# Estimation of Population Median under Robust Measures of an Auxiliary Variable

Authors: Muhammad Irfan; Maria Javed; Sandile C. Shongwe; Muhammad Zohaib; Sajjad Haider Bhatti
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a generalized class of estimators for the estimation of population median are proposed under simple random sampling without replacement (SRSWOR) through robust measures of the auxiliary variable. Three robust measures, decile mean, Hodges–Lehmann estimator, and trimean of an auxiliary variable, are used. Mathematical properties of the proposed estimators such as bias, mean squared error (MSE), and minimum MSE are derived up to first order of approximation. We considered various real-life datasets and a simulation study to check the potentiality of the proposed estimators over the competitors. Robustness is also examined through a real dataset. Based on the fascinating results, the researchers are encouraged to use the proposed estimators for population median under SRSWOR.


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## Body

## 1. Introduction

Extensive work has been done on the estimation of the population mean, proportion, variance, regression coefficient, and so forth; but very little attention has been made to propose the efficient estimators of the median. In many situations, researchers are often interested in dealing with variables such as income, expenditure, taxes, consumption, and production; and the latter variables have highly skewed distributions. In such situations, the median is considerably a more appropriate measure of location than the mean. The problem of estimation of median under simple random sampling scheme has been discussed by Gross [1], Sedransk and Meyer [2], and Smith and Sedransk [3]. Kuk and Mak [4] were the first authors to investigate the estimation of the median using auxiliary information. After Kuk and Mak’s [4] estimator, Singh et al. [5], Aladag and Cingi [6], Solanki and Singh [7], Shabbir and Gupta [8], Baig et al. [9], and Shabbir et al. [10] have developed different estimators for estimating finite population median based on the known conventional measures of the auxiliary variable under different sampling schemes. A brief explanation of Kuk and Mak’s [4] estimator is as described as follows.LetYandX be the study and the auxiliary variables selected from a finite population Θ=Θ1,Θ2,Θ3,…,ΘNof size “N” under simple random sampling without replacement (SRSWOR) subject to the constraint n<N. Further let YiandXi,i=1,2,…,N&yiandxi,i=1,2,…,n be the values of the ith units of the population and sample, respectively. Let MYandMX be the population median of the study and auxiliary variables with the probability density functions given by fYMYandfXMX, respectively. We further assume that fYMYandfXMX are positive.Suppose thaty1≤y2≤y3≤⋯≤yn are the y values of sample units in ascending order; furthermore, let s be the integer such that Ys≤MY≤Ys+1 and p=s/n are the proportion of Y values in the sample which are less than or equal to MY. Kuk and Mak [4] considered a two-way classification pij as given in Table 1.Table 1
Matrix of proportions.
Y≤MYY>MYTotalX≤MXp11p21p⋅1X>MXp12p22p⋅2Totalp1⋅p2⋅1Suppose thatM^Y and M^X are the sample estimators of MYandMX; then the correlation coefficient between M^Y,M^X is ρM^Y,M^Xorρcorρ=4p11−1 ranging from −1 to +1 as p11 increases from 0 to 0.5, wherep11 is the proportion of units in the population with X≤MXandY≤MY. Gross [1] proved that M^Y is consistent and asymptotically normally distributed with mean MY and variance(1)M^Y=1−f4n−1fYMY2,where f=n/N is the sampling fraction.Efficiency of the ratio, product, and regression type estimators are ambiguous in the presence of the extreme values/outlier(s) in the dataset. In our present study, the problem under consideration is to estimate the median for finite population and suggest some generalized classes of estimators by utilizing known robust measures of an auxiliary variable under SRSWOR. The novelty of this work is as follows:(i)
Robust measures (i.e., decile mean, Hodges–Lehmann estimator, and trimean) of an auxiliary variable are utilized for the first time to investigate the progressive estimation of the population median(ii)
A variety of estimators can be generated through the proposed generalized estimator(iii)
Robustness study is examined to check the performance of the proposed generalized estimator in the presence of outlierThe following relative error terms and notations are used to obtain the mathematical properties such as bias, mean squared error (MSE), and minimum MSE of various estimators:e0=M^Y−MY/MY,e1=M^X−MX/MX such that Ee0=Ee1=0.(2)Ee02=λCMY2,Ee12=λCMX2,Eeoe1=λCMYX=ρCMYCMX,where(3)CMY=1MYfYMYbe the population coefficient of variation ofY,CMX=1MXfXMXbe the population coefficient of variation ofX,λ=141n−1Nbe the finite population correctionf.p.cfactor.The rest of the article is organized in the following way. Section2 gives comprehensive details of existing estimators for the population median. Section 3 proposes generalized classes of estimators for estimating population median using robust measures of an auxiliary variable. Bias, mean squared error (MSE), and minimum MSE of generalized classes of estimators are derived up to the first degree of approximation in the same section. Four real-life datasets and a simulation study are performed in Section 4 to check the potential of the new estimators as compared to the existing ones. Robustness of the proposed estimators is evaluated by carrying out a real-life dataset in Section 5. Section 6 contains the concluding remarks and some recommendations.


## 2. Existing Median Estimators

The major drawback of all the suggested estimators for estimating population median is that they are based on the usual conventional measures of an auxiliary variable. In this section, we discuss the usual and well-known estimators for estimating population median under SRSWOR as suggested by different authors.Kuk and Mak [4] suggested a ratio-type estimator by assuming the known median of the X variable.(4)M^R=M^YMXM^X.The expression for mean square error ofM^R estimator is given as(5)MSEM^R≅λMY2CMY2+CMX2−2CMYX.The exponential ratio-type estimator for estimating median is given as(6)M^EX=M^YexpMX−M^XMX+M^X.The MSE ofM^EX up to the first degree of approximation is given by(7)MSEM^EX≅λMY2CMY2+14CMX2−CMYX.Singh [11] developed an unbiased difference estimator which is given by(8)M^D=M^Y+dMX−M^X,where d is an unknown constant whose value needs to be determined.Minimum MSE ofM^D up to the first degree of approximation is as follows:(9)MSEM^Dmin≅λMY2CMY21−ρ2.Remark 1.
The MSE ofM^D is always smaller than the MSE of M^Y,M^RandM^EX if ρ>0,CMX−ρCMX2>0and0.5CMX−ρCMX2>0, respectively.
Rao [12] and Gupta et al. [13], respectively, suggested three difference types of estimators for estimating median as(10)M^D1=d1M^Y+d2MX−M^X,M^D2=d3M^Y+d4MX−M^XMXM^X,M^D3=d5M^Y+d6MX−M^XMX−M^XMX+M^X,where d1,d2,d3,d4,d5,andd6 are unknown constants.
The minimum MSE ofM^D1 at optimum values of d1opt=1/1+λCMY21−ρ2andd2opt=MY/MXρCMY/CMX/1+λCMY21−ρ2 is given by(11)MSEM^D1min≅λMY2CMY21−ρ21+λCMY21−ρ2.
The minimum MSE ofM^D2 at optimum values of d3opt=1/1−λCMX2+λCMY21−ρ2andd4opt=MY/MX1+d3optρCMY/CMY−2 is given by(12)MSEM^D2min≅MY21−λCMX2λCMY21−ρ21−λCMX2+λCMY21−ρ2.
The minimum MSE ofM^D3 at optimum values of d5opt=1−λCMX2/5/1+λCMY21−ρ2andd6opt=MY/MX1/2+d5optρCMY/CMX−1 is given by(13)MSEM^D3min≅MY2λCMY21−ρ2−1/64λ2CMX4−1/4λ2CMY2CMX21−ρ21−λCMX2+λCMY21−ρ2.
Shabbir and Gupta [8] suggested a generalized difference type estimator for the estimation of median as(14)M^SG=d7M^Y+d8MX−M^XaMX+baM^X+bα1expα2aMX−M^Xaγ−1MX+M^X+2b,where d7andd8 are unknown constants whose values need to be determined, aandb are the known population parameters, and α1,α2andγ are the scalar quantities.Remark 2.
By substitution of the scalar quantities asα1=b=0,α2=γ=a=1, equation (14) becomes(15)M^SG=d7M^Y+d8MX−M^XexpMXM^X−1.
The minimum MSE ofM^SG at optimum values of d7opt=1−1/2λCMX2/1+λCMY21−ρ2andd8opt=MY/MX1+d7optρCMY/CMX−2 is given by(16)MSEM^SGmin≅MY2λCMY21−ρ2−1/4λ2CMX4−λ2CMY2CMX21−ρ21+λCMY21−ρ2.


## 3. Proposed Generalized Estimator

One eminent disadvantage of existing estimators/class of estimators is that they are typically based on conventional measures. Efficiency of the estimators is uncertain in the occurrence of the extreme values in the dataset. In this section, we define a generalized class of estimators for the estimation of population median using robust measures of an auxiliary variable with the linear combination of nonconventional measures: quartile deviation, midrange, interquartile range, and quartile average. We included three robust measures: decile mean suggested by Rana et al. [14], Hodges–Lehmann estimator suggested by Hettmansperger and McKean [15], and the trimean suggested by Wang et al. [16]. For more details of these robust measures, see the works of Irfan et al. [17, 18].A generalized estimator for the estimation of population median is(17)Tid=M^Ym1ψM^X+δψMX+δα3expMX−M^XMX+M^X+m2ψMX+δψM^X+δα4,where m1andm2 are suitably chosen constants, and αii=3and4 takes on the values 1,−1,2,−2 for designing new estimators. Note that ψ and δ may be any constant values or functions of the known robust measures as well as nonconventional measures associated with X variable.Remark 3.
Robust measures related toX are the following:Trimean:TM=Q1+2Q2+Q3/4Hodges–Lehmann:HL=Medianxj+xk/2,1≤j≤k≤NDecile mean:DM=∑i=19Di/9Remark 4.
The nonconventional measures (i.e., interquartile range, midrange, quartile average, and quartile deviation) of an auxiliary variable can be defined as follows:Interquartile range:QR=Q3−Q1Midrange:MR=x1+xN/2Quartile average:QA=Q3+Q1/2Quartile deviation:QD=Q3−Q1/2Remark 5.
By putting different values ofαii=3and4 in equation (17), we get the following families of estimators:(i)
Putα3=1andα4=2; proposed family of estimators reduces to(18)Tid⊖=M^Ym1ψM^X+δψMX+δexpMX−M^XMX+M^X+m2ψMX+δψM^X+δ2.(ii)
Putα3=−1andα4=−1; proposed family of estimators reduces to(19)Tid⊕=M^Ym1ψMX+δψM^X+δexpMX−M^XMX+M^X+m2ψM^X+δψMX+δ.(iii)
Putα3=−1andα4=−2; proposed family of estimators reduces to(20)Tid⊗=M^Ym1ψMX+δψM^X+δexpMX−M^XMX+M^X+m2ψM^X+δψMX+δ2.(iv)
Putα3=2andα4=2; proposed family of estimators reduces to(21)Tid⊛=M^Ym1ψM^X+δψMX+δ2expMX−M^XMX+M^X+m2ψMX+δψM^X+δ2.(v)
Putα3=−2andα4=−1; proposed family of estimators reduces to(22)Tid⊚=M^Ym1ψMX+δψM^X+δ2expMX−M^XMX+M^X+m2ψM^X+δψMX+δ.Remark 6.
When we put robust measures of auxiliary variable with the linear combination of median, quartile deviation, midrange, interquartile range, and quartile average of an auxiliary variable in equation (17), we obtain different series of estimators such as Tid⊖,Tid⊕,Tid⊗,Tid⊛andTid⊚. Some members of the class of estimator Tid⊚ are presented in Table 2. Placing the same values of ψ and δ in Tid⊖,Tid⊕,Tid⊗andTid⊛, we obtain a number of estimators.Table 2
Some members ofTid⊚.
ψδiEstimatorTid⊚=M^Ym1ψMX+δ/ψM^X+δ2expMX−M^X/MX+M^X+m2ψM^X+δ/ψMX+δQATM1T1d⊚=M^Ym1QAMX+TM/QAMX+TM2expMX−M^X/MX+M^X+m2QAM^X+TM/QAMX+TMMRTM2T2d⊚=M^Ym1MRMX+TM/MRM^X+TM2expMX−M^X/MX+M^X+m2MRM^X+TM/MRMX+TMHLTM3T3d⊚=M^Ym1HLMX+TM/HLM^X+TM2expMX−M^X/MX+M^X+m2HLM^X+TM/HLMX+TMTMHL4T4d⊚=M^Ym1TMMX+HL/TMM^X+HL2expMX−M^X/MX+M^X+m2T⋅MM^X+HL/T⋅MMX+HLQAHL5T5d⊚=M^Ym1QAMX+HL/QAMX+HL2expMX−M^X/MX+M^X+m2QAM^X+HL/QAMX+HLHLQR6T6d⊚=M^Ym1HLMX+QR/HLM^X+QR2expMX−M^X/MX+M^X+m2HLM^X+QR/HLMX+QRMRHL7T7d⊚=M^Ym1MRMX+HL/MRM^X+HL2expMX−M^X/MX+M^X+m2MRM^X+HL/MRMX+HLHLDM8T8d⊚=M^Ym1HLMX+DM/HLM^X+DM2expMX−M^X/MX+M^X+m2HLM^X+DM/HLMX+DMDMTM9T9d⊚=M^Ym1DMMX+TM/DMM^X+TM2expMX−M^X/MX+M^X+m2DMM^X+TM/DMMX+TMMRDM10T10d⊚=M^Ym1MRMX+DM/MRM^X+DM2expMX−M^X/MX+M^X+m2MRM^X+DM/MRMX+DMQDDM11T11d⊚=M^Ym1QDMX+DM/QDM^X+DM2expMX−M^X/MX+M^X+m2QDM^X+DM/QDMX+DMDMQA12T12d⊚=M^Ym1DMMX+QA/DMM^X+QA2expMX−M^X/MX+M^X+m2DMM^X+QA/DMMX+QARemark 7.
Putting appropriate constants or known conventional parameters of the auxiliary variable in place ofψ and δ in equation (17), we can get many optimal estimators. Conventional parameters associated with auxiliary variable X are variance, standard deviation, coefficient of variation, coefficient of skewness, coefficient of kurtosis, coefficient of correlation, and so forth.


### 3.1. Bias, MSE, and Minimum MSE ofTid

The suggested generalized class of estimatorsTid in terms of e0ande1 is expressed as follows:(23)Tid=MY1+e0m11+ψMXe1ψMX+δα3exp−e121+e12−1+m21+ψMXe1ψMX+δ−α4.After some simplification of equation (23), we have(24)Tid=MY1+e0m11+ϑe1α3+12α3α3−1ϑe12exp−e12+e124+m21−ϑe1α4+12α4α4+1ϑe12,where ϑ=ψMX/ψMX+δ.SubtractingMY from both sides of equation (24), we get(25)Tid−MY=MYm11−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e1+m21−ϑe1α4+12α4α4+1ϑ2e12+e0−α4ϑe0e1−1.The bias of the proposed estimators,Tid, is defined as(26)BiasTid≅ETid−MY.Taking expectations on both sides of equation (25), we get the bias of generalized class of estimators Tid:(27)BiasTid=MYm11+λCMX2234−α3ϑ+α3α3−1ϑ2+λρCMYCMXα3ϑ−12+m21+α4α4+1ϑ2λCMX22−α4ϑλρCMYCMX−1.The MSE of the proposed estimators,Tid, is defined as(28)MSETid≅ETid−MY2.Squaring both sides of equation (25), we have(29)Tid−MY2=MY2m121−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e12+m221−ϑe1α4+α4α4+1ϑ2e122+e0−α4ϑe0e12+1+2m1m21−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e11−ϑe1α4+α4α4+1ϑ2e122+e0−α4ϑe0e1−2m1−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e1−2m21−ϑe1α4+α4α4+1ϑ2e122+e0−α4ϑe0e1.Taking expectations on both sides of equation (29), we get the MSE of proposed estimators up to the first order of approximation as(30)MSETid=MY21+m12A1+m22A2+2m1m2A3−2m1A4−2m2A5,where(31)A1=1+λCMY2+CMX21+2α32ϑ2−α3ϑ2−2α3ϑ−2pCMYCMX1−2α3ϑ,A2=1+λCMY2+ϑ2CMX22α42+α4−4α4ϑpCMXCMY,A3=1+λCMY2−pCMYCMX2ϑα4−α3+1−CMX2α3α4ϑ2−38+α3ϑ2−α4ϑ2−α4α4+1ϑ22−12α3α3−1ϑ2,A4=1+λpCMYCMXα3ϑ−12+λCMX2234−α3ϑ+α3α3−1ϑ2,A5=1+12α4α4+1ϑ2λCMX2−α4ϑλpCMYCMX.Partially differentiating equation (30) with respect to m1andm2 and equating them to zero, we get the optimal values of m1andm2 as follows:(32)m1opt=A2A4−A3A5A1A2−A32,m2opt=A1A5−A3A4A1A2−A32.Placing these optimal values in equation (30), we obtained the minimum MSE as given by(33)MSETidmin≅MY21−A2A42+A1A52−2A3A4A5A1A2−A32.


## 3.1. Bias, MSE, and Minimum MSE ofTid

The suggested generalized class of estimatorsTid in terms of e0ande1 is expressed as follows:(23)Tid=MY1+e0m11+ψMXe1ψMX+δα3exp−e121+e12−1+m21+ψMXe1ψMX+δ−α4.After some simplification of equation (23), we have(24)Tid=MY1+e0m11+ϑe1α3+12α3α3−1ϑe12exp−e12+e124+m21−ϑe1α4+12α4α4+1ϑe12,where ϑ=ψMX/ψMX+δ.SubtractingMY from both sides of equation (24), we get(25)Tid−MY=MYm11−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e1+m21−ϑe1α4+12α4α4+1ϑ2e12+e0−α4ϑe0e1−1.The bias of the proposed estimators,Tid, is defined as(26)BiasTid≅ETid−MY.Taking expectations on both sides of equation (25), we get the bias of generalized class of estimators Tid:(27)BiasTid=MYm11+λCMX2234−α3ϑ+α3α3−1ϑ2+λρCMYCMXα3ϑ−12+m21+α4α4+1ϑ2λCMX22−α4ϑλρCMYCMX−1.The MSE of the proposed estimators,Tid, is defined as(28)MSETid≅ETid−MY2.Squaring both sides of equation (25), we have(29)Tid−MY2=MY2m121−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e12+m221−ϑe1α4+α4α4+1ϑ2e122+e0−α4ϑe0e12+1+2m1m21−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e11−ϑe1α4+α4α4+1ϑ2e122+e0−α4ϑe0e1−2m1−e12+3e128+α3ϑe1−α3ϑe122+α3α3−1ϑ2e122+e0−12e0e1+α3ϑe0e1−2m21−ϑe1α4+α4α4+1ϑ2e122+e0−α4ϑe0e1.Taking expectations on both sides of equation (29), we get the MSE of proposed estimators up to the first order of approximation as(30)MSETid=MY21+m12A1+m22A2+2m1m2A3−2m1A4−2m2A5,where(31)A1=1+λCMY2+CMX21+2α32ϑ2−α3ϑ2−2α3ϑ−2pCMYCMX1−2α3ϑ,A2=1+λCMY2+ϑ2CMX22α42+α4−4α4ϑpCMXCMY,A3=1+λCMY2−pCMYCMX2ϑα4−α3+1−CMX2α3α4ϑ2−38+α3ϑ2−α4ϑ2−α4α4+1ϑ22−12α3α3−1ϑ2,A4=1+λpCMYCMXα3ϑ−12+λCMX2234−α3ϑ+α3α3−1ϑ2,A5=1+12α4α4+1ϑ2λCMX2−α4ϑλpCMYCMX.Partially differentiating equation (30) with respect to m1andm2 and equating them to zero, we get the optimal values of m1andm2 as follows:(32)m1opt=A2A4−A3A5A1A2−A32,m2opt=A1A5−A3A4A1A2−A32.Placing these optimal values in equation (30), we obtained the minimum MSE as given by(33)MSETidmin≅MY21−A2A42+A1A52−2A3A4A5A1A2−A32.


## 4. Application

In this section, comparison of theTid estimators with other existing estimators under study is given by using real-life application and simulated datasets.


### 4.1. Real-Life Application

We evaluated the performance of proposed class of estimators as compared to other competing estimators in terms of the MSE. For this purpose, we selected four real-life datasets:Population 1: source: Singh [11].Y=number of fish caught in the year1995X = number of fish caught by the marine recreation fishermen in the previous year 1994.Population 2: source: Koyuncu and Kadilar [19].Y=number of teacher’s staffX=number of enrolled studentsPopulation 3: source: Singh [11].Y=number of fish caught in the year1995X = number of fish caught by the marine recreation fishermen in the previous year 1993.Population 4: source: Murthy [20].Y=number of householdsX=area in square milesTable3 presents the detailed descriptions of each of the abovementioned populations.Table 3
Data statistics of four different populations.
ValuesPopulation 1Population 2Population 3Population 4N6992369128n171801745MY20681712068686MX2011412323074.715fYMY0.000140.0026760.000140.00092fXMX0.000140.000094090.000140.1154ρ0.1510.8550.3140.468β1X2.603.942.670.89β2X7.6118.727.610.40MR19019.589530170308.270QD19684141.519752.309QA29565870.530025.316QR3936828339364.618TM4043772637775.385HL3004550229355.045DM3853.47348.33615.25.378We calculated the MSE and minimum MSE of all the estimators, that is,M^R,M^EX,M^D,M^D1,M^D2,M^D3,M^SG,Tid⊖,Tid⊕,Tid⊗,Tid⊛,andTid⊚, for populations 1–4. Expressions for the MSE of all the existing and proposed estimators are given in Sections 2 and 3 in detail. All empirical results are summarized in Tables 4–9 and the important deductions are as follows:(i)
M^SGperforms better than all existing estimators, that is, M^R,M^EX,M^D,M^D1,M^D2,M^D3(ii)
All the proposed estimators, that is,Tid⊖,Tid⊕,Tid⊗,Tid⊛andTid⊚, have minimum MSE as compared to all existing estimators(iii)
A deep insight of columns ofTid⊚ provides the least MSE among all other classes of proposed estimatorsTable 4
MSE and minimum MSE of the existing estimators.
EstimatorsPopulation 1Population 2Population 3Population 4M^R988372.7658.627746752.565361.578M^EX627420.2153.068524362.053377.321M^D552636.1342.128508766.023323.944M^D1489395.2442.067454675.783300.661M^D2480458.2942.067447982.613300.375M^D3471131.7641.964439763.443289.539M^SG402459.2841.352384146.793243.201Table 5
Minimum MSE of estimatorsTid⊖.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊖369076.340.289359823.23234.103T2d⊖368888.540.287359714.23221.120T3d⊖369072.840.289359826.23235.205T4d⊖368974.840.288359772.63231.655T5d⊖369019.240.289359793.73232.035T6d⊖369067.040.290359831.93230.939T7d⊖368879.040.287359709.63219.404T8d⊖369062.540.389359820.43235.729T9d⊖369024.640.289359800.83233.736T10d⊖368886.940.287359713.23221.085T11d⊖369172.140.290359883.23261.507T12d⊖368978.740.288359778.23233.329Table 6
Minimum MSE of estimatorsTid⊕.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊕354935.740.653358584.13216.740T2d⊕354803.440.652358509.73205.636T3d⊕354933.240.653358586.23218.205T4d⊕354864.240.653358549.63214.608T5d⊕354895.540.653358564.03214.955T6d⊕354929.140.654358590.13213.988T7d⊕354797.140.654358506.13204.203T8d⊕354926.040.654358582.23218.168T9d⊕354899.240.653358568.83216.419T10d⊕354802.340.652358509.03205.607T11d⊕355003.240.654358625.23242.254T12d⊕354866.240.653358553.43216.064Table 7
Minimum MSE of estimatorsTid⊗.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊗232289.539.237283982.23142.874T2d⊗232013.639.235283834.33117.669T3d⊗232284.339.238284177.83146.160T4d⊗232140.439.236283913.23138.074T5d⊗232205.639.237283942.23138.857T6d⊗232275.839.238283948.63136.674T7d⊗232000.639.235283827.33114.381T8d⊗232269.239.237283978.43146.077T9d⊗232213.539.237283951.73142.153T10d⊗232011.239.235283833.03117.602T11d⊗232430.339.238284063.73198.588T12d⊗232146.139.236283921.23141.355Table 8
Minimum MSE of estimatorsTid⊛.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊛206469.737.512256261.43140.691T2d⊛206038.137.507206023.53102.652T3d⊛206987.637.513256619.53145.516T4d⊛206236.437.509256151.03133.585T5d⊛206338.537.511256197.03134.749T6d⊛206448.237.514256280.43131.500T7d⊛206017.737.514256012.23097.565T8d⊛206438.037.513256255.33145.396T9d⊛206350.837.511256212.43139.628T10d⊛206034.437.506256021.33102.549T11d⊛206689.637.515256392.53217.482T12d⊛206245.437.510256163.23138.450Table 9
Minimum MSE of estimatorsTid⊚.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊚216068.836.883254850.73116.231T2d⊚215737.336.879254667.03086.071T3d⊚216062.636.884255093.83120.166T4d⊚215889.636.882254765.53110.485T5d⊚215968.036.883254801.03111.423T6d⊚216052.336.885254865.43108.809T7d⊚215721.636.879254658.23082.140T8d⊚216044.536.884254846.03120.068T9d⊚215977.536.883254812.93115.368T10d⊚215734.436.879254665.33085.991T11d⊚216238.036.886254952.03183.047T12d⊚215896.536.882254774.93114.413


### 4.2. Simulation Study

A Monte Carlo simulation study is conducted to assess the performance of the proposed generalized estimators through a real population. We consider a real-life application of primary and secondary schools for 923 districts of Turkey in 2007, considering number of teachers as study variable and number of enrolled students as auxiliary variable (source: [19]). The following are some important measures of the dataset:(34)Y¯=436.4344,X¯=11440.498,ρyx=0.855,QD=4141.5,QA=5870.5,TM=7726,MR=89530,HL=5502,DM=7348.3,QR=8283.The following steps are made to carry out the simulation study:Step 1: select a SRSWOR of sizen from the population of size NStep 2: use sample data from step 1 to find the MSE of all the existing and proposed estimatorsStep 3: perform 20,000 iterations to conduct step 1 and step 2Step 4: get 20,000 values for MSE of all existing and proposed estimatorsStep 5: take the average of 20,000 values obtained in step 4 to get the simulated MSE of each estimatorThe following is revealed from Table10:(i)
M^SG performs better than all existing estimators, that is, M^R,M^EX,M^D,M^D1,M^D2,M^D3(ii)
Minimum MSE of all the proposed estimators is the least as compared to all the existing estimators under study(iii)
As sample size increases, there is a decrease in the minimum MSE of all the proposed estimatorsTable 10
The MSE and minimum MSE of the existing and proposed estimators based on simulation study.
EstimatorsSample sizen=180n=200n=230M^R58.93651.43342.951M^EX53.38646.77338.902M^D41.99536.77830.654M^D141.93436.73130.621M^D241.93336.73030.621M^D342.14536.89330.734M^GPP41.20736.17630.237T1d⊖40.25835.46629.763T2d⊖40.27635.48629.769T3d⊖40.25835.45729.744T4d⊖40.22935.48329.766T5d⊖40.29935.47729.789T6d⊖40.22735.42929.744T7d⊖40.26335.45329.763T8d⊖40.28735.48729.712T9d⊖40.29635.45129.789T10d⊖40.28935.45529.776T11d⊖40.27835.48629.741T12d⊖40.25335.45029.750T1d⊕40.23235.75329.951T2d⊕40.27835.79629.911T3d⊕40.28935.74129.938T4d⊕40.25635.71129.987T5d⊕40.28435.78929.963T6d⊕40.26935.73529.945T7d⊕40.28735.74629.987T8d⊕40.23535.74529.966T9d⊕40.25335.78929.988T10d⊕40.25835.75929.970T11d⊕40.27835.71129.983T12d⊕40.23535.73629.955T1d⊗39.16834.63929.180T2d⊗39.16334.68829.177T3d⊗39.14734.64029.145T4d⊗39.15834.66729.144T5d⊗39.14934.68429.189T6d⊗39.13634.62229.167T7d⊗39.19634.63729.149T8d⊗39.14734.69929.156T9d⊗39.15534.68029.148T10d⊗39.18934.64429.155T11d⊗39.13834.65529.147T12d⊗39.14934.62829.156T1d⊛37.41133.29128.251T2d⊛37.41533.28428.298T3d⊛37.47833.23628.278T4d⊛37.46933.25928.263T5d⊛37.45533.24728.241T6d⊛37.43633.28928.258T7d⊛37.48933.21328.266T8d⊛37.47333.25828.254T9d⊛37.42533.23628.219T10d⊛37.45833.23328.258T11d⊛37.47933.24928.247T12d⊛37.46933.25828.259T1d⊚36.79932.81827.912T2d⊚36.79432.81027.916T3d⊚36.80932.80827.910T4d⊚36.83232.80327.909T5d⊚36.80832.81327.907T6d⊚36.78032.80327.922T7d⊚36.80932.81227.907T8d⊚36.80532.82227.916T9d⊚36.78932.80527.919T10d⊚36.80232.80127.907T11d⊚36.81332.81727.919T12d⊚36.79932.81127.909It is concluded that our generalized estimator impeccably performs the best in the presence of extreme value(s).


## 4.1. Real-Life Application

We evaluated the performance of proposed class of estimators as compared to other competing estimators in terms of the MSE. For this purpose, we selected four real-life datasets:Population 1: source: Singh [11].Y=number of fish caught in the year1995X = number of fish caught by the marine recreation fishermen in the previous year 1994.Population 2: source: Koyuncu and Kadilar [19].Y=number of teacher’s staffX=number of enrolled studentsPopulation 3: source: Singh [11].Y=number of fish caught in the year1995X = number of fish caught by the marine recreation fishermen in the previous year 1993.Population 4: source: Murthy [20].Y=number of householdsX=area in square milesTable3 presents the detailed descriptions of each of the abovementioned populations.Table 3
Data statistics of four different populations.
ValuesPopulation 1Population 2Population 3Population 4N6992369128n171801745MY20681712068686MX2011412323074.715fYMY0.000140.0026760.000140.00092fXMX0.000140.000094090.000140.1154ρ0.1510.8550.3140.468β1X2.603.942.670.89β2X7.6118.727.610.40MR19019.589530170308.270QD19684141.519752.309QA29565870.530025.316QR3936828339364.618TM4043772637775.385HL3004550229355.045DM3853.47348.33615.25.378We calculated the MSE and minimum MSE of all the estimators, that is,M^R,M^EX,M^D,M^D1,M^D2,M^D3,M^SG,Tid⊖,Tid⊕,Tid⊗,Tid⊛,andTid⊚, for populations 1–4. Expressions for the MSE of all the existing and proposed estimators are given in Sections 2 and 3 in detail. All empirical results are summarized in Tables 4–9 and the important deductions are as follows:(i)
M^SGperforms better than all existing estimators, that is, M^R,M^EX,M^D,M^D1,M^D2,M^D3(ii)
All the proposed estimators, that is,Tid⊖,Tid⊕,Tid⊗,Tid⊛andTid⊚, have minimum MSE as compared to all existing estimators(iii)
A deep insight of columns ofTid⊚ provides the least MSE among all other classes of proposed estimatorsTable 4
MSE and minimum MSE of the existing estimators.
EstimatorsPopulation 1Population 2Population 3Population 4M^R988372.7658.627746752.565361.578M^EX627420.2153.068524362.053377.321M^D552636.1342.128508766.023323.944M^D1489395.2442.067454675.783300.661M^D2480458.2942.067447982.613300.375M^D3471131.7641.964439763.443289.539M^SG402459.2841.352384146.793243.201Table 5
Minimum MSE of estimatorsTid⊖.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊖369076.340.289359823.23234.103T2d⊖368888.540.287359714.23221.120T3d⊖369072.840.289359826.23235.205T4d⊖368974.840.288359772.63231.655T5d⊖369019.240.289359793.73232.035T6d⊖369067.040.290359831.93230.939T7d⊖368879.040.287359709.63219.404T8d⊖369062.540.389359820.43235.729T9d⊖369024.640.289359800.83233.736T10d⊖368886.940.287359713.23221.085T11d⊖369172.140.290359883.23261.507T12d⊖368978.740.288359778.23233.329Table 6
Minimum MSE of estimatorsTid⊕.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊕354935.740.653358584.13216.740T2d⊕354803.440.652358509.73205.636T3d⊕354933.240.653358586.23218.205T4d⊕354864.240.653358549.63214.608T5d⊕354895.540.653358564.03214.955T6d⊕354929.140.654358590.13213.988T7d⊕354797.140.654358506.13204.203T8d⊕354926.040.654358582.23218.168T9d⊕354899.240.653358568.83216.419T10d⊕354802.340.652358509.03205.607T11d⊕355003.240.654358625.23242.254T12d⊕354866.240.653358553.43216.064Table 7
Minimum MSE of estimatorsTid⊗.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊗232289.539.237283982.23142.874T2d⊗232013.639.235283834.33117.669T3d⊗232284.339.238284177.83146.160T4d⊗232140.439.236283913.23138.074T5d⊗232205.639.237283942.23138.857T6d⊗232275.839.238283948.63136.674T7d⊗232000.639.235283827.33114.381T8d⊗232269.239.237283978.43146.077T9d⊗232213.539.237283951.73142.153T10d⊗232011.239.235283833.03117.602T11d⊗232430.339.238284063.73198.588T12d⊗232146.139.236283921.23141.355Table 8
Minimum MSE of estimatorsTid⊛.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊛206469.737.512256261.43140.691T2d⊛206038.137.507206023.53102.652T3d⊛206987.637.513256619.53145.516T4d⊛206236.437.509256151.03133.585T5d⊛206338.537.511256197.03134.749T6d⊛206448.237.514256280.43131.500T7d⊛206017.737.514256012.23097.565T8d⊛206438.037.513256255.33145.396T9d⊛206350.837.511256212.43139.628T10d⊛206034.437.506256021.33102.549T11d⊛206689.637.515256392.53217.482T12d⊛206245.437.510256163.23138.450Table 9
Minimum MSE of estimatorsTid⊚.
EstimatorsPopulation 1Population 2Population 3Population 4T1d⊚216068.836.883254850.73116.231T2d⊚215737.336.879254667.03086.071T3d⊚216062.636.884255093.83120.166T4d⊚215889.636.882254765.53110.485T5d⊚215968.036.883254801.03111.423T6d⊚216052.336.885254865.43108.809T7d⊚215721.636.879254658.23082.140T8d⊚216044.536.884254846.03120.068T9d⊚215977.536.883254812.93115.368T10d⊚215734.436.879254665.33085.991T11d⊚216238.036.886254952.03183.047T12d⊚215896.536.882254774.93114.413


## 4.2. Simulation Study

A Monte Carlo simulation study is conducted to assess the performance of the proposed generalized estimators through a real population. We consider a real-life application of primary and secondary schools for 923 districts of Turkey in 2007, considering number of teachers as study variable and number of enrolled students as auxiliary variable (source: [19]). The following are some important measures of the dataset:(34)Y¯=436.4344,X¯=11440.498,ρyx=0.855,QD=4141.5,QA=5870.5,TM=7726,MR=89530,HL=5502,DM=7348.3,QR=8283.The following steps are made to carry out the simulation study:Step 1: select a SRSWOR of sizen from the population of size NStep 2: use sample data from step 1 to find the MSE of all the existing and proposed estimatorsStep 3: perform 20,000 iterations to conduct step 1 and step 2Step 4: get 20,000 values for MSE of all existing and proposed estimatorsStep 5: take the average of 20,000 values obtained in step 4 to get the simulated MSE of each estimatorThe following is revealed from Table10:(i)
M^SG performs better than all existing estimators, that is, M^R,M^EX,M^D,M^D1,M^D2,M^D3(ii)
Minimum MSE of all the proposed estimators is the least as compared to all the existing estimators under study(iii)
As sample size increases, there is a decrease in the minimum MSE of all the proposed estimatorsTable 10
The MSE and minimum MSE of the existing and proposed estimators based on simulation study.
EstimatorsSample sizen=180n=200n=230M^R58.93651.43342.951M^EX53.38646.77338.902M^D41.99536.77830.654M^D141.93436.73130.621M^D241.93336.73030.621M^D342.14536.89330.734M^GPP41.20736.17630.237T1d⊖40.25835.46629.763T2d⊖40.27635.48629.769T3d⊖40.25835.45729.744T4d⊖40.22935.48329.766T5d⊖40.29935.47729.789T6d⊖40.22735.42929.744T7d⊖40.26335.45329.763T8d⊖40.28735.48729.712T9d⊖40.29635.45129.789T10d⊖40.28935.45529.776T11d⊖40.27835.48629.741T12d⊖40.25335.45029.750T1d⊕40.23235.75329.951T2d⊕40.27835.79629.911T3d⊕40.28935.74129.938T4d⊕40.25635.71129.987T5d⊕40.28435.78929.963T6d⊕40.26935.73529.945T7d⊕40.28735.74629.987T8d⊕40.23535.74529.966T9d⊕40.25335.78929.988T10d⊕40.25835.75929.970T11d⊕40.27835.71129.983T12d⊕40.23535.73629.955T1d⊗39.16834.63929.180T2d⊗39.16334.68829.177T3d⊗39.14734.64029.145T4d⊗39.15834.66729.144T5d⊗39.14934.68429.189T6d⊗39.13634.62229.167T7d⊗39.19634.63729.149T8d⊗39.14734.69929.156T9d⊗39.15534.68029.148T10d⊗39.18934.64429.155T11d⊗39.13834.65529.147T12d⊗39.14934.62829.156T1d⊛37.41133.29128.251T2d⊛37.41533.28428.298T3d⊛37.47833.23628.278T4d⊛37.46933.25928.263T5d⊛37.45533.24728.241T6d⊛37.43633.28928.258T7d⊛37.48933.21328.266T8d⊛37.47333.25828.254T9d⊛37.42533.23628.219T10d⊛37.45833.23328.258T11d⊛37.47933.24928.247T12d⊛37.46933.25828.259T1d⊚36.79932.81827.912T2d⊚36.79432.81027.916T3d⊚36.80932.80827.910T4d⊚36.83232.80327.909T5d⊚36.80832.81327.907T6d⊚36.78032.80327.922T7d⊚36.80932.81227.907T8d⊚36.80532.82227.916T9d⊚36.78932.80527.919T10d⊚36.80232.80127.907T11d⊚36.81332.81727.919T12d⊚36.79932.81127.909It is concluded that our generalized estimator impeccably performs the best in the presence of extreme value(s).


## 5. Robustness ofTid

In this section, robustness is examined to check the performance of the proposed generalized estimator as compared to other existing estimators under study. If the estimator performs efficiently in the presence of the extreme values, the estimator is called a robust estimator. For this purpose, we consider a real-life application taken from Punjab development statistics (PDS) for the year of 2012 [21]. For the deep study regarding robustness, different sample sizes are taken n=6,9and12. The following are the important statistics of the data:(35)Y=teaching staff in govt middle schools for boys and girls in36districts,X=number of enrolled students in govt middle schools for boys and girls in36districts,N=36,MY=2033,MX=54559,fYMY=0.0004219,fXMX=0.0000141,ρ=0.8888,TM=58995,MR=86569,QD=17600.5,QA=60775.5,QR=35201,HL=61491.5,DM=58502.9.Scatter plot confirms the presence of the extreme value in the dataset. Scatter plot can be seen in Figure1. Therefore, we can access the robustness of the generalized estimator for this dataset. Numerical results based on the robustness study are reported in Table 11. It is revealed from Table 11 that the minimum MSE of all the proposed estimators is the least as compared to all the existing estimators under study. Moreover, as the sample size increases, the minimum MSE of all the proposed estimators decreases. Therefore, it is concluded that our proposed estimator performs impeccably in the presence of the extreme value(s).Figure 1
Scatter plot of teaching staff versus students.Table 11
The MSE and minimum MSE of the existing and proposed estimators based on the robustness study.
EstimatorsSample sizen=6n=9n=12M^R50949.19030569.5120379.68M^EX623884.2137430.5324953.68M^D40971.2524582.7516388.50M^D140569.0924437.4016323.77M^D240544.2824432.1316322.22M^D339753.8824142.7716192.57M^GPP34666.6722304.1615373.79T1d⊖27237.8619454.2314065.72T2d⊖27236.5619520.4614065.67T3d⊖27237.8519454.1914065.71T4d⊖27237.9319454.0114065.73T5d⊖27237.9019454.1214065.72T6d⊖27237.4519453.9214065.65T7d⊖27237.5919507.0514065.67T8d⊖27237.8419454.1514065.71T9d⊖27237.9019454.1014065.72T10d⊖27237.5619515.1714065.66T11d⊖27240.2619454.2314066.13T12d⊖27237.9319454.1114065.73T1d⊕30863.3120815.2714683.88T2d⊕30863.1420853.5414683.85T3d⊕30863.3120815.2514683.88T4d⊕30863.3520815.1514683.89T5d⊕30863.3420815.2114683.89T6d⊕30863.0820815.1014683.84T7d⊕30863.1620845.7914683.86T8d⊕30863.3020815.2314683.88T9d⊕30863.3420815.2014683.89T10d⊕30863.1420850.4814683.85T11d⊕30864.7020815.2814684.12T12d⊕30863.3520815.2014683.89T1d⊗21617.1917310.3513083.88T2d⊗21616.8317392.1513083.82T3d⊗21617.1817310.3013083.88T4d⊗21617.2817310.0813083.90T5d⊗21617.2417310.2213083.89T6d⊗21616.7017309.9713083.79T7d⊗21616.8717375.5813083.82T8d⊗21617.1717310.2513083.88T9d⊗21617.2417310.1813083.89T10d⊗21616.1417385.6113083.82T11d⊗21620.1117310.3513084.40T12d⊗21617.2817310.2013083.90T1d⊛8921.6112290.3110740.24T2d⊛8920.8812461.1610740.10T3d⊛8921.5812290.2210740.23T4d⊛8921.7912289.7610740.27T5d⊛8921.7112290.0410740.26T6d⊛8920.6112289.5210740.05T7d⊛8920.9512426.5810740.12T8d⊛8921.5612290.1210740.23T9d⊛8921.7112289.9710740.26T10d⊛8920.8712447.5210740.10T11d⊛8927.5412290.3210741.34T12d⊛8921.7812290.0010740.27T1d⊚2994.919992.489678.24T2d⊚2994.2910135.559678.13T3d⊚2994.889992.409678.23T4d⊚2995.059992.029678.26T5d⊚2994.999992.269678.25T6d⊚2994.069992.829678.08T7d⊚2994.3610106.569678.14T8d⊚2994.879992.329678.23T9d⊚2994.989992.199678.25T10d⊚2994.2810124.129678.12T11d⊚2999.879992.489679.15T12d⊚2995.059992.229678.26


## 6. Concluding Remarks and Recommendations

We proposed the generalized classes of estimators for estimating population median under simple random sampling using robust measures of an auxiliary variable. Bias, mean squared error, and minimum mean squared error of the proposed generalized classes are derived up to the first degree of approximation. Four real-life datasets are used to check the numerical performance of the new estimators. A simulation study through a real dataset is also conducted to assess the potential of suggested classes of estimators. Robustness is also examined through a real dataset. On the basis of numerical findings, it is concluded that the new generalized classes can generate optimum estimators. Therefore, use of the proposed generalized class is recommended for future applications.The possible extensions of this work are to estimate the following: (1) finite population median under other sampling designs like stratified random sampling, double sampling, rank set sampling, and so forth; (2) other unknown finite population parameters including mean, variance, and proportions; and (3) population median in the presence of nonsampling errors.



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