# A Simulation-Based Study for Progressive Estimation of Population Mean through Traditional and Nontraditional Measures in Stratified Random Sampling

Authors: Maria Javed; Muhammad Irfan; Sajjad Haider Bhatti; Ronald Onyango
Journal: Journal of Mathematics
 (2021)

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## Abstract

This study suggests a new optimal family of exponential-type estimators for estimating population mean in stratified random sampling. These estimators are based on the traditional and nontraditional measures of auxiliary information. Expressions for the bias, mean square error, and minimum mean square error of the proposed estimators are derived up to first order of approximation. It is observed that proposed estimators perform better than the traditional estimators (unbiased, combined ratio, and combined regression) and other recent estimators. A real dataset is used to highlight the applicability of proposed estimators. In addition, a simulation study is carried out to assess the performance of new family as compared to other estimators.


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## Body

## 1. Introduction

Nowadays, it is common practice to use the auxiliary/ancillary information to boost the efficiency of estimators in survey sampling. Most of the researchers only deal with the traditional information of auxiliary variable(s) such as standard deviation, coefficient of variation, coefficient of skewness, coefficient of kurtosis, and coefficient of correlation. Having edge of this traditional information, many authors have been trying to explore new optimal estimators and families of estimators for estimating population mean under stratified random sampling. Stratified random sampling has often proved needful in improving the precision of estimators over simple random sampling, for instance, see works of Kadilar and Cingi [1, 2], Koyuncu and Kadilar [3, 4], Singh and Vishwakarma [5–7], Shabbir and Gupta [8], Haq and Shabbir [9], Singh and Solanki [10], Yadav et al. [11], Solanki and Singh [10, 12], Javed et al. [13], and Javed and Irfan [14].The motivation behind this article is to utilize the nontraditional information as well as the traditional information of the auxiliary variable to progress the estimation of population mean in stratified random sampling. This idea is initiated first time in this article under stratified random sampling.Nontraditional information includes quartile deviation, midrange, interquartile range, quartile average, decile mean, tri-mean, Hodges–Lehmann estimator, and L-moments of an auxiliary variable. L-moments are determined by linear combinations of the expected values of the order statistics (for detail, check the works of Hosking [15] and Shahzad et al. [16]). Furthermore, efficiency of the estimators is uncertain in the occurrence of the extreme values in the dataset. Some of the above nontraditional measures such as decile mean, Hodges–Lehmann estimator, and tri-mean are robust measures. Utilizing these measures, we can well cope with the extreme values/outliers in the dataset. In addition, L-moments also are used to reduce the negative effect of outliers on the estimators.Rest of the article is organized in the following way. Section2 presents the useful notations. Section 3 gives comprehensive detail of existing families of estimators. Section 4 suggests a new optimal family of estimators for estimating population mean using traditional and nontraditional measures of auxiliary variable. Expressions for the bias, mean squared error (MSE) and minimum MSE of this family are derived up to first degree of approximation in the same section. A real dataset is used in Section 5 to check the potential of new estimators as compared to existing ones. In Section 6, the performance of suggested family is evaluated by carrying out a simulation study using the same dataset used in Section 5. Section 7 contains the final discussion.


## 2. Useful Notations

Let us consider a finite populationU=U1,U2,U3,…,UN of size N, and it can be stratified into L homogenous strata with hth stratum containing Nh,h=1,2,…,L units subject to the restriction that ∑h=1LNh=N. A sample of size nh is drawn under simple random sampling without replacement (SRSWOR) from hth stratum such that ∑h=1Lnh=n. Consider the Nh pairs of observations yhi,xhi,i=1,2,3,…,Nh, made from hth stratum for the study and auxiliary variables, respectively.Furthermore, letY¯=Y¯st=∑h=1LWhY¯handy¯st=∑h=1LWhy¯h be the population and sample means of y, respectively, where Y¯h=∑i=1Nhyhi/Nh,y¯h=∑i=1nhyhi/nh,andWh=Nh/N are the population mean, sample mean, and the weight of hth stratum, respectively. Following the same lines, X¯st,x¯st,X¯handx¯h can be defined for the auxiliary variable x.To derive the expressions for the bias, mean square error (MSE), and minimum mean square error of the existing and proposed estimators, we consider the following relative error terms along with their expectations as(1)ζ0=y¯st−Y¯Y¯,ζ1=x¯st−X¯X¯,such that(2)Eζi=0fori=0,1,Vmn=∑h=1LWhm+nEy¯h−Y¯hmx¯h−X¯hnY¯mX¯n.From (2), we can write as below:(3)Eζ02=∑h=1LWh2ϕhSyh2Y¯2=V20,Eζ12=∑h=1LWh2ϕhSxh2X¯2=V02,Eζ0ζ1=∑h=1LWh2ϕhSyxhY¯X¯=V11,where(4)ϕh=1−fhnh,fh=nhNh,Syh2=∑i=1Nhyhi−Y¯h2Nh−1,Sxh2=∑i=1Nhxhi−X¯h2Nh−1,Syxh=∑i=1Nhyhi−Y¯hxhi−X¯hNh−1.Some other formulas forhth stratum, under stratified random sampling are listed below:(5)Coefficient of VaraiationCxh=SxhX¯h,Coefficient of Skewnessβ1xh=N∑i=1Nxih−X¯h3N−1N−2Sxh3,Coefficient of kurtosisβ2xh=NN+1∑i=1Nxih−X¯h4N−1N−2N−3Sxh4−3N−12N−2N−3,Quartile deviationQDxh=Q3h−Q1h2,MidrangeMRxh=x1h+xNh2,Interquartile rangeQRxh=Q3h−Q1h,Quartile averageQAxh=Q3h+Q1h2,Tri−meanTMxh=Q1h+2Q2h+Q3h4,Hodge−LehmannHLxh=Medianxjh+xkh2,1≤jh≤kh≤N,where Q1,Q2,andQ3 are the first, second, and third quartiles, respectively, x1 is the minimum value, and xN is the maximum value of the data.


## 3. Some Existing Estimators/Classes of Estimators

This section gives a brief introduction of some well-known estimators/classes of estimators from the literature.


### 3.1. Usual Estimators

In stratified random sampling usual unbiasedY¯^st, combined ratio Y¯^CR and combined regression Y¯^CReg estimators and their MSEs are detailed below:(6)Y¯^st=∑h=1LWhy¯h,Y¯^CR=y¯stX¯x¯st,Y¯^CReg=y¯st+b^X¯−x¯st,withb^=∑h=1LWh2ϕhSyxh∑h=1LWh2ϕhSxh2,MSEY¯^st=VarY¯^st=Y¯2V20,MSEY¯^CR=Y¯2V20+V02−2V11,MSEY¯^CReg=Y¯2V201−ρst2,where ρst=∑h=1LWhρyxh=∑h=1LWh2ϕhSyxh/∑h=1LWh2ϕhSyh2∑h=1LWh2ϕhSxh2=V11/V20V02.Bahl and Tuteja [17] suggested ratio and product exponential-type estimators for population mean under stratified random sampling as(7)Y¯^BT,Re=y¯stexpX¯−x¯stX¯+x¯st,(8)Y¯^BT,Pe=y¯stexpx¯st−X¯x¯st+X¯.Average of (7) and (8) can be written as(9)Y¯^BT,Avg=y¯st2expX¯−x¯stX¯+x¯st+expx¯st−X¯x¯st+X¯,MSEY¯^BT,Re≅Y¯2V20+V024−V11,MSEY¯^BT,Pe≅Y¯2V20+V024+V11.


### 3.2. Koyuncu and Kadilar [3]

To estimate the population mean under stratified random sampling, a family of ratio estimators was introduced by Koyuncu and Kadilar [3] as below:(10)Y¯^K=y¯stastX¯+bstγastx¯st+bst+1−γastX¯+bstg,where γ and g are suitable constants and ast≠0 and bst are either real numbers or functions of known parameters of the auxiliary variable such as coefficient of skewness, coefficient of kurtosis, coefficient of variation, and coefficient of correlation.Up to the first order of approximation, the bias and MSE ofY¯^K are given by(11)BiasY¯^K≅Y¯gg+12γ2η2V02−gγηV11,MSEY¯^K≅Y¯2V20+g2γ2η2V02−2gγηV11,whereη=astX¯astX¯+bst,Atgγηopt=V11V02,the minimum MSEY¯^Kis given as,MSEminY¯^K≅Y¯2V201−ρst2.


### 3.3. Koyuncu and Kadilar [4]

Koyuncu and Kadilar [4] considered the ratio estimator of Gupta and Shabbir [18] and suggested an improved estimator defined as below:(12)Y¯^KK=θ1y¯st+θ2X¯−x¯stastX¯+bstastx¯st+bst,where θ1 and θ2 are suitably chosen weights.Given below are the expressions, up to first degree of approximation, for the bias and MSE ofY¯^KK, respectively:(13)BiasY¯^KK≅−Y¯+X¯ηθ2V02+Y¯θ11+η2V02−ηV11,(14)MSEY¯^KK≅Y¯2+X¯θ2−2Y¯η+X¯θ2V02−2Y¯θ1Y¯+ηY¯η−2X¯θ2V02+−Y¯η+X¯θ2V11+Y¯2θ121+3η2V02−4ηV11+V20.The suitable weights ofθ1 and θ2 are given by(15)θ1=−1+η2V02V02η2V022+V112−V021+V20,θ2=−Y¯V11+ηη2V022−V112+V02−1−ηV11+V20X¯η2V022+V112−V021+V20.Inserting the above weights ofθ1 and θ2 in (14), we get the minimum MSE of Y¯^KK as(16)MSEminY¯^KK≅Y¯2−1+η2V02−V112+V02V20η2V022+V112−V021+V20.


### 3.4. Shabbir and Gupta [8]

Given below is a ratio-type estimator suggested by Shabbir and Gupta [8] in stratified random sampling:(17)Y¯^SG=θ3y¯st+θ4X¯−x¯stexpA¯−a¯stA¯+a¯st,where θ3 and θ4 are the constants to be determined. Also, we consider that(18)ahi=xhi+NX¯,a¯st=x¯st+NX¯,A¯=1+NX¯.Expressions for the bias and the MSE ofY¯^SG, respectively, are given below:(19)BiasY¯^SG≅41+N−21+NY¯+X¯θ4V02+Y¯θ33V02+41+N2+2N−V1181+N2,(20)MSEY¯^SG≅Y¯2θ3−12+Y¯2V11θ31+N−3Y¯2V02θ341+N2+Y¯2V02θ32−2Y¯2V11θ321+N+Y¯2V02θ321+N2−X¯Y¯V02θ41+N−2X¯Y¯V11θ3θ4+2X¯Y¯V02θ3θ41+N+X¯2V02θ42.The suitable weights ofθ3 and θ4 are given as(21)θ3=V0281+N2−V0281+N2−V112+V021+V20,θ4=Y¯V022+41+N22+2N−V11V11+1+NV02−V112+41+NV20−18X¯1+N3−V112+V021+V20.Putting weights ofθ3 and θ4 in (20), we have minimum MSE of Y¯^SG as(22)MSEminY¯^SG≅Y¯2V023+641+N4V112+161+N2V022V20−161+N2V02V112+41+N2V20641+N4V112−V021+V20.


### 3.5. Haq and Shabbir [9]

Haq and Shabbir [9] proposed two exponential ratio-type families of estimators detailed below:(23)Y¯^HS1=θ5y¯st+θ6X¯−x¯stexpastX¯+bstγastx¯st+bst+1−γastX¯+bst−1,Y¯^HS2=θ7y¯st+θ8X¯−x¯stexpastX¯+bstγastx¯st+bst+1−γastX¯+bst−1×12astX¯+bstγastx¯st+bst+1−γastX¯+bst+γastx¯st+bst+1−γastX¯+bstastX¯+bst2,where θ5,θ6,θ7,andθ8 are the suitable constants.Given below are the expressions for bias and MSE ofY¯^HS1andY¯^HS2, respectively:(24)BiasY¯^HS1≅−Y¯+X¯ψθ6V02+θ5Y¯+3Y¯ψ2V022−Y¯ψV11,BiasY¯^HS2≅−Y¯+X¯ψθ8V02+θ7Y¯+5Y¯ψ2V022−Y¯ψV11,(25)MSEY¯^HS1≅Y¯2+X¯θ6−2Y¯ψ+X¯θ6V02+Y¯θ5−2Y¯+ψ−3Y¯ψ+4X¯θ6V02+2Y¯ψ−X¯θ6V11+Y¯2θ521+4ψ2V02−4ψV11+V20,(26)MSEY¯^HS2≅Y¯2+X¯θ8−2Y¯ψ+X¯θ8V02+Y¯θ7−2Y¯+ψ−5Y¯ψ+4X¯θ8V02+2Y¯ψ−X¯θ8V11+Y¯2θ721+6ψ2V02−4ψV11+V20,where ψ=γη and η is defined earlier.The weights ofθ5,θ6,θ7,andθ8 are determined as below:(27)θ5=V022−ψ2V022−V112+V021+V20,θ6=Y¯2ψ3V022−2V11−1+ψV11−ψV022+ψV11−2V202X¯−V112+V021+V20,θ7=V022+ψ2V022−V112+V021+2ψ2V02+V20,θ8=Y¯2ψ3V022−2V11−1+ψV11+ψV02−2+ψV11+2V202X¯−V112+V021+2ψ2V02+V20.By substituting values ofθ5andθ6 in (25) and θ7andθ8 in (26), we get minimum MSE of Y¯^HS1 and Y¯^HS2, respectively:(28)MSEminY¯^HS1≅Y¯24V112+V02ψ4V022−4ψ2V112+4−1+ψ2V02V204V112−V021+V20,MSEminY¯^HS2≅Y¯24V112+V029ψ4V022−4ψ2V112+4−1+ψ2V02V204V112−V021+2ψ2V02+V20.


### 3.6. Singh and Solanki [19]

Singh and Solanki [19] proposed a family of estimators as given below:(29)Y¯^SS1=θ9y¯stγastx¯st+bst+1−γastX¯+bstastX¯+bstδ1+θ10y¯stastX¯+bstγastx¯st+bst+1−γastX¯+bstδ2,where δ1 and δ2 are suitable scalars and θ9andθ10 are the constants to be determined to make the MSE minimum. Assuming different values of δ1,δ2, and γ proposed family Y¯^SS1 reduces to the ratio-type Y¯^SS1R, product-type Y¯^SS1P, and ratio-cum-product-type Y¯^SS1RP estimators.For ratio-type, product-type, and ratio-cum-product-type estimators suitable values areδ1=0,δ2=1,γ=1,δ1=0,δ2=−1,γ=1,andδ1=1,δ2=1,γ=1, respectively.For bias and MSE ofY¯^SS1, we consider the expressions given below:(30)BiasY¯^SS1≅Y¯θ91+γδ1ηV11+δ1δ1−12γ2η2V02+θ101−γδ2ηV11+δ2δ2+12γ2η2V02−1,(31)MSEY¯^SS1≅Y¯21+θ92A1+θ102A2+2θ9θ10A3−2θ9A4−2θ10A5,where(32)A1=1+V20+4γδ1ηV11+δ12δ1−1γ2η2V02,A2=1+V20−4γδ2ηV11+δ22δ2+1γ2η2V02,A3=1+V20+2γδ1−δ2ηV11+γ2η22δ1−δ2δ1−δ2−1V02,A4=1+γδ1ηV11+δ1δ1−12γ2η2V02,A5=1−γδ2ηV11+δ2δ2+12γ2η2V02.The weights ofθ9andθ10 are determined as below:(33)θ9=A2A4−A5A3A2A1−A32,θ10=A1A5−A4A3A2A1−A32.Substituting the above weights in (31), we get the minimum MSE as given by(34)MSEminY¯^SS1≅Y¯21−A2A42−2A4A3A5+A52A1A2A1−A32.


### 3.7. Solanki and Singh [10]

Given below is the class of estimators suggested by Solanki and Singh [10]:(35)Y¯^SS2=θ11y¯stastX¯+bstγastx¯st+bst+1−γastX¯+bstδ3+θ12y¯stexpδ4astX¯+bst−astx¯st+bstastX¯+bst+astx¯st+bst,where θ11andθ12 are the feasible weights to be found such that the MSE is minimal. Here, γ=1; δ3 and δ4 being the constants take values (0, 1, −1) for obtaining different estimators like(i)
Ratio-type exponential estimatorsY¯^SS2R for δ3=0,δ4=1(ii)
Product-type exponential estimatorsY¯^SS2P for δ3=0,δ4=−1(iii)
Ratio-ratio-type exponential estimatorsY¯^SS2RR for δ3=1,δ4=1(iv)
Product-product-type exponential estimatorsY¯^SS2PP for δ3=−1,δ4=−1(v)
Ratio-product-type exponential estimatorsY¯^SS2RP for δ3=1,δ4=−1(vi)
Product-ratio-type exponential estimatorsY¯^SS2PR for δ3=−1,δ4=1For bias and MSE ofY¯^SS2, we consider the expressions given below:(36)BiasY¯^SS2≅Y¯θ111+γδ3η2V02γηδ3+1−2V11V02+θ121+δ4η8V02ηδ4+2−4V11V02−1,(37)MSEY¯^SS2≅Y¯21+θ112B1+θ122B2+2θ11θ12B3−2θ11B4−2θ12B5,where(38)B1=1+V20+γ2η22δ32+δ3V02−4γδ3ηV11,B2=1+V20+η2δ42+δ42V02−2δ4ηV11,B3=1+V20+η22γδ3+δ42+22γ2δ3+δ48V02−η2γδ3+δ4V11,B4=1+γ2η2δ32+δ32V02−γδ3ηV11,B5=1+η2δ42+2δ48V02−δ4η2V11.The suitable weights ofθ11andθ12 are as below:(39)θ11=B2B4−B5B3B2B1−B32,θ12=B1B5−B4B3B2B1−B32.Substituting these suitable weights in (37), we have the minimum MSE as given by(40)MSEminY¯^SS2≅Y¯21−B2B42−2B4B3B5+B52B1B2B1−B32.


### 3.8. Solanki and Singh [12]

Recently, Solanki and Singh [12] defined an improved estimation given as(41)Y¯^SS3=θ13y¯stX¯st∗x¯st∗δ5expδ6X¯st∗−x¯st∗X¯st∗+x¯st∗+θ14y¯stx¯st∗X¯st∗δ7expδ8x¯st∗−X¯st∗X¯st∗+x¯st∗,where θ13andθ14 are the suitably chosen weights to get minimum MSE. X¯st∗=∑h=1LWhahX¯h+bhandx¯st∗=∑h=1LWhahx¯h+bh,ah≠0,bh, are real number to parameters related to auxiliary variate x. Here, δ5,δ6,δ7, and δ8 being the constants take values (−1, 0, 1) for obtaining different classes of estimators Y¯^SS3i:(i)
Y¯^SS31, for δ5=−1,δ6=1,δ7=1,andδ8=1(ii)
Y¯^SS32, for δ5=−1,δ6=1,δ7=1,andδ8=0(iii)
Y¯^SS33, for δ5=+1,δ6=1,δ7=1,andδ8=1For bias and MSE ofY¯^SS3, we consider the expressions given below:(42)BiasY¯^SS3≅θ13+θ14−1Y¯−t1θ132X¯st∗∑h=1LWh2ϕhahSxh2t3+t1−24ahR∗+t2θ142X¯st∗∑h=1LWh2ϕhahSxh2t3−t2−24ahR∗,where t1=2δ5+δ6,t2=2δ7+δ8,andt3=Syxh/Sxh2andR∗=Y¯/X¯st∗:(43)MSEY¯^SS3≅Y¯21+θ132C1+θ142C2+2θ13θ14C3−2θ13C4−2θ14C5,where(44)C1=1+1Y¯2∑h=1LWh2ϕhSyh2−2t1ahR∗Syxh+t1t1+12ah2R∗2Sxh2,C2=1+1Y¯2∑h=1LWh2ϕhSyh2+2t2ahR∗Syxh+t2t2−12ah2R∗2Sxh2,C3=1+1Y¯2∑h=1LWh2ϕhSyh2+t2−t1ahR∗Syxh+t2−t1t2−t1−28ah2R∗2Sxh2,C4=1−t12Y¯∑h=1LWh2ϕhR∗ahSyxh−t1+24ah2R∗2Sxh2,C5=1+t22Y¯∑h=1LWh2ϕhR∗ahSyxh+t2−24ah2R∗2Sxh2.Given below are the weights ofθ13andθ14 for minimizing the MSE:(45)θ13=C2C4−C5C3C2C1−C32,θ14=C1C5−C4C3C2C1−C32.Thus, the minimum MSE by putting the above values ofθ13andθ14 in (43) is given by(46)MSEminY¯^SS3≅Y¯21−C1C52−2C4C3C5+C42C2C2C1−C32.


## 3.1. Usual Estimators

In stratified random sampling usual unbiasedY¯^st, combined ratio Y¯^CR and combined regression Y¯^CReg estimators and their MSEs are detailed below:(6)Y¯^st=∑h=1LWhy¯h,Y¯^CR=y¯stX¯x¯st,Y¯^CReg=y¯st+b^X¯−x¯st,withb^=∑h=1LWh2ϕhSyxh∑h=1LWh2ϕhSxh2,MSEY¯^st=VarY¯^st=Y¯2V20,MSEY¯^CR=Y¯2V20+V02−2V11,MSEY¯^CReg=Y¯2V201−ρst2,where ρst=∑h=1LWhρyxh=∑h=1LWh2ϕhSyxh/∑h=1LWh2ϕhSyh2∑h=1LWh2ϕhSxh2=V11/V20V02.Bahl and Tuteja [17] suggested ratio and product exponential-type estimators for population mean under stratified random sampling as(7)Y¯^BT,Re=y¯stexpX¯−x¯stX¯+x¯st,(8)Y¯^BT,Pe=y¯stexpx¯st−X¯x¯st+X¯.Average of (7) and (8) can be written as(9)Y¯^BT,Avg=y¯st2expX¯−x¯stX¯+x¯st+expx¯st−X¯x¯st+X¯,MSEY¯^BT,Re≅Y¯2V20+V024−V11,MSEY¯^BT,Pe≅Y¯2V20+V024+V11.


## 3.2. Koyuncu and Kadilar [3]

To estimate the population mean under stratified random sampling, a family of ratio estimators was introduced by Koyuncu and Kadilar [3] as below:(10)Y¯^K=y¯stastX¯+bstγastx¯st+bst+1−γastX¯+bstg,where γ and g are suitable constants and ast≠0 and bst are either real numbers or functions of known parameters of the auxiliary variable such as coefficient of skewness, coefficient of kurtosis, coefficient of variation, and coefficient of correlation.Up to the first order of approximation, the bias and MSE ofY¯^K are given by(11)BiasY¯^K≅Y¯gg+12γ2η2V02−gγηV11,MSEY¯^K≅Y¯2V20+g2γ2η2V02−2gγηV11,whereη=astX¯astX¯+bst,Atgγηopt=V11V02,the minimum MSEY¯^Kis given as,MSEminY¯^K≅Y¯2V201−ρst2.


## 3.3. Koyuncu and Kadilar [4]

Koyuncu and Kadilar [4] considered the ratio estimator of Gupta and Shabbir [18] and suggested an improved estimator defined as below:(12)Y¯^KK=θ1y¯st+θ2X¯−x¯stastX¯+bstastx¯st+bst,where θ1 and θ2 are suitably chosen weights.Given below are the expressions, up to first degree of approximation, for the bias and MSE ofY¯^KK, respectively:(13)BiasY¯^KK≅−Y¯+X¯ηθ2V02+Y¯θ11+η2V02−ηV11,(14)MSEY¯^KK≅Y¯2+X¯θ2−2Y¯η+X¯θ2V02−2Y¯θ1Y¯+ηY¯η−2X¯θ2V02+−Y¯η+X¯θ2V11+Y¯2θ121+3η2V02−4ηV11+V20.The suitable weights ofθ1 and θ2 are given by(15)θ1=−1+η2V02V02η2V022+V112−V021+V20,θ2=−Y¯V11+ηη2V022−V112+V02−1−ηV11+V20X¯η2V022+V112−V021+V20.Inserting the above weights ofθ1 and θ2 in (14), we get the minimum MSE of Y¯^KK as(16)MSEminY¯^KK≅Y¯2−1+η2V02−V112+V02V20η2V022+V112−V021+V20.


## 3.4. Shabbir and Gupta [8]

Given below is a ratio-type estimator suggested by Shabbir and Gupta [8] in stratified random sampling:(17)Y¯^SG=θ3y¯st+θ4X¯−x¯stexpA¯−a¯stA¯+a¯st,where θ3 and θ4 are the constants to be determined. Also, we consider that(18)ahi=xhi+NX¯,a¯st=x¯st+NX¯,A¯=1+NX¯.Expressions for the bias and the MSE ofY¯^SG, respectively, are given below:(19)BiasY¯^SG≅41+N−21+NY¯+X¯θ4V02+Y¯θ33V02+41+N2+2N−V1181+N2,(20)MSEY¯^SG≅Y¯2θ3−12+Y¯2V11θ31+N−3Y¯2V02θ341+N2+Y¯2V02θ32−2Y¯2V11θ321+N+Y¯2V02θ321+N2−X¯Y¯V02θ41+N−2X¯Y¯V11θ3θ4+2X¯Y¯V02θ3θ41+N+X¯2V02θ42.The suitable weights ofθ3 and θ4 are given as(21)θ3=V0281+N2−V0281+N2−V112+V021+V20,θ4=Y¯V022+41+N22+2N−V11V11+1+NV02−V112+41+NV20−18X¯1+N3−V112+V021+V20.Putting weights ofθ3 and θ4 in (20), we have minimum MSE of Y¯^SG as(22)MSEminY¯^SG≅Y¯2V023+641+N4V112+161+N2V022V20−161+N2V02V112+41+N2V20641+N4V112−V021+V20.


## 3.5. Haq and Shabbir [9]

Haq and Shabbir [9] proposed two exponential ratio-type families of estimators detailed below:(23)Y¯^HS1=θ5y¯st+θ6X¯−x¯stexpastX¯+bstγastx¯st+bst+1−γastX¯+bst−1,Y¯^HS2=θ7y¯st+θ8X¯−x¯stexpastX¯+bstγastx¯st+bst+1−γastX¯+bst−1×12astX¯+bstγastx¯st+bst+1−γastX¯+bst+γastx¯st+bst+1−γastX¯+bstastX¯+bst2,where θ5,θ6,θ7,andθ8 are the suitable constants.Given below are the expressions for bias and MSE ofY¯^HS1andY¯^HS2, respectively:(24)BiasY¯^HS1≅−Y¯+X¯ψθ6V02+θ5Y¯+3Y¯ψ2V022−Y¯ψV11,BiasY¯^HS2≅−Y¯+X¯ψθ8V02+θ7Y¯+5Y¯ψ2V022−Y¯ψV11,(25)MSEY¯^HS1≅Y¯2+X¯θ6−2Y¯ψ+X¯θ6V02+Y¯θ5−2Y¯+ψ−3Y¯ψ+4X¯θ6V02+2Y¯ψ−X¯θ6V11+Y¯2θ521+4ψ2V02−4ψV11+V20,(26)MSEY¯^HS2≅Y¯2+X¯θ8−2Y¯ψ+X¯θ8V02+Y¯θ7−2Y¯+ψ−5Y¯ψ+4X¯θ8V02+2Y¯ψ−X¯θ8V11+Y¯2θ721+6ψ2V02−4ψV11+V20,where ψ=γη and η is defined earlier.The weights ofθ5,θ6,θ7,andθ8 are determined as below:(27)θ5=V022−ψ2V022−V112+V021+V20,θ6=Y¯2ψ3V022−2V11−1+ψV11−ψV022+ψV11−2V202X¯−V112+V021+V20,θ7=V022+ψ2V022−V112+V021+2ψ2V02+V20,θ8=Y¯2ψ3V022−2V11−1+ψV11+ψV02−2+ψV11+2V202X¯−V112+V021+2ψ2V02+V20.By substituting values ofθ5andθ6 in (25) and θ7andθ8 in (26), we get minimum MSE of Y¯^HS1 and Y¯^HS2, respectively:(28)MSEminY¯^HS1≅Y¯24V112+V02ψ4V022−4ψ2V112+4−1+ψ2V02V204V112−V021+V20,MSEminY¯^HS2≅Y¯24V112+V029ψ4V022−4ψ2V112+4−1+ψ2V02V204V112−V021+2ψ2V02+V20.


## 3.6. Singh and Solanki [19]

Singh and Solanki [19] proposed a family of estimators as given below:(29)Y¯^SS1=θ9y¯stγastx¯st+bst+1−γastX¯+bstastX¯+bstδ1+θ10y¯stastX¯+bstγastx¯st+bst+1−γastX¯+bstδ2,where δ1 and δ2 are suitable scalars and θ9andθ10 are the constants to be determined to make the MSE minimum. Assuming different values of δ1,δ2, and γ proposed family Y¯^SS1 reduces to the ratio-type Y¯^SS1R, product-type Y¯^SS1P, and ratio-cum-product-type Y¯^SS1RP estimators.For ratio-type, product-type, and ratio-cum-product-type estimators suitable values areδ1=0,δ2=1,γ=1,δ1=0,δ2=−1,γ=1,andδ1=1,δ2=1,γ=1, respectively.For bias and MSE ofY¯^SS1, we consider the expressions given below:(30)BiasY¯^SS1≅Y¯θ91+γδ1ηV11+δ1δ1−12γ2η2V02+θ101−γδ2ηV11+δ2δ2+12γ2η2V02−1,(31)MSEY¯^SS1≅Y¯21+θ92A1+θ102A2+2θ9θ10A3−2θ9A4−2θ10A5,where(32)A1=1+V20+4γδ1ηV11+δ12δ1−1γ2η2V02,A2=1+V20−4γδ2ηV11+δ22δ2+1γ2η2V02,A3=1+V20+2γδ1−δ2ηV11+γ2η22δ1−δ2δ1−δ2−1V02,A4=1+γδ1ηV11+δ1δ1−12γ2η2V02,A5=1−γδ2ηV11+δ2δ2+12γ2η2V02.The weights ofθ9andθ10 are determined as below:(33)θ9=A2A4−A5A3A2A1−A32,θ10=A1A5−A4A3A2A1−A32.Substituting the above weights in (31), we get the minimum MSE as given by(34)MSEminY¯^SS1≅Y¯21−A2A42−2A4A3A5+A52A1A2A1−A32.


## 3.7. Solanki and Singh [10]

Given below is the class of estimators suggested by Solanki and Singh [10]:(35)Y¯^SS2=θ11y¯stastX¯+bstγastx¯st+bst+1−γastX¯+bstδ3+θ12y¯stexpδ4astX¯+bst−astx¯st+bstastX¯+bst+astx¯st+bst,where θ11andθ12 are the feasible weights to be found such that the MSE is minimal. Here, γ=1; δ3 and δ4 being the constants take values (0, 1, −1) for obtaining different estimators like(i)
Ratio-type exponential estimatorsY¯^SS2R for δ3=0,δ4=1(ii)
Product-type exponential estimatorsY¯^SS2P for δ3=0,δ4=−1(iii)
Ratio-ratio-type exponential estimatorsY¯^SS2RR for δ3=1,δ4=1(iv)
Product-product-type exponential estimatorsY¯^SS2PP for δ3=−1,δ4=−1(v)
Ratio-product-type exponential estimatorsY¯^SS2RP for δ3=1,δ4=−1(vi)
Product-ratio-type exponential estimatorsY¯^SS2PR for δ3=−1,δ4=1For bias and MSE ofY¯^SS2, we consider the expressions given below:(36)BiasY¯^SS2≅Y¯θ111+γδ3η2V02γηδ3+1−2V11V02+θ121+δ4η8V02ηδ4+2−4V11V02−1,(37)MSEY¯^SS2≅Y¯21+θ112B1+θ122B2+2θ11θ12B3−2θ11B4−2θ12B5,where(38)B1=1+V20+γ2η22δ32+δ3V02−4γδ3ηV11,B2=1+V20+η2δ42+δ42V02−2δ4ηV11,B3=1+V20+η22γδ3+δ42+22γ2δ3+δ48V02−η2γδ3+δ4V11,B4=1+γ2η2δ32+δ32V02−γδ3ηV11,B5=1+η2δ42+2δ48V02−δ4η2V11.The suitable weights ofθ11andθ12 are as below:(39)θ11=B2B4−B5B3B2B1−B32,θ12=B1B5−B4B3B2B1−B32.Substituting these suitable weights in (37), we have the minimum MSE as given by(40)MSEminY¯^SS2≅Y¯21−B2B42−2B4B3B5+B52B1B2B1−B32.


## 3.8. Solanki and Singh [12]

Recently, Solanki and Singh [12] defined an improved estimation given as(41)Y¯^SS3=θ13y¯stX¯st∗x¯st∗δ5expδ6X¯st∗−x¯st∗X¯st∗+x¯st∗+θ14y¯stx¯st∗X¯st∗δ7expδ8x¯st∗−X¯st∗X¯st∗+x¯st∗,where θ13andθ14 are the suitably chosen weights to get minimum MSE. X¯st∗=∑h=1LWhahX¯h+bhandx¯st∗=∑h=1LWhahx¯h+bh,ah≠0,bh, are real number to parameters related to auxiliary variate x. Here, δ5,δ6,δ7, and δ8 being the constants take values (−1, 0, 1) for obtaining different classes of estimators Y¯^SS3i:(i)
Y¯^SS31, for δ5=−1,δ6=1,δ7=1,andδ8=1(ii)
Y¯^SS32, for δ5=−1,δ6=1,δ7=1,andδ8=0(iii)
Y¯^SS33, for δ5=+1,δ6=1,δ7=1,andδ8=1For bias and MSE ofY¯^SS3, we consider the expressions given below:(42)BiasY¯^SS3≅θ13+θ14−1Y¯−t1θ132X¯st∗∑h=1LWh2ϕhahSxh2t3+t1−24ahR∗+t2θ142X¯st∗∑h=1LWh2ϕhahSxh2t3−t2−24ahR∗,where t1=2δ5+δ6,t2=2δ7+δ8,andt3=Syxh/Sxh2andR∗=Y¯/X¯st∗:(43)MSEY¯^SS3≅Y¯21+θ132C1+θ142C2+2θ13θ14C3−2θ13C4−2θ14C5,where(44)C1=1+1Y¯2∑h=1LWh2ϕhSyh2−2t1ahR∗Syxh+t1t1+12ah2R∗2Sxh2,C2=1+1Y¯2∑h=1LWh2ϕhSyh2+2t2ahR∗Syxh+t2t2−12ah2R∗2Sxh2,C3=1+1Y¯2∑h=1LWh2ϕhSyh2+t2−t1ahR∗Syxh+t2−t1t2−t1−28ah2R∗2Sxh2,C4=1−t12Y¯∑h=1LWh2ϕhR∗ahSyxh−t1+24ah2R∗2Sxh2,C5=1+t22Y¯∑h=1LWh2ϕhR∗ahSyxh+t2−24ah2R∗2Sxh2.Given below are the weights ofθ13andθ14 for minimizing the MSE:(45)θ13=C2C4−C5C3C2C1−C32,θ14=C1C5−C4C3C2C1−C32.Thus, the minimum MSE by putting the above values ofθ13andθ14 in (43) is given by(46)MSEminY¯^SS3≅Y¯21−C1C52−2C4C3C5+C42C2C2C1−C32.


## 4. Suggested Family of Estimators

Following the lines of Shabbir et al. [20], a generalized estimator for the estimation of population mean is proposed using some traditional and nontraditional measures of an auxiliary variable. For more details of these nontraditional measures, see the works of Hettmansperger and McKean [21], Wang et al. [22], and Irfan et al. [23–26]:(47)Y¯^P=Y¯^BT,Avg+θ15X¯−x¯st+θ16y¯stexpastX¯+bstastx¯st+bst−1,where θ15 and θ16 are the suitably chosen weights and ast≠0 and bst are either real numbers or functions of known parameters of the auxiliary variable such as standard deviation Sxst, coefficient of skewness β1xst, coefficient of kurtosis β2xst, coefficient of variation Cxst, coefficient of correlation ρst, quartile deviation QDxst, midrange MRxst, interquartile range QRxst, quartile average QAxst, tri-mean TMxst, and Hodge–Lehmann estimator HLxst.Remark 1.
Many more estimators can be generated by placing different available parameters of auxiliary variable in place ofast and bst. Some of them are presented in Table 1.
Using (1), the suggested class of estimators Y¯^P can be rewritten as(48)Y¯^P=Y¯1+ζ02exp−ζ121+ζ12−1+expζ121+ζ12−1−θ15X¯ζ1+θ16Y¯1+ζ0exp−ηζ11+ηζ1−1×122+2ηζ1+η2ζ121+ηζ1−12.
As defined earlier,η=astX¯/astX¯+bst.
Expanding the right-hand side of (48), up to first order of approximation and subtracting Y¯ from both sides, we obtain(49)Y¯^P−Y¯=Y¯ζ0+Y¯ζ128−θ15X¯ζ1+θ16Y¯+θ16Y¯ζ0−ηY¯ζ1−ηY¯ζ0ζ1+θ15ηX¯ζ12−θ16ηY¯ζ1−θ16ηY¯ζ0ζ1+32η2Y¯ζ12+52θ16η2Y¯ζ12+η2Y¯ζ12.
Up to first order of approximation, the bias and the MSE are given by(50)BiasY¯^P≅Y¯V021+20η28−ηV11+X¯ηθ15V02+θ16Y¯+5Y¯η2V022−Y¯ηV11,(51)MSEY¯^P≅Y¯2η2V02+Y¯2V20−2Y¯2ηV11+X¯2θ152V02+2X¯Y¯ηθ15V02−2X¯Y¯θ15V11+Y¯θ164X¯ηθ15V02−2X¯θ15V11+7η2Y¯V02−6ηY¯V11+2Y¯V20+14Y¯V02+Y¯2θ1621+6η2V02−4ηV11+V20.
By minimizing (51), suitable weights of θ15 and θ16 are obtained as below:(52)θ15=Y¯2ηV022η2+1/4+2V111−ηV11−ηV022−ηV11−2V20−1/4V11V022X¯−V112+V021+2η2V02+V20,θ16=2V112−V023η2V02+2V20+V02/42−V112+V021+2η2V02+V20.
Putting the optimal weights ofθ15 and θ16 in (51), we have the minimum MSE given by(53)MSEminY¯^P≅Y¯24V112+V023η2V0223η2+1/2−V1121+4η2+1/16V022+4−1+V02η2+1/4V204V112−V021+2η2V02+V20.Table 1
Some members of the proposed family.
EstimatorY¯^Piastbst1Y¯^P1102Y¯^P21ρst=∑h=1LWhρh3Y¯^P31Cxst=∑h=1LWhCxh4Y¯^P4Cxst=∑h=1LWhCxhβ2xst=∑h=1LWhβ2xh5Y¯^P5β1xst=∑h=1LWhβ1xhCxst=∑h=1LWhCxh6Y¯^P6β1xst=∑h=1LWhβ1xhρst=∑h=1LWhρh7Y¯^P71β2xst=∑h=1LWhβ2xh8Y¯^P81β1xst=∑h=1LWhβ1xh9Y¯^P9TMxst=∑h=1LWhTMxh110Y¯^P10TMxst=∑h=1LWhTMxhQRxst=∑h=1LWhQRxh11Y¯^P11QAxst=∑h=1LWhQAxhSxst=∑h=1LWhSxh12Y¯^P12HLxst=∑h=1LWhHLxhρst=∑h=1LWhρh13Y¯^P13HLxst=∑h=1LWhHLxhQRxst=∑h=1LWhQRxh14Y¯^P14MRxst=∑h=1LWhMRxhQRxst=∑h=1LWhQRxh15Y¯^P15MRxst=∑h=1LWhMRxhQDxst=∑h=1LWhQDxh


## 5. Application to a Dataset

To examine the performance of the proposed class of estimators, we considered a real data of Turkey (2007) used by Koyuncu and Kadilar [3] given in Table 2. In this data, let y be the number of teachers (study variable) and x be the number of students (auxiliary variable) which are recorded for primary and secondary schools at 6 regions for N=923 districts. A total sample of size n=180 is selected through Neyman allocation from 6 strata. (source: Ministry of Education, Republic of Turkey).Table 2
Data statistics.
Valueshth stratum123456Nh127117103170205201nh312129382239Y¯h703.74413.00573.17424.66267.03393.84X¯h20804.599211.7914309.39478.855569.9512997.59Syh883.835644.9221033.467810.585403.654711.723Sxh30486.75115180.76927549.69718218.9318497.77623094.141Syxh25237153.529747942.8528294397.0414523885.533393591.7515864573.97Cxh1.4651.6481.9251.9221.5261.777ρh0.9360.9960.9940.9830.9890.965β2hx4.59318.54315.44610.16221.94723.114β1hx2.1643.8673.7483.1214.0844.411QDxh12736.83023.55698.752742.252025.55545.5MRxh77193.55464485780.556386.53405789530.5QRxh25473.5604711397.55484.5405111091QAxh15489.35104.57704.253899.253468.58105.5TMxh10853.14705.755994.133321.383137.757116.25HLxh7913.754883.257124.53899.53173.258074We have computedMSEminY¯^K,MSEminY¯^KK,MSEminY¯^SG,MSEminY¯^HS1, MSEminY¯^HS2,MSEminY¯^SS1,MSEminY¯^SS2,MSEminY¯^SS3,andMSEminY¯^P for the population dataset given in Table 2 and are reported in Tables 3 and 4.Table 3
Minimum MSEs of different estimators based on real data.
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P10216.4168194.0826194.0852183.4821115.8697193.4183175.9254192.8467102.95731∑h=1LWhρh216.3768194.0826194.0852183.4853115.8949193.4192175.9254192.8484102.98431∑h=1LWhCxh216.3470194.0826194.0852183.4877115.9136193.4198175.9254192.8496103.0045∑h=1LWhCxh∑h=1LWhβ2xh216.0228194.0826194.0852183.5136116.1178193.4264175.9254192.8633103.2247∑h=1LWhβ1xh∑h=1LWhCxh216.3976194.0826194.0852183.4836115.8818193.4187175.9254192.8475102.9702∑h=1LWhβ1xh∑h=1LWhρh216.4058194.0826194.0852183.4830115.8766193.4186175.9254192.8472102.96471∑h=1LWhβ2xh215.7475194.0826194.0852183.5357116.2921193.4320175.9254192.8749103.41261∑h=1LWhβ1xh216.2676194.0826194.0852183.4940115.9635193.4214175.9254192.8530103.0583∑h=1LWhTMxh1216.4168194.0826194.0852183.4821115.8697193.4183175.9254192.8467102.9573∑h=1LWhTMxh∑h=1LWhQRxh216.3449194.0826194.0852183.4878115.9149193.4198175.9254192.8497103.0059∑h=1LWhQAxh∑h=1LWhSxh216.3012194.0826194.0852183.4913115.9423193.4207175.9254192.8516103.0355∑h=1LWhHLxh∑h=1LWhρh216.4168194.0826194.0852183.4821115.8697193.4183175.9254192.8467102.9573∑h=1LWhHLxh∑h=1LWhQRxh216.3457194.0826194.0852183.4878115.9144193.4198175.9254192.8497103.0054∑h=1LWhMRxh∑h=1LWhQRxh216.4105194.0826194.0852183.4826115.8737193.4185175.9254192.8470102.9615∑h=1LWhMRxh∑h=1LWhQDxh216.4136194.0826194.0852183.4823115.8717193.4184175.9254192.8468102.9594Bold values indicate minimum MSE.Table 4
Minimum MSEs of different estimators based on real data.
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P10194.1562185.5207193.6484143.7752193.6484191.9754108.8591143.7752172.0985102.95731∑h=1LWhρh194.1560185.5200193.6489143.7805193.6489191.9743108.8704143.7804172.1103102.98431∑h=1LWhCxh194.1558185.5195193.6492143.7844193.6492191.9735108.8793143.7844172.1196103.0045∑h=1LWhCxh∑h=1LWhβ2xh194.1539185.5141193.6532143.8273193.6532191.9647107.0285132.7917161.5885103.2247∑h=1LWhβ1xh∑h=1LWhCxh194.1561185.5203193.6486143.7777193.1521191.9749178.5658214.5583231.1787102.9702∑h=1LWhβ1xh∑h=1LWhρh194.1562185.5205193.6485143.7767193.1515191.9751178.5631214.5571231.1754102.96471∑h=1LWhβ2xh194.1522185.5095193.6566143.8638193.6566191.9571109.0538143.8638172.3019103.41261∑h=1LWhβ1xh194.1554185.5182193.6502143.7949193.1561191.9714108.9023143.7949172.1437103.0583∑h=1LWhTMxh1194.1562185.5207193.6484143.7752193.1515191.9754676.9615703.3570656.1403102.9573∑h=1LWhTMxh∑h=1LWhQRxh194.1558185.5195193.6493143.7847193.1537191.9735676.9765703.3632656.1786103.0059∑h=1LWhQAxh∑h=1LWhSxh194.1556185.5187193.6498143.7904193.155191.9723820.1003844.5858772.9658103.0355∑h=1LWhHLxh∑h=1LWhρh194.1562185.5207193.6484143.7752193.1515191.9754476.1121506.1138486.5211102.9573∑h=1LWhHLxh∑h=1LWhQRxh194.1558185.5195193.6493143.7846193.1537191.9735476.1289506.1211486.5537103.0054∑h=1LWhMRxh∑h=1LWhQRxh194.1562185.5206193.6485143.7761193.1517191.9753383.5274415.2221407.5903102.9615∑h=1LWhMRxh∑h=1LWhQDxh194.1562185.5206193.6484143.7756193.1516191.9753383.5266415.2217407.5889102.9594Bold values indicate minimum MSE.


### 5.1. Important Findings

(i)
The values of MSE of usual unbiased, combined ratio, and combined regression estimators in stratified random sampling are computed as given below:(54)MSEY¯^st=2229.2662,MSEY¯^CR=216.4168,MSEY¯^CReg=194.2832.(ii)
It is obvious from Table3 that MSEs of Koyuncu and Kadilar [3] class of estimators Y¯^K round about the value of MSEY¯^CR.(iii)
value ofMSEY¯^CReg is less than MSEY¯^CR and MSEminY¯^K.(iv)
It is observed from Tables3 and 4 that the MSEs of Koyuncu and Kadilar [4] class of estimators Y¯^KK, Shabbir and Gupta [8] class of estimators Y¯^SG, Haq and Shabbir [9] classes of estimators Y¯^HSi,i=1,2, and Singh and Solanki [10, 12] classes of estimators Y¯^SSj,j=1,2,3, are less than the MSE of combined regression estimator Y¯^CReg.(v)
Again, from Tables3 and 4, it is valuable to mention that the proposed family of estimators Y¯^P have the least MSEs as compared to all other classes of estimators against different values of ast and bst.


## 5.1. Important Findings

(i)
The values of MSE of usual unbiased, combined ratio, and combined regression estimators in stratified random sampling are computed as given below:(54)MSEY¯^st=2229.2662,MSEY¯^CR=216.4168,MSEY¯^CReg=194.2832.(ii)
It is obvious from Table3 that MSEs of Koyuncu and Kadilar [3] class of estimators Y¯^K round about the value of MSEY¯^CR.(iii)
value ofMSEY¯^CReg is less than MSEY¯^CR and MSEminY¯^K.(iv)
It is observed from Tables3 and 4 that the MSEs of Koyuncu and Kadilar [4] class of estimators Y¯^KK, Shabbir and Gupta [8] class of estimators Y¯^SG, Haq and Shabbir [9] classes of estimators Y¯^HSi,i=1,2, and Singh and Solanki [10, 12] classes of estimators Y¯^SSj,j=1,2,3, are less than the MSE of combined regression estimator Y¯^CReg.(v)
Again, from Tables3 and 4, it is valuable to mention that the proposed family of estimators Y¯^P have the least MSEs as compared to all other classes of estimators against different values of ast and bst.


## 6. Simulation Study

In this section, we carried out a simulation study using R statistical software to evaluate the behavior of proposed estimatorsY¯^Pi in comparison with Y¯^st,Y¯^CR,Y¯^CReg,Y¯^K,Y¯^KK,Y¯^SG,Y¯^HS1,Y¯^HS2, Y¯^SS1,Y¯^SS2,andY¯^SS3. Real population presented in Table 3 is used for the simulation study. Three different sample sizes n=180,250,and350 are taken from this population on the basis of proportional allocation.The following steps summarize the procedure of finding the average MSE of an estimator.Step 1: select a bivariate stratified sample of sizen using simple random sampling without replacement from the bivariate stratified normal populationStep 2: use sample data from Step 1 to find the MSE of all the estimators under the studyStep 3: repeat Step 1 and Step 2 10,000 times and obtain 10,000 values for MSEsStep 4: Average of 10,000 values obtained in Step 3 are the MSE of each estimator


### 6.1. Findings

MSEs ofY¯^st,Y¯^CR,andY¯^CReg for different sample sizes are under Table 5.Table 5
MSEs ofY¯^st,Y¯^CR,andY¯^CReg for different sample sizes.
MSEY¯^stMSEY¯^CRMSEY¯^CRegn=1802234.0170208.6489191.4028n=2501600.3790155.6986136.6418n=350971.940995.614784.1167MSEs of the other estimators under the study are presented in Tables6–11, and the following important considerations are made from them.(i)
It is quite obvious that the MSE values ofY¯^Pi,i=1,2,3,…,15, as compared to the MSEs of all other estimators are minimum under different sample sizes taken from the population. It proves that all the proposed estimators are more efficient.(ii)
It is also shown that, by increasing the sample size selected from the population, the MSEs decrease.Table 6
Minimum MSEs of different estimators based on the simulation study (n=180).
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P10208.7024182.4218182.4243171.9060104.0812181.7569163.3593181.133491.10381∑h=1LWhρh208.9233182.8256182.8281172.3738105.0166182.1669163.8692181.550392.12831∑h=1LWhCxh208.1726182.0508182.0533171.5916104.1605181.3923163.0362180.775191.2563∑h=1LWhCxh∑h=1LWhβ2xh206.1250180.3394180.3419169.9015102.4944179.6982161.1364179.098189.5657∑h=1LWhβ1xh∑h=1LWhCxh207.2181181.6087181.6112171.2167104.2205180.9596162.6734180.352591.4022∑h=1LWhβ1xh∑h=1LWhρh208.0968181.9132181.9157171.4806104.2386181.2527163.0222180.633591.37371∑h=1LWhβ2xh207.6863181.5076181.5100171.1143104.1185180.8584162.5738180.252091.26281∑h=1LWhβ1xh208.9447182.9924182.9949172.5532105.3055182.3371164.0466181.725392.4328∑h=1LWhTMxh1208.6489182.8849182.8874172.3985104.8140182.2277163.8243181.614291.8843∑h=1LWhTMxh∑h=1LWhQRxh208.9039182.8772182.8797172.4226105.0771182.2188163.9184181.603292.1900∑h=1LWhQAxh∑h=1LWhSxh209.1184182.5700182.5725172.0577104.2553181.9039163.516181.278891.2757∑h=1LWhHLxh∑h=1LWhρh208.7805182.6141182.6166172.1151104.4292181.9516163.5377181.331491.4787∑h=1LWhHLxh∑h=1LWhQRxh207.4471181.8892181.8914173.1457117.5424181.6167162.7754181.3831105.7468∑h=1LWhMRxh∑h=1LWhQRxh208.2513181.6575181.66171.1876103.6541180.9891162.7212180.360990.7320∑h=1LWhMRxh∑h=1LWhQDxh207.2467181.1462181.1487170.6938103.2264180.4891162.1121179.872490.3161Bold values indicate minimum MSE.Table 7
Minimum MSEs of different estimators based on the simulation study (n=180).
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P10182.3466173.2652182.0005130.4476181.4683179.931394.73656130.4476160.54591.10381∑h=1LWhρh182.7497173.7173182.4081131.1713181.8815180.342295.68471131.166161.114392.12831∑h=1LWhCxh181.9698172.9086181.6336130.2683181.1066179.550594.70196130.261160.324491.2563∑h=1LWhCxh∑h=1LWhβ2xh180.2450171.0819179.9328128.2003179.4204177.779792.33457128.1317158.612589.5657∑h=1LWhβ1xh∑h=1LWhCxh181.5270172.4999181.1971130.0811180.6785179.11294.72428130.0791160.074891.4022∑h=1LWhβ1xh∑h=1LWhρh181.8389172.8398181.4947130.4089180.9661179.444295.01959130.4074160.22191.37371∑h=1LWhβ2xh181.4267172.4007181.0958129.978180.5776179.012794.42132129.8881159.776191.26281∑h=1LWhβ1xh182.9171173.8882182.5766131.3816182.0538180.508295.89918131.3617161.305592.4328∑h=1LWhTMxh1182.8072173.7227182.4678130.9729181.9436180.382795.33219130.9729161.142491.8843∑h=1LWhTMxh∑h=1LWhQRxh182.8017173.7682182.4597131.2176181.9338180.393095.71564131.208161.165992.1900∑h=1LWhQAxh∑h=1LWhSxh182.4934173.4168182.1482130.6215181.6146180.078994.8953130.6059160.659491.2757∑h=1LWhHLxh∑h=1LWhρh182.5333173.444182.1942130.6498181.6645180.108494.9620130.6483160.808891.4787∑h=1LWhHLxh∑h=1LWhQRxh181.6013172.2849181.7109132.9205181.5076178.668894.22969129.8961160.2185105.7468∑h=1LWhMRxh∑h=1LWhQRxh181.5821172.5648181.2344129.9974180.6984179.187694.48528129.9965159.830690.7320∑h=1LWhMRxh∑h=1LWhQDxh181.0637171.9921180.7301129.3164180.2037178.642793.73929129.316159.423890.3161Bold values indicate minimum MSE.Table 8
Minimum MSEs of different estimators based on the simulation study (n=250).
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P10155.6986130.9425130.9435125.183187.54561130.5094121.2922130.093280.34671∑h=1LWhρh155.6335130.9868130.9878125.240487.70134130.5545121.3769130.1480.52041∑h=1LWhCxh155.188130.6177130.6186124.872987.3303130.1873120.9742129.774480.1478∑h=1LWhCxh∑h=1LWhβ2xh155.2285130.7593130.7602125.026887.56625130.3357121.0766129.930880.3851∑h=1LWhβ1xh∑h=1LWhCxh155.3389130.8714130.8724125.113787.48645130.4407121.2061130.028180.2890∑h=1LWhβ1xh∑h=1LWhρh155.6699131.0058131.0068125.24487.60117130.572121.3724130.156180.40111∑h=1LWhβ2xh155.0061130.2404130.2413124.523587.1653129.8145120.6219129.406579.99851∑h=1LWhβ1xh155.2016130.5104130.5113124.772887.28445130.0795120.8933129.665980.1095∑h=1LWhTMxh1155.5360131.0069131.0078125.257387.69095130.5764121.3634130.163880.5060∑h=1LWhTMxh∑h=1LWhQRxh155.8067131.277131.278125.570188.33712130.8469121.7497130.434981.2152∑h=1LWhQAxh∑h=1LWhSxh154.6152130.1677130.1686124.459787.16355129.7406120.5834129.331180.0265∑h=1LWhHLxh∑h=1LWhρh155.7181130.6542130.6551124.906387.34791130.2172121.0742129.796680.1639∑h=1LWhHLxh∑h=1LWhQRxh155.0446130.6161130.6169125.835695.03376130.4127120.9906130.242488.5004∑h=1LWhMRxh∑h=1LWhQRxh155.8021131.0947131.0957125.36587.96272130.6616121.5326130.245880.8098∑h=1LWhMRxh∑h=1LWhQDxh154.8427130.0278130.0287124.295686.82202129.5947120.4377129.17879.6540Bold values indicate minimum MSE.Table 9
Minimum MSEs of different estimators based on the simulation study (n=250).
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P10130.9376126.3729130.671104.1672130.317129.810885.72231104.1672118.259880.34671∑h=1LWhρh130.9843126.4391130.7156104.3089130.3628129.864485.91894104.3061118.326480.52041∑h=1LWhCxh130.612126.0493130.3477103.8761129.9964129.483885.45359103.8723117.971980.1478∑h=1LWhCxh∑h=1LWhβ2xh130.751126.1653130.4933103.9616130.1484129.6185.45669103.9258118.112580.3851∑h=1LWhβ1xh∑h=1LWhCxh130.8666126.2933130.6011104.0666130.2499129.736385.6053104.0656118.20780.2890∑h=1LWhβ1xh∑h=1LWhρh131.0036126.4477130.7337104.2572130.3797129.881885.81913104.2564118.311680.40111∑h=1LWhβ2xh130.2341125.6814129.9731103.5868129.6259129.105885.13934103.5396117.570279.99851∑h=1LWhβ1xh130.5056125.9563130.2402103.8306129.8883129.38285.4301103.8202117.857780.1095∑h=1LWhTMxh1131.0025126.4398130.7368104.2592130.3856129.875285.83846104.2592118.362380.5060∑h=1LWhTMxh∑h=1LWhQRxh131.2748126.7694131.0072104.829130.6564130.163486.59144104.824118.714581.2152∑h=1LWhQAxh∑h=1LWhSxh130.162125.6269129.8997103.5951129.5513129.039985.28246103.587117.599880.0265∑h=1LWhHLxh∑h=1LWhρh130.6518126.1239130.3805104.0301130.0228129.5485.66646104.0293117.957680.1639∑h=1LWhHLxh∑h=1LWhQRxh130.5198125.8475130.4821105.5124130.3329129.13585.57302103.9372118.026688.5004∑h=1LWhMRxh∑h=1LWhQRxh131.0921126.571130.8231104.5418130.4693129.978386.23798104.5414118.474880.8098∑h=1LWhMRxh∑h=1LWhQDxh130.0233125.4884129.7565103.4035129.4021128.906385.05503103.4033117.383179.6540Bold values indicate minimum MSE.Table 10
Minimum MSEs of different estimators based on the simulation study (n=350).
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P1095.614780.916580.916778.778364.722380.754977.433980.602362.03821∑h=1LWhρh95.797081.178181.178379.041965.010181.016877.703080.864962.33041∑h=1LWhCxh95.838181.208781.208979.072265.036981.047577.731980.895762.3564∑h=1LWhCxh∑h=1LWhβ2xh95.932481.325681.325879.199165.237481.166677.854081.017362.5658∑h=1LWhβ1xh∑h=1LWhCxh95.439080.888080.888378.758264.763280.727777.417080.576662.0907∑h=1LWhβ1xh∑h=1LWhρh95.574680.998280.998478.867264.865680.837577.527980.686162.19201∑h=1LWhβ2xh95.469780.930480.930678.816764.935880.773177.470780.625662.27761∑h=1LWhβ1xh95.225980.789980.790178.660264.662480.631077.302680.481561.9877∑h=1LWhTMxh195.449180.978480.978678.843564.815380.818377.493180.667662.1367∑h=1LWhTMxh∑h=1LWhQRxh95.470880.942780.942978.807964.778480.782577.458080.631762.0986∑h=1LWhQAxh∑h=1LWhSxh95.849381.213981.214179.075665.026181.053077.729280.901562.3421∑h=1LWhHLxh∑h=1LWhρh95.261880.654980.655178.521764.496480.494277.174880.342661.8179∑h=1LWhHLxh∑h=1LWhQRxh95.335180.871880.872079.098467.626180.797177.413280.736965.1968∑h=1LWhMRxh∑h=1LWhQRxh95.349080.772080.772278.645264.670880.611477.310380.460162.0023∑h=1LWhMRxh∑h=1LWhQDxh95.429580.922880.923078.799964.858080.762877.467180.612262.1960Bold values indicate minimum MSE.Table 11
Minimum MSEs of different estimators based on the simulation study (n=350).
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P1080.923479.272580.814571.253480.684280.523464.660871.253476.176662.03821∑h=1LWhρh81.185679.538281.076371.533980.946480.786664.950871.532976.445062.33041∑h=1LWhCxh81.216279.567981.106971.560780.977180.816964.974771.559376.474362.3564∑h=1LWhCxh∑h=1LWhβ2xh81.332779.685981.225171.703781.097480.932065.116371.690876.599262.5658∑h=1LWhβ1xh∑h=1LWhCxh80.895179.249380.786871.259880.657680.495964.691571.259576.172462.0907∑h=1LWhβ1xh∑h=1LWhρh81.005479.360280.896771.370080.767380.606664.800971.369876.278962.19201∑h=1LWhβ2xh80.937079.295280.830971.348980.704780.536664.784171.332076.227262.27761∑h=1LWhβ1xh80.796179.141580.689571.130280.561780.393064.539671.126476.076761.9877∑h=1LWhTMxh180.985379.332380.877271.315480.748480.583764.726471.315476.257562.1367∑h=1LWhTMxh∑h=1LWhQRxh80.949679.296980.841571.280780.712680.548364.688171.278876.215962.0986∑h=1LWhQAxh∑h=1LWhSxh81.221279.568981.112371.547880.982880.820464.947971.544976.474162.3421∑h=1LWhHLxh∑h=1LWhρh80.661679.011580.553571.002880.423980.261364.419171.002575.928661.8179∑h=1LWhHLxh∑h=1LWhQRxh80.848079.164580.821971.842680.768780.356564.731571.276976.172565.1968∑h=1LWhMRxh∑h=1LWhQRxh80.779379.138380.670771.166180.541380.381864.611671.165976.060462.0023∑h=1LWhMRxh∑h=1LWhQDxh80.930179.291780.821871.335680.693080.532864.794971.335576.225162.1960Bold values indicate minimum MSE.


## 6.1. Findings

MSEs ofY¯^st,Y¯^CR,andY¯^CReg for different sample sizes are under Table 5.Table 5
MSEs ofY¯^st,Y¯^CR,andY¯^CReg for different sample sizes.
MSEY¯^stMSEY¯^CRMSEY¯^CRegn=1802234.0170208.6489191.4028n=2501600.3790155.6986136.6418n=350971.940995.614784.1167MSEs of the other estimators under the study are presented in Tables6–11, and the following important considerations are made from them.(i)
It is quite obvious that the MSE values ofY¯^Pi,i=1,2,3,…,15, as compared to the MSEs of all other estimators are minimum under different sample sizes taken from the population. It proves that all the proposed estimators are more efficient.(ii)
It is also shown that, by increasing the sample size selected from the population, the MSEs decrease.Table 6
Minimum MSEs of different estimators based on the simulation study (n=180).
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P10208.7024182.4218182.4243171.9060104.0812181.7569163.3593181.133491.10381∑h=1LWhρh208.9233182.8256182.8281172.3738105.0166182.1669163.8692181.550392.12831∑h=1LWhCxh208.1726182.0508182.0533171.5916104.1605181.3923163.0362180.775191.2563∑h=1LWhCxh∑h=1LWhβ2xh206.1250180.3394180.3419169.9015102.4944179.6982161.1364179.098189.5657∑h=1LWhβ1xh∑h=1LWhCxh207.2181181.6087181.6112171.2167104.2205180.9596162.6734180.352591.4022∑h=1LWhβ1xh∑h=1LWhρh208.0968181.9132181.9157171.4806104.2386181.2527163.0222180.633591.37371∑h=1LWhβ2xh207.6863181.5076181.5100171.1143104.1185180.8584162.5738180.252091.26281∑h=1LWhβ1xh208.9447182.9924182.9949172.5532105.3055182.3371164.0466181.725392.4328∑h=1LWhTMxh1208.6489182.8849182.8874172.3985104.8140182.2277163.8243181.614291.8843∑h=1LWhTMxh∑h=1LWhQRxh208.9039182.8772182.8797172.4226105.0771182.2188163.9184181.603292.1900∑h=1LWhQAxh∑h=1LWhSxh209.1184182.5700182.5725172.0577104.2553181.9039163.516181.278891.2757∑h=1LWhHLxh∑h=1LWhρh208.7805182.6141182.6166172.1151104.4292181.9516163.5377181.331491.4787∑h=1LWhHLxh∑h=1LWhQRxh207.4471181.8892181.8914173.1457117.5424181.6167162.7754181.3831105.7468∑h=1LWhMRxh∑h=1LWhQRxh208.2513181.6575181.66171.1876103.6541180.9891162.7212180.360990.7320∑h=1LWhMRxh∑h=1LWhQDxh207.2467181.1462181.1487170.6938103.2264180.4891162.1121179.872490.3161Bold values indicate minimum MSE.Table 7
Minimum MSEs of different estimators based on the simulation study (n=180).
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P10182.3466173.2652182.0005130.4476181.4683179.931394.73656130.4476160.54591.10381∑h=1LWhρh182.7497173.7173182.4081131.1713181.8815180.342295.68471131.166161.114392.12831∑h=1LWhCxh181.9698172.9086181.6336130.2683181.1066179.550594.70196130.261160.324491.2563∑h=1LWhCxh∑h=1LWhβ2xh180.2450171.0819179.9328128.2003179.4204177.779792.33457128.1317158.612589.5657∑h=1LWhβ1xh∑h=1LWhCxh181.5270172.4999181.1971130.0811180.6785179.11294.72428130.0791160.074891.4022∑h=1LWhβ1xh∑h=1LWhρh181.8389172.8398181.4947130.4089180.9661179.444295.01959130.4074160.22191.37371∑h=1LWhβ2xh181.4267172.4007181.0958129.978180.5776179.012794.42132129.8881159.776191.26281∑h=1LWhβ1xh182.9171173.8882182.5766131.3816182.0538180.508295.89918131.3617161.305592.4328∑h=1LWhTMxh1182.8072173.7227182.4678130.9729181.9436180.382795.33219130.9729161.142491.8843∑h=1LWhTMxh∑h=1LWhQRxh182.8017173.7682182.4597131.2176181.9338180.393095.71564131.208161.165992.1900∑h=1LWhQAxh∑h=1LWhSxh182.4934173.4168182.1482130.6215181.6146180.078994.8953130.6059160.659491.2757∑h=1LWhHLxh∑h=1LWhρh182.5333173.444182.1942130.6498181.6645180.108494.9620130.6483160.808891.4787∑h=1LWhHLxh∑h=1LWhQRxh181.6013172.2849181.7109132.9205181.5076178.668894.22969129.8961160.2185105.7468∑h=1LWhMRxh∑h=1LWhQRxh181.5821172.5648181.2344129.9974180.6984179.187694.48528129.9965159.830690.7320∑h=1LWhMRxh∑h=1LWhQDxh181.0637171.9921180.7301129.3164180.2037178.642793.73929129.316159.423890.3161Bold values indicate minimum MSE.Table 8
Minimum MSEs of different estimators based on the simulation study (n=250).
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P10155.6986130.9425130.9435125.183187.54561130.5094121.2922130.093280.34671∑h=1LWhρh155.6335130.9868130.9878125.240487.70134130.5545121.3769130.1480.52041∑h=1LWhCxh155.188130.6177130.6186124.872987.3303130.1873120.9742129.774480.1478∑h=1LWhCxh∑h=1LWhβ2xh155.2285130.7593130.7602125.026887.56625130.3357121.0766129.930880.3851∑h=1LWhβ1xh∑h=1LWhCxh155.3389130.8714130.8724125.113787.48645130.4407121.2061130.028180.2890∑h=1LWhβ1xh∑h=1LWhρh155.6699131.0058131.0068125.24487.60117130.572121.3724130.156180.40111∑h=1LWhβ2xh155.0061130.2404130.2413124.523587.1653129.8145120.6219129.406579.99851∑h=1LWhβ1xh155.2016130.5104130.5113124.772887.28445130.0795120.8933129.665980.1095∑h=1LWhTMxh1155.5360131.0069131.0078125.257387.69095130.5764121.3634130.163880.5060∑h=1LWhTMxh∑h=1LWhQRxh155.8067131.277131.278125.570188.33712130.8469121.7497130.434981.2152∑h=1LWhQAxh∑h=1LWhSxh154.6152130.1677130.1686124.459787.16355129.7406120.5834129.331180.0265∑h=1LWhHLxh∑h=1LWhρh155.7181130.6542130.6551124.906387.34791130.2172121.0742129.796680.1639∑h=1LWhHLxh∑h=1LWhQRxh155.0446130.6161130.6169125.835695.03376130.4127120.9906130.242488.5004∑h=1LWhMRxh∑h=1LWhQRxh155.8021131.0947131.0957125.36587.96272130.6616121.5326130.245880.8098∑h=1LWhMRxh∑h=1LWhQDxh154.8427130.0278130.0287124.295686.82202129.5947120.4377129.17879.6540Bold values indicate minimum MSE.Table 9
Minimum MSEs of different estimators based on the simulation study (n=250).
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P10130.9376126.3729130.671104.1672130.317129.810885.72231104.1672118.259880.34671∑h=1LWhρh130.9843126.4391130.7156104.3089130.3628129.864485.91894104.3061118.326480.52041∑h=1LWhCxh130.612126.0493130.3477103.8761129.9964129.483885.45359103.8723117.971980.1478∑h=1LWhCxh∑h=1LWhβ2xh130.751126.1653130.4933103.9616130.1484129.6185.45669103.9258118.112580.3851∑h=1LWhβ1xh∑h=1LWhCxh130.8666126.2933130.6011104.0666130.2499129.736385.6053104.0656118.20780.2890∑h=1LWhβ1xh∑h=1LWhρh131.0036126.4477130.7337104.2572130.3797129.881885.81913104.2564118.311680.40111∑h=1LWhβ2xh130.2341125.6814129.9731103.5868129.6259129.105885.13934103.5396117.570279.99851∑h=1LWhβ1xh130.5056125.9563130.2402103.8306129.8883129.38285.4301103.8202117.857780.1095∑h=1LWhTMxh1131.0025126.4398130.7368104.2592130.3856129.875285.83846104.2592118.362380.5060∑h=1LWhTMxh∑h=1LWhQRxh131.2748126.7694131.0072104.829130.6564130.163486.59144104.824118.714581.2152∑h=1LWhQAxh∑h=1LWhSxh130.162125.6269129.8997103.5951129.5513129.039985.28246103.587117.599880.0265∑h=1LWhHLxh∑h=1LWhρh130.6518126.1239130.3805104.0301130.0228129.5485.66646104.0293117.957680.1639∑h=1LWhHLxh∑h=1LWhQRxh130.5198125.8475130.4821105.5124130.3329129.13585.57302103.9372118.026688.5004∑h=1LWhMRxh∑h=1LWhQRxh131.0921126.571130.8231104.5418130.4693129.978386.23798104.5414118.474880.8098∑h=1LWhMRxh∑h=1LWhQDxh130.0233125.4884129.7565103.4035129.4021128.906385.05503103.4033117.383179.6540Bold values indicate minimum MSE.Table 10
Minimum MSEs of different estimators based on the simulation study (n=350).
astbstY¯^KY¯^KKY¯^SGY¯^HS1Y¯^HS2Y¯^SS1RY¯^SS1PY¯^SS1RPY¯^P1095.614780.916580.916778.778364.722380.754977.433980.602362.03821∑h=1LWhρh95.797081.178181.178379.041965.010181.016877.703080.864962.33041∑h=1LWhCxh95.838181.208781.208979.072265.036981.047577.731980.895762.3564∑h=1LWhCxh∑h=1LWhβ2xh95.932481.325681.325879.199165.237481.166677.854081.017362.5658∑h=1LWhβ1xh∑h=1LWhCxh95.439080.888080.888378.758264.763280.727777.417080.576662.0907∑h=1LWhβ1xh∑h=1LWhρh95.574680.998280.998478.867264.865680.837577.527980.686162.19201∑h=1LWhβ2xh95.469780.930480.930678.816764.935880.773177.470780.625662.27761∑h=1LWhβ1xh95.225980.789980.790178.660264.662480.631077.302680.481561.9877∑h=1LWhTMxh195.449180.978480.978678.843564.815380.818377.493180.667662.1367∑h=1LWhTMxh∑h=1LWhQRxh95.470880.942780.942978.807964.778480.782577.458080.631762.0986∑h=1LWhQAxh∑h=1LWhSxh95.849381.213981.214179.075665.026181.053077.729280.901562.3421∑h=1LWhHLxh∑h=1LWhρh95.261880.654980.655178.521764.496480.494277.174880.342661.8179∑h=1LWhHLxh∑h=1LWhQRxh95.335180.871880.872079.098467.626180.797177.413280.736965.1968∑h=1LWhMRxh∑h=1LWhQRxh95.349080.772080.772278.645264.670880.611477.310380.460162.0023∑h=1LWhMRxh∑h=1LWhQDxh95.429580.922880.923078.799964.858080.762877.467180.612262.1960Bold values indicate minimum MSE.Table 11
Minimum MSEs of different estimators based on the simulation study (n=350).
astbstY¯^SS2RY¯^SS2PY¯^SS2RRY¯^SS2PPY¯^SS2RPY¯^SS2PRY¯^SS31Y¯^SS32Y¯^SS33Y¯^P1080.923479.272580.814571.253480.684280.523464.660871.253476.176662.03821∑h=1LWhρh81.185679.538281.076371.533980.946480.786664.950871.532976.445062.33041∑h=1LWhCxh81.216279.567981.106971.560780.977180.816964.974771.559376.474362.3564∑h=1LWhCxh∑h=1LWhβ2xh81.332779.685981.225171.703781.097480.932065.116371.690876.599262.5658∑h=1LWhβ1xh∑h=1LWhCxh80.895179.249380.786871.259880.657680.495964.691571.259576.172462.0907∑h=1LWhβ1xh∑h=1LWhρh81.005479.360280.896771.370080.767380.606664.800971.369876.278962.19201∑h=1LWhβ2xh80.937079.295280.830971.348980.704780.536664.784171.332076.227262.27761∑h=1LWhβ1xh80.796179.141580.689571.130280.561780.393064.539671.126476.076761.9877∑h=1LWhTMxh180.985379.332380.877271.315480.748480.583764.726471.315476.257562.1367∑h=1LWhTMxh∑h=1LWhQRxh80.949679.296980.841571.280780.712680.548364.688171.278876.215962.0986∑h=1LWhQAxh∑h=1LWhSxh81.221279.568981.112371.547880.982880.820464.947971.544976.474162.3421∑h=1LWhHLxh∑h=1LWhρh80.661679.011580.553571.002880.423980.261364.419171.002575.928661.8179∑h=1LWhHLxh∑h=1LWhQRxh80.848079.164580.821971.842680.768780.356564.731571.276976.172565.1968∑h=1LWhMRxh∑h=1LWhQRxh80.779379.138380.670771.166180.541380.381864.611671.165976.060462.0023∑h=1LWhMRxh∑h=1LWhQDxh80.930179.291780.821871.335680.693080.532864.794971.335576.225162.1960Bold values indicate minimum MSE.


## 7. Discussion

In this study, we proposed a new optimal family of estimators for estimating population mean under stratified random sampling. Bias, MSE, and minimum MSE of this family of estimators are derived up to first degree of approximation. The proposed family is compared with some well-known estimators/classes of estimators under stratified random sampling such as the works of Koyuncu and Kadilar [3, 4], Shabbir and Gupta [8], Haq and Shabbir [9], Singh and Solanki [19], and Solanki and Singh [10, 12]. It is numerically inferred that the proposed family behaves optimal as compared to other estimators. A simulation study is also carried out in support of efficient proposed estimators. So, to get more enhanced results in practice under stratified random sampling, our suggested family of estimators is recommended.The possible extensions of this work are to estimate the (1) finite population mean using robust quantile regression and L-moments characteristics of an auxiliary information under stratified ranked set sampling, (2) finite population parameters including median, variance, and proportions using L-moments under different sampling designs, and (3) population mean in the presence of nonsampling errors using L-moments and calibration approach.



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*Source: 9038126-2021-12-20.xml*