful.When the study and the auxiliary variables are correlated, then the rank of the auxiliary variable, distribution function, median, etc. are also correlated with the study variable. When there exists a positive or negative correlation between the study and the auxiliary variables, ratio and product estimators can improve the precision of estimates. By making use of the auxiliary information the researcher can explore these research findings by looking in, Ahmad and Shabbir [1]; Kadilar and Cingi [2]; Grover and Kaur [3]; Haq et al. [4]; and Al-Marzouki et al. [5].The dual use of the auxiliary information to estimate finite population distribution function is rarely used. The issue of estimating the finite population distribution function arises when our interest lies in finding out the proportion of the values of the study variable which is less than or equal to some threshold. In certain situations, the need of cumulative distribution function is much more important. Many authors have estimated the distribution function by using information on single or more auxiliary variables. First of all, Chamber and Dunstan [6] suggested a procedure for estimating the finite population distribution function. Kuk [7] presented a classical as well as a prediction approach in estimating the distribution function from a survey data. Researchers can investigate articles related to distribution function (DF) such as research studies conducted by Chamber et al. [8]; Dorfman [9]; Diana [10]; Rao [11]; Diana and Perri [12]; Ahmad and Abu-Dayyeh [13]; Rueda and Arcos [14]; Singh and Kumar [15]; Dorfman [16]; Diana and perri [12]; Husaain et al. [17]. They proposed two new estimators for estimating the finite population distribution function based on simple and stratified random sampling schemes using supplementary information.Taking motivation fromℱ∧R,DY, ℱ∧SY, and average of ℱ∧BT,RY and ℱ∧BT,PY, we proposed new family of estimators for estimating finite population distribution function under stratified sampling scheme.The rest of the article is organized as follows. Section2 presents the notation and symbols of stratified random sampling. In Section 3, the existing estimators were studied. We proposed a new family of estimators for estimating finite population distribution function under stratified random sampling in Section 4. Empirical study is conducted in Section 5. We also conduct a simulation study for the support of our proposed family of estimators under stratified random sampling in Section 6. Finally, conclusion of this paper is drawn in Section 7.


## 2. Notations in Stratified Random Sampling

LetΩ=1,2,…,N be a finite population of N units, which is divided into L homogeneous strata, where the size of hth stratum is Nh, for h=1,2,…,L, in such a manner that ∑h=1LNh=N. Let Yih and Xih be the characteristics of the study variable (Y) and the auxiliary variable (X), respectively, where i=1,2,…,Nh and h=1,2,…,L. A sample of size nh is drawn from variable such that ∑h=1Lnh=n, where n is the sample size.Letℱsty=ℱy=∑h=1LWhℱhy and ℱstx=ℱx=∑h=1LWhℱhx, ℱ∧sty=ℱ∧y=∑h=1LWhℱ∧hy and ℱ∧stx=ℱ∧x=∑h=1LWhℱ∧hx be the population and sample distribution function, respectively, of Y and X under stratified random sampling, where Wh=Nh/N, ℱhy=∑i=1NhIYih≤Y/Nh, ℱ∧hy=∑i=1nhIYih≤Y/nh, ℱhx=∑i=1NhIXih≤X/Nh, and ℱ∧hx=∑i=1nhIXih≤X/nh. Let Z¯st=Z¯=∑h=1LWhZ¯h and Z¯^st=Z¯=∑h=1LWhZ¯h be the population and sample means of Z under stratified random sampling, respectively, where Z¯h=∑i=1NhZih/Nh, Z¯^h=∑i=1nhZih/nh.Letρ1h2=∑i=1NhIYi≤Y−ℱY2/N−1, ρ2h2=∑i=1NhIXi≤X−ℱX2/N−1, ρ3h2=∑i=1NhZi−Z¯2/N−1, CFy=ρ1/ℱy, CFx=ρ2/ℱx, Crx=ρ3/Z¯, ρ12h=σ12/σ1σ2, ρ13h=σ13/σ1σ3, ρ23h=σ23/σ2σ3, ρ12h=∑i=1NhIYi≤Y−ℱyIXi≤X−ℱx/N−1, ρ13h=∑i=1NhIYi≤y−ℱyXi−X¯/N−1, and ρ23h=∑i=1NhIXi≤x−ℱxZi−Z¯/N−1. Similarly, let R1.23h2=ρ12h2+ρ13h2−2ρ12hρ13hρ23h/1−ρ23h2 be the coefficient of multiple determination of IY≤Y on IX≤X and Z.To find the properties of the existing and proposed estimators ofℱy, we consider the following relative error terms under stratified random sampling.Lete1=ℱ∧y−ℱy/ℱy, e2=ℱ∧stx−ℱx/ℱx, and e3=Z¯^st−Z¯/Z¯, such that Eei=0 for i=1,2,3, where E⋅ is the mathematical expectation of ⋅.Letφrstu=Ee1re2se3t, where r,s,t=1,2,3. Here,(1)Ee12=∑h=1LWh2λh2CFyh2=φ200,Ee22=∑h=1LWh2λh2CFxh2=φ020,Ee32=∑h=1LWh2λh2Czh2=φ002,Ee1e2=∑h=1LWh2λh2ρ12hCFyhCFxh=φ110,Ee1e3=∑h=1LWh2λh2ρ13hCFyhCzh=φ101,Ee2e3=∑h=1LWh2λh2ρ23hCFxhCzh=φ011,where λh=1/nh−1/Nh.


## 3. Existing Estimators

In this section, some estimators of finite population mean are adapted for estimating the finite CDF under stratified random sampling.(1)
The traditional unbiased estimator ofℱy is(2)ℱ∧st∗y=1n∑i=1nIYi≤Y.The variance ofℱ∧y is(3)Varℱ∧st∗y=ℱ2yφ200.(2)
Cochran [18] ratio estimator of ℱy is given by(4)ℱ∧stR∗Y=ℱ∧styℱxℱ∧stx.The bias and MSE ofℱ∧stR∗Y are given by(5)Bℱ∧stR∗Y≅ℱYφ020−φ110,MSEℱ∧stR∗Y≅ℱ2Yφ200+φ020−2φ110.(3)
The usual product estimator ofℱy is(6)ℱ∧stP∗Y=ℱ∧styℱ∧stxℱx.The bias and MSE ofℱ∧stP∗Y are given by(7)Bℱ∧stP∗Y=ℱyφ110,MSEℱ∧stP∗Y≅ℱ2yφ200+φ020+2φ110.(4)
Following Bahl and Tuteja [19], exponential estimators of ℱy, respectively, are(8)ℱ∧stBT,R∗Y=ℱ∧styexpℱx−ℱ∧stxℱ∧stx+ℱx,ℱ∧stBT,P∗Y=ℱ∧styexpℱ∧stx−ℱxℱ∧stx+ℱx.The bias and MSE ofℱ∧stBT,R∗Y and ℱ∧stBT,P∗Y are given by(9)Bℱ∧stBT,R∗Y≅ℱy38φ020−12φ110,MSEℱ∧stBT,R∗Y≅ℱ2y44φ200+φ020−4φ110,Bℱ∧stBT,P∗Y≅ℱy12φ110−18φ020,MSEℱ∧stBT,P∗Y≅ℱ2y44φ200+φ020+4φ110.(5)
The regression type estimator ofℱy is given by(10)ℱ∧stReg∗Y=ℱ∧sty+wℱx−ℱ∧stx,wherew is constant. Here, ℱ∧stReg∗Y is an unbiased estimator of ℱ∧sty. The minimum variance of ℱ∧stReg∗Y at the optimum value is wopt=ℱyφ110/ℱxφ020:(11)Varminℱ∧stReg∗Y=ℱ2yφ200φ020−φ1102φ020.We can also write (11)as(12)Varminℱ∧stReg∗Y=ℱ2yφ2001−ρ122.(6)
The usual difference estimator ofℱy is(13)ℱ∧stR,D∗Y=w1ℱ∧sty+w2ℱx−ℱ∧stx,wherew1 and w2 are unknown constants. The bias and MSE of ℱ∧stR,D∗Y are given as(14)Bℱ∧stR,D∗Y=ℱyw1−1,MSEℱ∧stR,D∗Y≅ℱ2y−2w1ℱ2y+w12ℱ2y+w12ℱ2yφ200,−2w1w2ℱyℱxφ110+w22F2xφ020.The optimum values ofw1 and w2, determined by minimizing (14), are(15)w1opt=φ020φ020φ200−φ1102+φ020,w2opt=ℱyφ110ℱxφ200φ020−φ1102+φ020.The minimum MSE ofℱ∧st∗Y at the optimum values of w1 and w2 is(16)MSEminℱ∧stR,D∗Y=ℱ2yφ200φ020−φ1102φ200φ020−φ1102+φ020.Equation (16) may also be written as(17)MSEminℱ∧stR,D∗Y=ℱ2yφ2001−ρ1221+φ2001−ρ122.(7)
Singh et al. [20] generalized ratio type exponential estimator of ℱy as(18)ℱ∧stS∗Y=ℱ∧styexpαℱx−ℱ∧stxαℱx+ℱ∧stx+2β,wherea and b are known constants. The properties of ℱ∧stS∗Y are given as(19)Bℱ∧stS∗Y≅ℱy38Θ2φ020−12Θφ110,MSEℱ∧stS∗Y≅ℱ2y44φ200+Θ2φ020−4Θφ110,whereΘ=αℱx/αℱx+β.(8)
Grover and Kaur [21] generalized class of ratio type exponential estimator of ℱy as(20)ℱ∧stG,K∗Y=w3ℱ∧sty+w4ℱx−ℱ∧stxexpαℱx−ℱ∧xαℱx+ℱ∧x+2β,where w3 and w4 are constants. The bias and MSE of ℱ∧stG,K∗Y, to the first order of approximation, are(21)Bℱ∧stG,K∗Y≅ℱyw3−1+38Θ2w3ℱy+12Θw4ℱxφ020−12Θℱyφ110,MSEℱ∧stG,K∗Y≅w42ℱ2xφ020+w32ℱ2yφ200+2Θw3w4ℱyℱxφ020,−2w3w4ℱyℱxφ110+ℱ2y−2w3ℱ2y+Θw32ℱ2y,+w3ℱ2yφ110−Θw4ℱyℱxφ020−2Θw32ℱ2yφ110,−34Θ2w3ℱ2yφ020+Θ2w32ℱ2yφ020.The optimum values ofw3 and w4, determined by minimizing (21), are given as(22)w3opt=φ020Θ2φ020−88−φ200φ020+φ1102−φ020,w4opt=ℱyΘ3φ0202−Θ2φ020φ110+4Θφ200φ020−4Θφ1102−4Θφ020+8φ1108ℱxφ200φ020−φ1102+φ020.The minimum MSE ofℱ∧stG,K∗Y at the optimum values of w3 and w4 is given as(23)MSEminℱ∧stG,K∗Y≅F2y6464−16Θ2φ020−φ020−8+Θ2φ0202φ0201+φ200−V1102.Here, (23) may be written as(24)MSEminℱ∧stG,K∗Y≅Varminℱ∧stReg∗Y−ℱ2yΘ2φ0202−8φ1102+8φ020φ200264φ02021+φ2001−ρ122,which shows that ℱ∧stG,K∗Y is more precise than ℱ∧stReg∗Y.


## 4. Proposed Family of Estimators

The use of auxiliary variables may enhance the accuracy of an estimator either at the design stage or at the estimation stage. When a correlation exists between the study variable and the auxiliary variable, the order of the auxiliary variable is also correlated to the study variable. Thus, the rank of the auxiliary variable can be treated as a new auxiliary variable, and it is helpful in increasing the efficiency of an estimator. On the lines ofℱ∧R,Dy, ℱ∧Sy, and average of ℱ∧BT,Ry and ℱ∧BT,Py, we proposed a new family of estimator say ℱy:(25)ℱ∧stprop∗Y=12ℱ∧styexpℱx−ℱ∧stxℱ∧stx+ℱx+expℱ∧stx−ℱxℱ∧stx+ℱx+w5ℱx−ℱ∧stx+w6ℱ∧sty+w7Z¯st−Z¯^expαℱx−ℱ∧stxαℱx+ℱ∧stx+2β,where w5, w6, and w7 are unknown constants, α≠0 and β are either real numbers or functions of known population parameters of IX≤x, which may be ρ12, β2st=∑i=1LWhβ2hx (coefficient of kurtosis), CFx=∑i=1LWhCFxh, etc.The estimatorℱ∧stprop∗Y can also be written as(26)ℱ∧stprop∗Y=ℱy1+e01+w6−w5e1−w7e2+18Θ2ℱye121−12Θe1+38Θ2ξ12+⋯.Expressing (26) up to first order of approximation,(27)ℱ∧stprop∗Y−ℱy≅w6ℱy+ℱye0+w6ℱye0−12Θℱye1+12Θ2ℱye12−12Θℱye0e1−w5e1+12Θw5e12−12Θw6ℱye1+38Θ2w6ℱye12−12Θw6ℱye0e1−w7e2+12Θw7e1e2.The properties ofℱ∧stprop∗Y are given as(28)Bℱ∧stprop∗Y≅12Θ2ℱyφ020−12Θℱyφ110+12w5Θφ020+w6ℱy+38w6Θ2ℱyφ020−12w6Θℱyφ110+12w7Θφ011,MSEℱ∧stprop∗Y≅−Θℱ2yφ110+32w6Θ2ℱ2yφ020+w62Θ2ℱ2yφ020+w5Θℱyφ020−2w6w7ℱyφ101+14Θ2ℱ2yφ020+2w6ℱ2yφ200+w52φ020−2w62Θℱ2yφ110+ℱ2yφ200+w62ℱ2y+w7Θℱyφ011+2w5w6Θℱyφ020−2w5w6ℱyφ110+2w5w7φ011−3w6Θℱ2yφ110−2w5ℱyφ110−2w7ℱyφ101+w62ℱ2yφ200+m72φ002+2w6w7Θℱyφ011.The optimal values ofw5, w6, and w7 are given by(29)w5opt=ℱyΘ3φ0203/2ρ232−1+φ2001/2−4+Θ2φ020ρ12−ρ23ρ13+2Θφ0201/2ρ232−1−1+φ2001−R1.2328φ0201/2ρ232−1−1+φ2001−R1.232,w6opt=4φ220−2ρ1122ρ113ρ2223+ρ11222+ρ1132+ρ2232−1+Θ2φ022ρ2232−1−4ρ2232−11+φ2201−R1122,w7opt=ℱyφ2201/2Θ2φ022−ρ13+ρ122ρ223+4ρ13−ρ122ρ223−4φ0021/2ρ2232−11+φ2201−R1.222.The minimum MSE ofℱ∧stprop∗Y at the optimum values of w5, w6, and w7 is(30)MSEminℱ∧stprop∗Y≅ℱ2y16φ2001−R1.232−Θ4φ0202−8Θ2φ020φ2001−R1.232161+φ2001−R1.232,where R1.232=φ1102φ002+φ1012φ020−2φ101φ110φ011/φ200φ020φ002−φ0112.Table 18 shows some members of the proposed family of estimators for different choices ofαand β (Table 1).Table 1
Some members of the proposed and existing estimators.
αβℱ∧stG,K∗Yℱ∧stprop∗Y1CFxℱ∧stG,K∗1Yℱ∧stprop∗1Y1β2stℱ∧stG,K∗2Yℱ∧stprop∗2Yβ2stCFxℱ∧stG,K∗3Yℱ∧stprop∗3tYCFxβ2stℱ∧stG,K∗4Yℱ∧stprop∗4Y1ρ12ℱ∧stG,K∗5Yℱ∧stprop∗5YCFxρ12ℱ∧stG,K∗6Yℱ∧stprop∗6Yρ12CFxℱ∧stG,K∗7Yℱ∧stprop∗7Yβ2stρ12ℱ∧stG,K∗8Yℱ∧stprop∗8Yρ12β2stℱ∧stG,K∗9Yℱ∧stprop∗9Y1Nℱxℱ∧stG,K∗10Yℱ∧stprop∗10Y


## 5. Empirical Study

To show the dominance of the proposed estimators over the existing estimators, we conduct a numerical study to investigate the performances of the existing and proposed CDF estimators. For this purpose, three populations are considered. The datasets are given inTables2–4. We use the following expression to find the percentage relative efficiency (PRE).(31)PREℱ∧iY,ℱ∧sty=Varℱ∧styMSEminℱ∧iY×100,where i  =  stR, stP, stBT,R, stBT,P, stReg, stR,D, stG,K (Tables 2–4).Population I (source: Koyuncu and Kadilar [22]):Y: number of teachers.X: number of teachers in 2007 for 923 districts.Population II (source: Koyuncu and Kadilar [22]):Y: number of teachers.X: number of classes in 2007 for 923 districts.Population III (source: Kadilar and Cingi [23]):Y: apple production in 1999.X: number of apples in 1999.Table 2
Summary statistics for population I.
hNhnhWhλhZ¯h1127310.13750.0244164.002117210.12680.0390759.003103290.11160.0247752.004170380.18420.0204385.505205220.22210.04058101.006201390.21780.02067101.00ℱyhY¯hQ1hyY˜hQ3hyX˜h0.606300.165350.354330.590550.370080.769230.205130.418800.760680.470090.679610.213590.427180.660190.427180.823530.376470.576470.800000.588240.848780.307320.614630.839020.614630.796020.208960.502490.761190.45274ρ12hY¯hQ1hyY˜hQ3hyX˜h0.904320.891440.898290.05592− 0.811440.873860.927480.866570.88428− 0.838170.830290.882020.960320.82525− 0.851900.834520.887260.927730.91233− 0.849080.894060.851760.876430.92659− 0.828510.868150.861830.845060.97263− 0.83431ρ23hY¯hQ1hyY˜hQ3hyX¯h− 0.85632− 0.66705− 0.83632− 0.10979− 202.287− 0.77947− 0.72980− 0.86453− 0.74316− 118.897− 0.83105− 0.68518− 0.85685− 0.65670− 185.585− 0.73474− 0.82455− 0.85247− 0.76926− 146.296− 0.65175− 0.80264− 0.84299− 0.82344− 79.540− 0.75371− 0.68496− 0.86217− 0.68993− 117.590σ13hY¯hQ1hyY˜hQ3hyX¯h− 15.194− 8.774− 14.341− 15.321− 15.643− 10.371− 9.547− 14.086− 10.569− 11.948− 10.995− 8.569− 12.652− 11.397− 11.971− 12.328− 19.849− 20.71− 11.586− 15.385− 11.186− 21.225− 21.978− 11.821− 14.583− 16.278− 16.485− 24.325− 18.300−19.125Table 3
Summary statistics for population II.
hNhnhWhλhZ¯h1127310.13750.0244164.002117210.12680.0390759.003103290.11160.0247752.004170380.18420.0204385.505205220.22210.04058101.006201390.21780.02067101.00ℱyhY¯hQ1hyY˜hQ3hyX¯hX˜h0.606300.165350.354330.590550.606300.377950.769230.205130.418800.760680.794870.487180.679610.213590.427180.660190.718450.466020.823530.376470.576470.800000.835290.611760.848780.307320.614630.839020.873170.653660.796020.796020.502490.761190.721390.35323ρ12hY¯hQ1hyY˜hQ3hyX¯hX˜h0.967010.942950.916420.95114− 0.84186− 0.815460.927480.905690.870930.92938− 0.72738− 0.843790.865490.833460.924400.84718− 0.79496− 0.848910.959260.848850.880540.94450− 0.65748− 0.841660.862010.098910.877150.88864− 0.61847− 0.830520.814560.236450.714540.83866− 0.68348− 0.81252ρ23hY¯hQ1hyY˜hQ3hyX¯hX˜h− 0.84625− 0.64348−0.83986− 0.849110.232660.21422− 0.69942− 0.73903− 0.86577− 0.709990.159150.21662− 0.77904− 0.73226− 0.86406− 0.801580.183420.23034− 0.64246− 0.82774− 0.84413− 0.668820.136440.21330− 0.57640− 0.14737− 0.82413− 0.603630.103280.20411− 0.77651− 0.66415− 0.82789− 0.799970.147890.17162σ13hY¯hQ1hyY˜hQ3hyX¯hX˜h− 15.20− 8.79− 14.41− 15.41− 15.28− 15.05− 10.44− 9.47− 14.18− 10.71− 9.62− 14.74− 11.14− 8.55− 12.61− 11.48− 10.52− 12.94− 12.37− 19.66− 20.53− 11.65− 11.76− 20.31− 11.18− 21.46− 24.04− 11.82− 11.41− 21.32− 16.06− 16.06− 21.69− 18.11− 20.30− 21.08Table 4
Summary statistics for population III.
hNhnhWhλhZ¯h110690.124120.1016851.502106170.124120.0493951.50394380.110070.0156847.504171670.200230.0090886.00520470.238880.13796102.50617320.202580.4942287.00ℱyhY¯hQ1hyY˜hQ3hyX¯hX˜h0.584910.905660.198110.726420.547170.188680.518870.896230.292450.660380.566040.301890.329790.691490.159570.457450.340430.191490.368420.853800.181290.497080.380120.181290.465690.946080.132350.642160.431370.132350.705200.971100.508670.809250.722540.49133ρ12hY¯hQ1hyY˜hQ3hyX¯hX˜h0.772170.650520.607240.75509− 0.76652− 0.54922.833090.626550.751790.70235− 0.82847− 0.73476.785390.850400.895370.66305− 0.75098− 0.63060.775540.654840.724190.64776− 0.75350− 0.60626.675020.586460.658510.62953− 0.72183− 0.50394.731920.665000.780880.63464− 0.72909− 0.82252ρ23hY¯hQ1hyY˜hQ3hyX¯hX˜h− 0.45227− 0.63562− 0.67771− 0.71397X¯hX˜h− 0.48161− 0.67774− 0.79523− 0.745300.191200.09560− 0.30868− 0.80006− 0.68157− 0.837390.208270.15849− 0.19359− 0.76273− 0.66730− 0.823400.176850.13040− 0.41292− 0.60487− 0.58696− 0.768630.182660.10812− 0.61019− 0.34130− 0.86593− 0.487420.167580.07599ρ24hY¯hQ1hyY˜hQ3hyX¯hX˜h− 11.67− 4.34− 6.76− 9.92− 11.26− 8.19− 12.79− 4.74− 10.32− 11.23− 11.14− 11.28− 9.68− 9.81− 6.33− 11.10− 10.67− 7.35− 18.05− 10.46− 11.60− 19.48− 20.26− 12.76− 21.31− 5.11− 10.11− 20.24− 25.14− 11.77− 16.70− 2.37− 20.65− 11.05− 17.44− 21.74From the numerical outcomes, given in Tables5–8, it is further noticed that the proposed family of estimators with different values of α and β performs more efficiently than existing estimators.Table 5
Efficiency result under populations I, II, and III whenx,y=X¯,Y¯,.
EstimatorsPopulation IPopulation IIPopulation IIIℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗1Yℱ∧stprop∗1Y423.81429.58461.93481.67166.25166.74ℱ∧stG,K∗2Yℱ∧stprop∗2Y423.76429.53461.88481.62166.19166.68ℱ∧stG,K∗3Yℱ∧stprop∗3Y423.87429.64462.00481.75166.32166.82ℱ∧stG,K∗4Yℱ∧stprop∗4Y423.76429.53461.88481.62166.19166.68ℱ∧stG,K∗5Yℱ∧stprop∗5Y423.79429.63461.91481.62166.23166.68ℱ∧stG,K∗6Yℱ∧stprop∗6Y423.77429.55461.89481.63166.20166.70ℱ∧stG,K∗7Yℱ∧stprop∗7Y423.80429.58461.93481.67166.23166.72ℱ∧stG,K∗8Yℱ∧stprop∗8Y423.84429.62461.97481.72166.31166.81ℱ∧stG,K∗9Yℱ∧stprop∗9Y423.76429.53461.88481.62166.19166.68ℱ∧stG,K∗10Yℱ∧stprop∗10Y423.75429.52461.88481.61166.19166.68ℱ∧sty100.00100.00100.00ℱ∧stR∗y337.47439.0181.55ℱ∧stP∗Y23.7526.7620.79ℱ∧stBT,R∗Y164.30271.46298.66ℱ∧stBT,P∗Y41.6647.1143.63ℱ∧RegY423.61461.74166.03ℱ∧stR,D∗Y423.75461.88166.19Table 6
Efficiency result using populations I, II, and III whenx,y=Q1x,Q1y.
EstimatorsPopulation IPopulation IIPopulation IIIℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗1Yℱ∧stprop∗1Y417.88447.00148.81261.62244.30271.46ℱ∧stG,K∗2Yℱ∧stprop∗2Y417.87446.99148.80261.61244.25271.41ℱ∧stG,K∗3Yℱ∧stprop∗3Y417.97447.10148.84261.68244.68271.88ℱ∧stG,K∗4Yℱ∧stprop∗4Y417.89447.01148.81261.62244.35271.51ℱ∧stG,K∗5Yℱ∧stprop∗5Y417.93447.01148.83261.62244.63271.51ℱ∧stG,K∗6Yℱ∧stprop∗6Y418.04447.18148.88261.75245.25272.52ℱ∧stG,K∗7Yℱ∧stprop∗7Y417.87446.99148.80261.61244.26271.42ℱ∧stG,K∗8Yℱ∧stprop∗8Y418.16447.30148.94261.85245.82273.15ℱ∧stG,K∗9Yℱ∧stprop∗9Y417.86446.98148.80261.61244.24271.39ℱ∧stG,K∗10Yℱ∧stprop∗10Y417.85446.97148.80261.61244.21271.36ℱ∧sty100.00100.00100.00ℱ∧stR∗Y382.69120.31205.34ℱ∧stP∗Y26.2633.3328.47ℱ∧stBT,R∗Y205.34145.64268.78ℱ∧stBT,P∗Y28.4756.4546.62ℱ∧RegY416.36147.31233.77ℱ∧stR,D∗Y417.85148.80244.21Table 7
Efficiency result using populations I, II, and III whenx,y=X˜,Y˜.
EstimatorsPopulation IPopulation IIPopulation IIIℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗Yℱ∧stprop∗Yℱ∧stG,K∗1Yℱ∧stprop∗1Y454.79502.91364.28428.83211.54241.14ℱ∧stG,K∗2Yℱ∧stprop∗2Y454.78502.90364.27428.82211.49241.09ℱ∧stG,K∗3Yℱ∧stprop∗3Y454.80502.92364.29428.85211.61241.22ℱ∧stG,K∗4Yℱ∧stprop∗4Y454.78502.90364.27428.82211.50241.10ℱ∧stG,K∗5Yℱ∧stprop∗5Y454.80502.90364.29428.82211.62241.10ℱ∧stG,K∗6Yℱ∧stprop∗6Y454.81502.92364.29428.85211.63241.24ℱ∧stG,K∗7Yℱ∧stprop∗7Y454.78502.90364.27428.82211.49241.08ℱ∧stG,K∗8Yℱ∧stprop∗8Y454.82502.93364.31428.87211.71241.34ℱ∧stG,K∗9Yℱ∧stprop∗9Y454.77502.89364.26428.81211.46241.05ℱ∧stG,K∗10Yℱ∧stprop∗10Y454.73502.84364.23428.78211.39240.97ℱ∧sty100.00100.00100.00ℱ∧stR∗Y427.32341.19181.83ℱ∧stP∗Y26.5327.3829.41ℱ∧stBT,R∗Y188.24247.93272.79ℱ∧stBT,P∗Y51.1248.0146.85ℱ∧RegY454.24361.74208.63ℱ∧stR,D∗Y454.73364.23211.39Table 8
Efficiency results using populations I, II, and III whenx,y=Q3x,Q3y.
EstimatorsPopulation IPopulation IIPopulation IIIℱ∧stG,K∗Yℱ∧stprop2∗Yℱ∧stG,K∗Yℱ∧stprop2∗Yℱ∧stG,K∗Yℱ∧stprop2∗Yℱ∧stG,K∗1Yℱ∧stprop∗1Y292.77340.09495.49517.85170.13220.22ℱ∧stG,K∗2Yℱ∧stprop∗2Y292.74340.06495.43517.79170.05220.11ℱ∧stG,K∗3Yℱ∧stprop∗3Y292.81340.14495.56517.93170.24220.36ℱ∧stG,K∗4Yℱ∧stprop∗4Y292.74340.05495.42517.78170.05220.11ℱ∧stG,K∗5Yℱ∧stprop∗5Y292.76340.05495.46517.78170.12220.11ℱ∧stG,K∗6Yℱ∧stprop∗6Y292.75340.07495.44517.80170.09220.16ℱ∧stG,K∗7Yℱ∧stprop∗7Y292.77340.09495.48517.84170.10220.18ℱ∧stG,K∗8Yℱ∧stprop∗8Y292.80340.12495.53517.89170.23220.18ℱ∧stG,K∗9Yℱ∧stprop∗9Y292.74340.055495.43517.79170.05220.11ℱ^stprop∗1Yℱ∧stprop∗10Y292.74340.05495.42517.78170.04220.10ℱ∧sty100.00100.00100.00ℱ∧stR∗Y267.68471.2637.09ℱ∧stP∗Y27.8926.6712.46ℱ∧stBT,R∗Y202.60277.40226.27ℱ∧stBT,P∗Y49.8446.9548.83ℱ∧RegY292.58495.27168.67ℱ∧stR,D∗Y292.74495.42170.04


## 6. Simulation Study

We have generated three populations of size 1,000 from multivariate normal distribution with different covariance matrices. All the populations have different correlations. Population I is negatively correlated, population II is positively correlated, and population III has strong positive correlation between X and Y variables. The population means and covariance matrices are given below:Population I:(32)μ1=55,∑1=4−9.0−9.064.N1=500 and N2=500.ρXY=−0.590220, ρ1XY=−0.597259, ρ2XY=−0.538706.Population II:(33)μ2=55,∑2=49.59.563,N1=500,N2=500.ρXY=0.612254, ρ1XY=0.638109, ρ2XY=0.633022.Population III:(34)μ3=55,∑3=24610.N1=500 and N2=500.(35)ρXY=0.902645,ρ1XY=0.910705,ρ2XY=0.901442.Relative efficiency (PRE) is calculated as(36)PREℱ∧iY,ℱ∧y=Varℱ∧yMSEℱ∧iY×100,where i  = stR, stP, stBT, R, stBT, P, stReg, stR, D, stGK.The results of MSE and PRE are given in Table 26 and 27. Here we can only point out the best results of MSEs and PREs in these tables whenΘ=αℱx/αℱx+β if α=1,β=CFxst (Tables 9 and 10).Table 9
MSEs of finite population distribution function estimators using simulation.
EstimatorPopulation IPopulation IIPopulation IIIℱ∧y0.00100160.00100060.0010013ℱ∧stR∗Y0.0029000.00121740.0005337ℱ∧stP∗Y0.00114520.00291630.0034462ℱ∧stBT,R∗Y0.00169550.00084240.000520ℱ∧stBT,P∗Y0.00169550.00084240.000520ℱ∧stReg∗Y0.00081310.00083140.000465ℱ∧stR,D∗Y0.00081050.00082880.0004641ℱ∧stG,K∗Y0.0008100.00082870.0004641ℱ∧stprop∗Y0.00076780.00075830.0004156Table 10
PREs of finite population distribution function using simulation.
EstimatorPopulation IPopulation IIPopulation IIIℱ∧y100100100ℱ∧stR∗Y34.5386982.19431187.60983ℱ∧stP∗Y87.4602534.3113929.05563ℱ∧stBT,R∗Y59.07314118.77536192.42847ℱ∧st∗Y59.07314118.77536192.42847ℱ∧stReg∗Y123.1813120.3492215.3049ℱ∧stR,D∗Y123.5772120.7309215.7119ℱ∧stG,K∗Y123.5916120.7450215.7371ℱ∧stprop∗Y130.4431131.9435240.9002


## 7. Conclusion

In this paper, we have proposed a new family of estimators to estimate the finite population distribution function (DF). Using simulation studies and actual datasets, it is observed that the proposed class of estimators gives better results than the existing estimates. Therefore, we recommend the use of the proposed estimators for future study. Based on the real datasets and simulation results, it can be seen that the proposed estimators perform better than all existing estimators. The percentage relative efficiency shows that the proposed family of estimators in stratified random sampling gives the best result whenX and Y variables have strong positive correlation [24–29].



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*Source: 3263022-2022-10-05.xml*