# Efficient Estimators of Finite Population Mean Based on Extreme Values in Simple Random Sampling

Authors: Anum Iftikhar; Hongbo Shi; Saddam Hussain; Mohsin Abbas; Kalim Ullah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The use of extreme values of the auxiliary variable is sometimes more beneficial to get the high efficiency of the estimators, and the study variable can have a correlation with the rank of the decently correlated auxiliary variable. As a result, it can be regarded as additional data for the study variable that can be used to improve the estimators’ efficiency. When the knowledge of the minimum and maximum values, as well as the rankings of the auxiliary variable, is known, various better estimators for calculating the finite population mean of the research variable based on extreme values under simple random sampling are proposed in this paper. The suggested estimators’ bias and mean squared error expressions are derived using first-order approximation. The recommended estimators have been compared mathematically to the current estimators. The suggested estimators are more exact in terms of relative efficiency than the other estimators addressed here, as shown by simulation and real datasets used to demonstrate the estimation of a limited population mean based on extreme values.


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## Body

## 1. Introduction

The purpose of survey sampling is to utilize the maximum amount of information about the characteristics of interest. Many fields of study require estimation of the finite population mean for a variable of interest. For example, average wheat production per acre, average income of households, mean weight of meat producing animals, etc. The mean per unit estimator is a base line estimator to estimate the finite population mean.Unusual observations can occur in sample survey data. The mean estimator is sensitive to very large and/or small values if included in the sample. It can provide biased results and, ultimately, tempted to delete the sample data. However, generally, when there are extreme values in the data, the efficiency of classical estimators declines. (for more details, see [1] and the reference cited therein).The use of supplementary information to enhance the precision of an estimator is a typical strategy in survey sampling. To improve their relative efficiency, the ratio, regression, and product-type estimators all require supplemental information on one or more auxiliary variables in addition to the information on the study variable. For example, when estimating the total household income, the household members and total expenditure may be used as two auxiliary variables. A significant amount of research work has been done to develop new and improved estimators of the population parameters, which include the population mean, total, CDF, median, etc. (for more details, see [2–9] and the references cited therein). To the best of our knowledge, [9–13] have done some recent work on the estimation of finite population mean using auxiliary information. However, because classical estimators are sensitive to extreme values, the outlier problem, which is the presence of extreme values in data, reduces efficiency (see [1] and the reference cited therein).When there are extreme values in the data, the efficiency of a classical ratio/product-type estimator declines in terms of relative efficiency (RE). Similarly, the regression-type estimators do not perform well in the presence of outliers as it is a well-known phenomenon that ordinary least square (OLS) estimators are sensitive in the presence of outliers. However, extreme values, if known, can be retained in the data and used as auxiliary information to increase the precision of the estimate (for more details, see [6, 9, 14–17] and the references cited therein, to name a few).In this study, we used extreme values of the auxiliary variable as auxiliary information and retained it in the data and suggested improved ratio/product-type estimators. On the lines of [17, 18] and 20 in simple random sampling (SRS), we introduced an improved class of estimators for predicting the finite population mean based on extreme values, using the lowest and maximum values of the auxiliary variable as auxiliary information.The rest of the study is as follows: in Section2, the methodology and notation of the study are described. In Section 3, existing estimators are discussed. In Section 4, we briefly discussed our proposed estimators. Section 5 and Section 6 contain the mathematical and numerical comparison. Finally, in Section 7, we discussed the main findings and concluded the study.


## 2. Methodology and Notation

Let us consider a finite populationδ=δ1,δ2,…,δN of size N. The values of the study variable Y and the auxiliary variable X, respectively, are yi and xi. Let ri be the values of the auxiliary variable R’s corresponding rankings for the i th i=1,2,3,…,N units. We use SRS without replacement to choose a sample of size n units from the population δ. Let Y¯=∑i=1Nyi/N, X¯=∑i=1Nxi/N, and R¯=∑i=1NRi/N be the population mean of study variable, auxiliary variable and the ranks of the auxiliary variables, respectively. It is further assumed that Sy2=∑i=1Nyi−Y¯2/N−1, Sx2=∑i=1Nxi−X¯2/N−1, and Sr2=∑i=1Nri−R¯2/N−1 be the corresponding population variances of Y, X and the ranks of the auxiliary variable R, respectively.LetCy=Sy/Y¯, Cx=Sx/X¯, and Cr=Sr/R¯ be the population coefficients of variation of the study variable, auxiliary variable, and the ranks of the auxiliary variable, respectively. ρyx, ρyr, and ρxr are the population correlation coefficients between the subscripts.Lety¯=∑i=1nyi/n, x˙=∑i=1nxi/n, and r¯=∑i=1nri/n be the sample means and S^y2=∑i=1nyi−y¯2/n−1, S^x2=∑i=1nxi−x¯2/n−1, and S^r2=∑i=1nri−r¯2/n−1 be the sample variance of the study variable, auxiliary variable, and the ranks of the auxiliary variable, respectively.We may use the following relative error terms to determine the biases and MSEs of the existing and proposed class ofY¯ estimators. Let(1)ξ0=y¯−Y¯Y¯,ξ1=x¯−X¯X¯,ξ2=r¯−R¯R¯,such that Eξ0=Eξ1=Eξ2=0.


## 3. Existing Estimator

In this section, we define the existing estimators of finite population means, which are to be compared with our proposed estimator.


### 3.1. Usual Unbiased Estimator

The unbiased estimator of a finite population mean with variance that is most commonly used is(2)y¯=1n∑i=1nyi,(3)Vary¯=θY¯2Cy2,respectively.


### 3.2. Cochran’s Ratio Estimator

Cochran [2] recommended a ratio type estimator by first estimating the finite population mean in SRS, which is obtained by employing auxiliary information.(4)y¯R=y¯X¯x¯.Mathematical expression upto the first-order of approximation for the bias and MSE ofy¯R is given by(5)Biasy¯R≈θY¯Cx2−ρyxCyCx,(6)MSEy¯R≈θY¯2Cy2+Cx2−2ρyxCyCx,respectively.


### 3.3. Classical Regression Estimator

The classical regression estimator forY¯ under SRS is given by(7)y¯lr=y¯−byxx¯−X¯,where byx=Syx/Sx2 is the regression coefficient between yi and xi for i=1,2,…,n. The MSE of y¯lr upto the first order of approximation is given as under(8)MSEy¯lr≈θSy21−ρyx2.


### 3.4. Mohanty and Sahoo Estimator

On the similar lines of [17, 18], as auxiliary information, we gave two finite population mean estimators based on the minimum and maximum values of the auxiliary variable as follows:(9)y¯t1=y¯V¯ν¯,y¯t2=y¯W¯w¯,respectively. Where ν=xi+xM/xM+xm and w=xi+xM/xM+xm. Here xm and xM are the minimum and maximum values, respectively.The expression for bias and MSE ofy¯t1 and y¯t2 are given by(10)Biasy¯t1≈θY¯t12Cx2−t1Cyx,(11)MSEy¯t1≈θY¯2Cy2−2t1ρyxCyCx+t12Cx2,(12)Biasy¯t2≈θY¯t22Cx2−t2Cyx,(13)MSEy¯t2≈θY¯2Cy2−2t2ρyxCyCx+t22Cx2,respectively. Where t1=X¯/X¯+XM and t2=X¯/X¯+XM


### 3.5. Walia et al. Estimator

Walia et al. [9] presented some estimators based on known knowledge about the auxiliary variable’s minimum and maximum values are provided. The following is the transformation:(14)Z=x+XMXm.The estimators of the finite population mean are listed as followsY¯:(15)y¯hm1=y¯Z¯z¯,y¯hm2=y¯Z¯+Czz¯+Cz,respectively. Where Cz=Sz/Z¯=Sx/X¯+XM+Xm and Sx2=Sz2.The bias and MSE of the above-modified estimatorsy¯hm1 and y¯hm2 are calculated as follows:(16)Biasy¯hm1≈θY¯C12Cx2−C1ρyxCyCx,(17)MSEy¯hm1≈θY¯2Cy2−2C1ρyxCyCx+C12Cx2,(18)Biasy¯hm2≈θY¯C22Cx2−C2ρyxCyCx,(19)MSEy¯hm2≈θY¯2Cy2−2C2ρyxCyCx+C22Cx2,respectively. Where C1=X¯/X¯+XM/Xm and C2=X¯X¯+XM/Xm/X¯+XM/Xm2+Sx.


## 3.1. Usual Unbiased Estimator

The unbiased estimator of a finite population mean with variance that is most commonly used is(2)y¯=1n∑i=1nyi,(3)Vary¯=θY¯2Cy2,respectively.


## 3.2. Cochran’s Ratio Estimator

Cochran [2] recommended a ratio type estimator by first estimating the finite population mean in SRS, which is obtained by employing auxiliary information.(4)y¯R=y¯X¯x¯.Mathematical expression upto the first-order of approximation for the bias and MSE ofy¯R is given by(5)Biasy¯R≈θY¯Cx2−ρyxCyCx,(6)MSEy¯R≈θY¯2Cy2+Cx2−2ρyxCyCx,respectively.


## 3.3. Classical Regression Estimator

The classical regression estimator forY¯ under SRS is given by(7)y¯lr=y¯−byxx¯−X¯,where byx=Syx/Sx2 is the regression coefficient between yi and xi for i=1,2,…,n. The MSE of y¯lr upto the first order of approximation is given as under(8)MSEy¯lr≈θSy21−ρyx2.


## 3.4. Mohanty and Sahoo Estimator

On the similar lines of [17, 18], as auxiliary information, we gave two finite population mean estimators based on the minimum and maximum values of the auxiliary variable as follows:(9)y¯t1=y¯V¯ν¯,y¯t2=y¯W¯w¯,respectively. Where ν=xi+xM/xM+xm and w=xi+xM/xM+xm. Here xm and xM are the minimum and maximum values, respectively.The expression for bias and MSE ofy¯t1 and y¯t2 are given by(10)Biasy¯t1≈θY¯t12Cx2−t1Cyx,(11)MSEy¯t1≈θY¯2Cy2−2t1ρyxCyCx+t12Cx2,(12)Biasy¯t2≈θY¯t22Cx2−t2Cyx,(13)MSEy¯t2≈θY¯2Cy2−2t2ρyxCyCx+t22Cx2,respectively. Where t1=X¯/X¯+XM and t2=X¯/X¯+XM


## 3.5. Walia et al. Estimator

Walia et al. [9] presented some estimators based on known knowledge about the auxiliary variable’s minimum and maximum values are provided. The following is the transformation:(14)Z=x+XMXm.The estimators of the finite population mean are listed as followsY¯:(15)y¯hm1=y¯Z¯z¯,y¯hm2=y¯Z¯+Czz¯+Cz,respectively. Where Cz=Sz/Z¯=Sx/X¯+XM+Xm and Sx2=Sz2.The bias and MSE of the above-modified estimatorsy¯hm1 and y¯hm2 are calculated as follows:(16)Biasy¯hm1≈θY¯C12Cx2−C1ρyxCyCx,(17)MSEy¯hm1≈θY¯2Cy2−2C1ρyxCyCx+C12Cx2,(18)Biasy¯hm2≈θY¯C22Cx2−C2ρyxCyCx,(19)MSEy¯hm2≈θY¯2Cy2−2C2ρyxCyCx+C22Cx2,respectively. Where C1=X¯/X¯+XM/Xm and C2=X¯X¯+XM/Xm/X¯+XM/Xm2+Sx.


## 4. Proposed Estimator

In this section, we develop two auxiliary information-based (AIB) classes of estimators, say ratio and exponential ratio, under the SRS technique, for calculating the mean of a finite populationY¯.


### 4.1. First Proposed Class of Estimator

We present a better class of estimators for estimatingY¯ under SRS utilizing known information about the auxiliary variable X’s lowest and maximum values, as motivated by [15]. The following is the improved class of estimator:(20)y¯Di=k1y¯X¯x¯α1+k2y¯X¯x¯α2expaU¯+b1−au¯+b1aU¯+b1+au¯+b1,where k1 and k2 are unknown constants whose values must be determined in order to calculate the bias and MSE of the y¯Di minimum and u¯=x¯+XM−Xm/XM+Xm. Further, a=1,b1=XM−Xm,b2=RM−Rm, and α1,α2 be the scalar quantities that may assume 0,−1,1 values. In addition, the sub-cases of the y¯Di are summarized in appendix (given in Table 1).Table 1
Subcases of the proposed estimator I.
Estimatorα1α2y¯D1=k1y¯+k2x¯/X¯L0−1y¯D2=k1y¯x¯/X¯+k2L−10y¯D3=k1y¯+k2X¯/x¯L01y¯D4=k1y¯X¯/x¯+k2L10y¯D5=k1y¯X¯/x¯+k2x¯/X¯L1−1y¯D6=k1y¯x¯/X¯+k2X¯/x¯L−11y¯D7=k1y¯X¯/x¯+k2X¯/x¯L11y¯D8=k1y¯X¯/x¯+k2x¯/X¯L−1−1In order to derive approximate mathematical expressions for the bias and MSE ofy¯Di, we can write y¯=Y¯1+ξ0 and x¯=X¯1+ξ1. Let us express the right-hand side (RHS) of (20) in terms of ξ’s to get(21)y¯Di=k1Y¯1+ξ01+ξ1−α1+k21+ξ1−α2exp−g1ξ121+g1ξ12−1,where g1=aX¯/aX¯+XM−Xm/XM+Xm+b1. Let us expand the RHS of equation (21) and retain terms up to 2nd power of ξ’s, we have(22)y¯Di−Y¯≈−Y¯+k1Y¯1+ξ0−ξ1α1+g12+ξ12α1g12+3g128+α1α1+12−ξ0ξ1α1+g12+k21−ξ1α2+g12+ξ12α2g12+3g128+α2α2+12.Let us take expectation on both sides of equation (22), which is provided by, to get the bias of y¯Di up to the first order of approximation(23)Biasy¯Di≈−Y¯+k1Y¯D+k2G,where(24)D=1+θCx2α1g12+3g128+α1α1+12−θCyxα1+g12,G=1+θCx2α2g12+3g128+α2α2+12.Taking square on both sides of equation (22) and then taking its expectation to get the MSE of y¯Di under first order of approximation, which is given by(25)MSEy¯Di≈Y¯2+Y¯2k12A+k22B−2Y¯2k1D−2Y¯2k2G+2Y¯k1k2F,where(26)A=1+θCy2+Cx2α1+g122+α1g1+3g128+α1α1+1−4Cyxα1+g12,B=1+θCx2α2+g122+α2g1+3g128+α2α2+1,F=1+θCx2α1+g12α2+g12+α1g1+3g128+α1α1+1+α2g1+3g128+α2α2+11+θ−Cyxα1+α2+g1+θCx2α1+g12α2+g12+α1g1+3g128+α1α1+1+α2g1+3g128+α2α2+1.By reducing equation (25) with regard to k1 and k2, the optimum values of k1 and k2 are determined(27)k1opt=BD−FGAB−F2,k2opt=Y¯AG−DFAB−F2.Substituting the optimum values ofk1 and k2 in equations (23) and (25), we get the minimum bias and MSE of y¯Di, respectively(28)Biasy¯Dimin≈−Y¯1−AB2+BD2−2DFGAB−F2,(29)MSEy¯Dimin≈Y¯21−AB2+BD2−2DFGAB−F2.


### 4.2. Second Proposed Estimator

On the similar lines of 6, we propose another improved class of exponential-type estimator for estimatingY¯ using supplementary information in terms of minimum and the maximum values of X under SRS scheme. The improved estimators are given by(30)y¯De=y¯expk3aU¯+b1−au¯+b1aU¯+b1+au¯+b1expk4aR¯+b2−ar¯+b2aR¯+b2+ar¯+b2,where k3 and k4 are unknown constants whose values must be chosen so that the biases and MSE of y¯Di are as small as possible and where k3 and k4 are unknown constants whose values must be set so that the biases and MSE of y¯Di are as little as feasible, and u¯=x¯+XM−Xm/XM+Xm and U¯=X¯+XM−Xm/XM+Xm. Further, a=1,b1=XM−Xm, and b2=RM−Rm be the known values.Let us express the RHS of equation (30) in terms of ξ’s to acquire the following approximate mathematical equations for the bias and MSE of y¯De as(31)y¯De=Y¯1+ξ0expk3−g1ξ121+g1ξ12−1expk4−g2ξ221+g2ξ22−1,where g2=aR¯/aR¯+RM−Rm/RM+Rm+b2. Using the Taylor series’ first-order approximation, we have(32)y¯De−Y¯≈ξ0−k3g1ξ12−k4g2ξ22+k3g124+k32g128ξ12+k4g224+k42g228ξ22k4g224+k42g228ξ22−k3g1ξ0ξ12−k4g2ξ0ξ22+k3k4g1g2ξ1ξ24.Simplifying and applying expectation on equation (32), we have the final expression of bias of y¯De, given by(33)Biasy¯De≈θY¯k3g124+α12g128Cx2+k4g224+α22g228Cr2−k3g12Cyx−k4g22Cyr+k3k4g1g22Cxr.By squaring and applying expectation on both sides of equation (32), we obtain the MSE up to first-order of approximation as(34)MSEy¯De≈θY¯2Cy2+k3g124Cx2+k4g224Cr2−k3g1Cyx−k4g2Cyr+k3k4g1g22Cxr.Thek3 and k4 optimum values are derived by minimizing the equation (34), respectively, given by(35)k3opt=2Cyg1Cxρyx−ρyrρxr1−ρxr2,k4opt=2Cyg2Crρyr−ρyxρxr1−ρxr2.We get the minimal bias and MSE ofy¯De substituting the best values for k3 and k3 in equations (33) and (34), respectively.(36)Biasy¯Demin≈θY¯Cyg1Cxρyx−ρyrρxr+g2Crρyr−ρyxρxr1−ρxr2−Ry,xr2,where Ry,xr2=ρyx2+ρxr2−2ρyxρxrρyr/1−ρyx2 is the coefficient of multiple determination of Y on X and R.(37)MSEy¯Demin≈θY¯2Cy21−Ry,xr2.


## 4.1. First Proposed Class of Estimator

We present a better class of estimators for estimatingY¯ under SRS utilizing known information about the auxiliary variable X’s lowest and maximum values, as motivated by [15]. The following is the improved class of estimator:(20)y¯Di=k1y¯X¯x¯α1+k2y¯X¯x¯α2expaU¯+b1−au¯+b1aU¯+b1+au¯+b1,where k1 and k2 are unknown constants whose values must be determined in order to calculate the bias and MSE of the y¯Di minimum and u¯=x¯+XM−Xm/XM+Xm. Further, a=1,b1=XM−Xm,b2=RM−Rm, and α1,α2 be the scalar quantities that may assume 0,−1,1 values. In addition, the sub-cases of the y¯Di are summarized in appendix (given in Table 1).Table 1
Subcases of the proposed estimator I.
Estimatorα1α2y¯D1=k1y¯+k2x¯/X¯L0−1y¯D2=k1y¯x¯/X¯+k2L−10y¯D3=k1y¯+k2X¯/x¯L01y¯D4=k1y¯X¯/x¯+k2L10y¯D5=k1y¯X¯/x¯+k2x¯/X¯L1−1y¯D6=k1y¯x¯/X¯+k2X¯/x¯L−11y¯D7=k1y¯X¯/x¯+k2X¯/x¯L11y¯D8=k1y¯X¯/x¯+k2x¯/X¯L−1−1In order to derive approximate mathematical expressions for the bias and MSE ofy¯Di, we can write y¯=Y¯1+ξ0 and x¯=X¯1+ξ1. Let us express the right-hand side (RHS) of (20) in terms of ξ’s to get(21)y¯Di=k1Y¯1+ξ01+ξ1−α1+k21+ξ1−α2exp−g1ξ121+g1ξ12−1,where g1=aX¯/aX¯+XM−Xm/XM+Xm+b1. Let us expand the RHS of equation (21) and retain terms up to 2nd power of ξ’s, we have(22)y¯Di−Y¯≈−Y¯+k1Y¯1+ξ0−ξ1α1+g12+ξ12α1g12+3g128+α1α1+12−ξ0ξ1α1+g12+k21−ξ1α2+g12+ξ12α2g12+3g128+α2α2+12.Let us take expectation on both sides of equation (22), which is provided by, to get the bias of y¯Di up to the first order of approximation(23)Biasy¯Di≈−Y¯+k1Y¯D+k2G,where(24)D=1+θCx2α1g12+3g128+α1α1+12−θCyxα1+g12,G=1+θCx2α2g12+3g128+α2α2+12.Taking square on both sides of equation (22) and then taking its expectation to get the MSE of y¯Di under first order of approximation, which is given by(25)MSEy¯Di≈Y¯2+Y¯2k12A+k22B−2Y¯2k1D−2Y¯2k2G+2Y¯k1k2F,where(26)A=1+θCy2+Cx2α1+g122+α1g1+3g128+α1α1+1−4Cyxα1+g12,B=1+θCx2α2+g122+α2g1+3g128+α2α2+1,F=1+θCx2α1+g12α2+g12+α1g1+3g128+α1α1+1+α2g1+3g128+α2α2+11+θ−Cyxα1+α2+g1+θCx2α1+g12α2+g12+α1g1+3g128+α1α1+1+α2g1+3g128+α2α2+1.By reducing equation (25) with regard to k1 and k2, the optimum values of k1 and k2 are determined(27)k1opt=BD−FGAB−F2,k2opt=Y¯AG−DFAB−F2.Substituting the optimum values ofk1 and k2 in equations (23) and (25), we get the minimum bias and MSE of y¯Di, respectively(28)Biasy¯Dimin≈−Y¯1−AB2+BD2−2DFGAB−F2,(29)MSEy¯Dimin≈Y¯21−AB2+BD2−2DFGAB−F2.


## 4.2. Second Proposed Estimator

On the similar lines of 6, we propose another improved class of exponential-type estimator for estimatingY¯ using supplementary information in terms of minimum and the maximum values of X under SRS scheme. The improved estimators are given by(30)y¯De=y¯expk3aU¯+b1−au¯+b1aU¯+b1+au¯+b1expk4aR¯+b2−ar¯+b2aR¯+b2+ar¯+b2,where k3 and k4 are unknown constants whose values must be chosen so that the biases and MSE of y¯Di are as small as possible and where k3 and k4 are unknown constants whose values must be set so that the biases and MSE of y¯Di are as little as feasible, and u¯=x¯+XM−Xm/XM+Xm and U¯=X¯+XM−Xm/XM+Xm. Further, a=1,b1=XM−Xm, and b2=RM−Rm be the known values.Let us express the RHS of equation (30) in terms of ξ’s to acquire the following approximate mathematical equations for the bias and MSE of y¯De as(31)y¯De=Y¯1+ξ0expk3−g1ξ121+g1ξ12−1expk4−g2ξ221+g2ξ22−1,where g2=aR¯/aR¯+RM−Rm/RM+Rm+b2. Using the Taylor series’ first-order approximation, we have(32)y¯De−Y¯≈ξ0−k3g1ξ12−k4g2ξ22+k3g124+k32g128ξ12+k4g224+k42g228ξ22k4g224+k42g228ξ22−k3g1ξ0ξ12−k4g2ξ0ξ22+k3k4g1g2ξ1ξ24.Simplifying and applying expectation on equation (32), we have the final expression of bias of y¯De, given by(33)Biasy¯De≈θY¯k3g124+α12g128Cx2+k4g224+α22g228Cr2−k3g12Cyx−k4g22Cyr+k3k4g1g22Cxr.By squaring and applying expectation on both sides of equation (32), we obtain the MSE up to first-order of approximation as(34)MSEy¯De≈θY¯2Cy2+k3g124Cx2+k4g224Cr2−k3g1Cyx−k4g2Cyr+k3k4g1g22Cxr.Thek3 and k4 optimum values are derived by minimizing the equation (34), respectively, given by(35)k3opt=2Cyg1Cxρyx−ρyrρxr1−ρxr2,k4opt=2Cyg2Crρyr−ρyxρxr1−ρxr2.We get the minimal bias and MSE ofy¯De substituting the best values for k3 and k3 in equations (33) and (34), respectively.(36)Biasy¯Demin≈θY¯Cyg1Cxρyx−ρyrρxr+g2Crρyr−ρyxρxr1−ρxr2−Ry,xr2,where Ry,xr2=ρyx2+ρxr2−2ρyxρxrρyr/1−ρyx2 is the coefficient of multiple determination of Y on X and R.(37)MSEy¯Demin≈θY¯2Cy21−Ry,xr2.


## 5. Mathematical Comparison

We compared the proposed estimators mathematically to the existing estimator in Section3 in this section.


### 5.1. First Proposed Estimator

Condition 1.
From equations (2) and (28)(38)Vary¯>MSEy¯DiminifonlyθCy2+BD2+AG2−2DFGAB−F2>1.Condition 2.
From equations (5) and (28)(39)MSEy¯R>MSEy¯Diminif,onlyθCy2+Cx2−2Cxy+BD2+AG2−2DFGAB−F2>1.Condition 3.
From equations (7) and (28)(40)MSEy¯lrmin>MSEy¯Diminif,onlyθCy21−ρyx2+BD2+AG2−2DFGAB−F2>1.Condition 4.
From equations (11) and (28)(41)MSEy¯t1>MSEy¯DiminifonlyθCy2+t12Cx2−2t1Cyx+BD2+AG2−2DFGAB−F2>1.Condition 5.
From equations (13) and (28)(42)MSEy¯t2>MSEy¯DiminifonlyθCy2+t22Cx2−2t2Cyx+BD2+AG2−2DFGAB−F2>1.Condition 6.
From equations (18) and (28)(43)MSEy¯hm1>MSEy¯Diminif,onlyθCy2+C12Cx2−2C1Cyx+BD2+AG2−2DFGAB−F2>1.Condition 7.
From equations (20) and (28)(44)MSEy¯hm2>MSEy¯Diminif,onlyθCy2+C22Cx2−2C2Cyx+BD2+AG2−2DFGAB−F2>1.


### 5.2. Second Proposed Estimator

Condition 8.
From equations (2) and (36)(45)Vary¯>MSEy¯Diminif,onlyRy,xr2>0.Condition 9.
From equations (5) and (36)(46)MSEy¯R>MSEy¯Deminif,onlyRy,xr2+Cx2−2Cyx>1.Condition 10.
From equations (7) and (36)(47)MSEy¯lrmin>MSEy¯Deminif,onlyRy,xr2−ρyx2>0.Condition 11.
From equations (11) and (36)(48)MSEy¯t1>MSEy¯Deminif,onlyCy2Ry,xr2+t12Cx2−2t1Cyx>0.Condition 12.
From equations (13) and (36)(49)MSEy¯t2>MSEy¯Deminif,onlyCy2Ry,xr2+t22Cx2−2t2Cyx>0.Condition 13.
From equations (18) and (36)(50)MSEy¯hm1>MSEy¯Deminif,onlyCy2Ry,xr2+C12Cx2−2ρyxC1CyCx>0.Condition 14.
From equations (20) and (36)(51)MSEy¯hm2>MSEy¯Deminif,onlyCy2Ry,xr2+C22Cx2−2C2Cyx>0.


## 5.1. First Proposed Estimator

Condition 1.
From equations (2) and (28)(38)Vary¯>MSEy¯DiminifonlyθCy2+BD2+AG2−2DFGAB−F2>1.Condition 2.
From equations (5) and (28)(39)MSEy¯R>MSEy¯Diminif,onlyθCy2+Cx2−2Cxy+BD2+AG2−2DFGAB−F2>1.Condition 3.
From equations (7) and (28)(40)MSEy¯lrmin>MSEy¯Diminif,onlyθCy21−ρyx2+BD2+AG2−2DFGAB−F2>1.Condition 4.
From equations (11) and (28)(41)MSEy¯t1>MSEy¯DiminifonlyθCy2+t12Cx2−2t1Cyx+BD2+AG2−2DFGAB−F2>1.Condition 5.
From equations (13) and (28)(42)MSEy¯t2>MSEy¯DiminifonlyθCy2+t22Cx2−2t2Cyx+BD2+AG2−2DFGAB−F2>1.Condition 6.
From equations (18) and (28)(43)MSEy¯hm1>MSEy¯Diminif,onlyθCy2+C12Cx2−2C1Cyx+BD2+AG2−2DFGAB−F2>1.Condition 7.
From equations (20) and (28)(44)MSEy¯hm2>MSEy¯Diminif,onlyθCy2+C22Cx2−2C2Cyx+BD2+AG2−2DFGAB−F2>1.


## 5.2. Second Proposed Estimator

Condition 8.
From equations (2) and (36)(45)Vary¯>MSEy¯Diminif,onlyRy,xr2>0.Condition 9.
From equations (5) and (36)(46)MSEy¯R>MSEy¯Deminif,onlyRy,xr2+Cx2−2Cyx>1.Condition 10.
From equations (7) and (36)(47)MSEy¯lrmin>MSEy¯Deminif,onlyRy,xr2−ρyx2>0.Condition 11.
From equations (11) and (36)(48)MSEy¯t1>MSEy¯Deminif,onlyCy2Ry,xr2+t12Cx2−2t1Cyx>0.Condition 12.
From equations (13) and (36)(49)MSEy¯t2>MSEy¯Deminif,onlyCy2Ry,xr2+t22Cx2−2t2Cyx>0.Condition 13.
From equations (18) and (36)(50)MSEy¯hm1>MSEy¯Deminif,onlyCy2Ry,xr2+C12Cx2−2ρyxC1CyCx>0.Condition 14.
From equations (20) and (36)(51)MSEy¯hm2>MSEy¯Deminif,onlyCy2Ry,xr2+C22Cx2−2C2Cyx>0.


## 6. Numerical Comparison

In this section, simulated and real datasets are considered, and the percentage relative efficiencies (PREs) of the proposed estimator are computed.


### 6.1. Study of Simulation

We undertake simulation research using the notion from [19] to compare the performance of our recommended estimators to the comparable current estimators. We used the following distributions to construct six datasets of size N=10,000 for the auxiliary variable X:(52)Population1:X∼Exponentialλ=2,Population2:X∼Exponentialλ=5,Population3:X∼Uniformb2=0,b3=1,Population4:X∼Uniformb2=0,b3=2,Population5:X∼Gammaα3=1,α4=2,Population6:X∼Gammaα3=2,α4=4,and the study variable Y by(53)Y=ryx2×X+ε,where ryx2 is the sample correlation coefficient between study and auxiliary variables, and ε is the random error term, which has μ=0 and σ2=1 and follows a conventional normal distribution.We considered the following steps inRStudio to get the results of mentioned estimators in this study:Step 1: the first step was to create different populations (as an auxiliary variable) of 1000 units using the different distributions, and then Y is computed using the model given in equation (53).Step 2: the unknown constants’ optimal values for the suggested estimators are obtained using the datasets computed inStep1.Step 3: we use SRS without replacement to draw a sample of sizen=250 and calculate Var/MSE for all the estimators covered in this research.Step 4: under the same environment, the variances and MSE of the mean estimators are computed by drawing 50 thousand samples from each population under SRS given inStep3. The variances/MSE of the proposed and existing estimators based on SRS are approximated by using the following formulae:(54)Vary¯≈1ψ−1∑t=1ψy¯t−Y¯2,MSEy¯Di≈1ψ−1∑t=1ψy¯Di,t−Y¯2,MSEy¯De≈1ψ−1∑t=1ψy¯De,t−Y¯2,where ψ=50,000 and i=1,2,…,8. On similar lines, the MSE of other estimators y¯R, y¯lr, y¯t1, etc., given in Section 3, are obtained. The PRE of MSEy¯Di with respect to Vary¯ is given by(55)PREMSEy¯Di,Vary¯=Vary¯MSEy¯Di×100.On similar lines, the PREs of the other estimator based on SRS may be computed. The REs of these proposed and existing estimators are reported in Table2. It can be seen that the proposed estimators are more efficient than usual unbiased estimators and existing estimators as well in terms of PRE, i.e., all values of the PREs are greater than a hundred. The effect of increasing the number of sample size n is precluded. However, generally, with an increase in the sample size, the PREs tend to increase and vice versa.Table 2
PREs of proposed class of estimators using populations 1–6.
EstimatorPopulation 1Population 2Population 3Population 4Population 5Population 6y¯100100100100100100y¯R146.7758101.3152103.5414129.8623120.5604107.7454y¯lr155.5963104.1005104.9525131.1766122.7052108.5657y¯t1154.7504195.4096116.0315134.3762121.4452106.4048y¯t2144.7504165.8892114.0315132.6784119.0782109.4256y¯hm1100.9828104.1228108.2917122.7292124.5897112.4089y¯hm2100.1832103.7872107.8894121.2048123.1645110.5284y¯D15158.9673002.2021619.323519.6661625.71291095.498y¯D2251313.9143590.043483.9124482.3844110.7247409.46y¯D33932.3871880.8681023.353605.6147710.3697910.2687y¯D4210009.3133592.932736.9210462.9129291.2835035.28y¯D54822.2393126.5541735.952673.8917695.28911201.333y¯D64815.0772162.5041343.679983.9214989.94921259.309y¯D73233.5231708.126810.6883330.1844439.2591668.9487y¯D86050.2813023.4011723.625660.7379761.13241211.036y¯De228.7653170.0789130.9133207.0891133.9556361.1317


### 6.2. Real-Life Data

We used three real datasets to compare the PREs of all these estimators to see how well they performed compared to the comparable existing estimators. These datasets’ descriptions and summary statistics are listed as follows.


#### 6.2.1. Population Ι

This dataset is taken from [20] page 226 and was conducted in Pakistan during the year 2012, which comprised 33 divisions. This dataset may be downloaded from the Pakistan Bureau of Statistics web page via the link: https://www.pbs.gov.pk/content/microdata. The study variable Y corresponds to the employment level by divisions in 2012 and the number of registered factories in 2012, respectively, while R corresponds to the rank number of registered factories in 2012. Here, our objective is to estimate the finite population mean under extreme values in SRS. The population constants are N=33;n=10;Y¯=27.4909;X¯=72.5455;R¯=17;Sy=10.1308;Sx=10.5770;Sr=9.638;XM=95;Xm=58;RM=33;Rm=1.5;Cy=0.3685;Cx=0.1459;Cr=0.5669;g1=0.6608;g2=0.1151;ρyx=0.2522;ρyr=0.1950;ρxr=0.9810;andθ=0.0697


#### 6.2.2. Population ΙΙ

Another dataset is taken from [20] page 135, conducted in Pakistan during the year 2012, which comprised 33 divisions. This dataset may be downloaded from the Pakistan Bureau of Statistics web page via the link: https://www.pbs.gov.pk/content/microdata. The number of pupils enrolled in each division and the total number of government primary and secondary schools for boys and girls in each division are the research variables Y and X in 2012, respectively, and R correspond to the rank of auxiliary variable X. Here, our objective is to estimate the finite population mean under extreme values in SRS. The population constants are N=36;n=10;Y¯=148718.7;X¯=1054.39;R¯=18.5;Sy=182315.1;Sx=402.6098;Sr=10.535;XM=2370;Xm=388;RM=36;Rm=1;Cy=1.2259;Cx=0.3818;Cr=0.5695;g1=0.3472;g2=0.3498;ρyx=0.1799;ρyr=0.188;ρxr=0.9378;andθ=0.0722


#### 6.2.3. Population ΙΙΙ

This dataset is taken from [2], which comprises 36 units of food cost and weekly income of families. The study variable Y and auxiliary variable X are the food cost of families’ employment and weekly income of families, respectively, and R correspond to the rank of weekly income of families. For more detail, we can refer to [2] page 24. To estimate the finite population mean under extreme values, population constants are N=36;n=10;Y¯=148718.7;X¯=1054.39;R¯=18.5;Sy=182315.1;Sx=402.6098;Sr=10.535;XM=2370;Xm=388;RM=36;Rm=1;Cy=1.2259;Cx=0.3818;Cr=0.5695;g1=0.3472;g2=0.3498;ρyx=0.1799;ρyr=0.188;ρxr=0.9378;andθ=0.0722.On the abovementioned datasets, the PREs of these proposed and current estimators are provided in Table3. In terms of PRE, it can be seen that the proposed estimators are more efficient than the standard unbiased estimator and existing estimators, i.e., all values are more than one hundred.Table 3
Using real datasets, PRE of the proposed class of estimators.
EstimatorPopulation IPopulation IIPopulation IIIy¯100100100y¯R117.6365101.5271104.4765y¯lr117.651103.3431106.7879y¯t1117.5989103.1005106.6716y¯t2104.1569102.5963106.0517y¯hm1116.9265101.5758104.7442y¯hm2116.9108101.5783104.7662y¯D1812.30341519.9681604.261y¯D2263419.446122.156313.68y¯D31460.4421004.19405.2918y¯D4139601.236976.194762.583y¯D55208.6031061.294603.8153y¯D62848.7161292.207765.8887y¯D71014.97806.8052382.3507y¯D8917.26861539.1841683.153y¯De119.3596118.1962115.8183


## 6.1. Study of Simulation

We undertake simulation research using the notion from [19] to compare the performance of our recommended estimators to the comparable current estimators. We used the following distributions to construct six datasets of size N=10,000 for the auxiliary variable X:(52)Population1:X∼Exponentialλ=2,Population2:X∼Exponentialλ=5,Population3:X∼Uniformb2=0,b3=1,Population4:X∼Uniformb2=0,b3=2,Population5:X∼Gammaα3=1,α4=2,Population6:X∼Gammaα3=2,α4=4,and the study variable Y by(53)Y=ryx2×X+ε,where ryx2 is the sample correlation coefficient between study and auxiliary variables, and ε is the random error term, which has μ=0 and σ2=1 and follows a conventional normal distribution.We considered the following steps inRStudio to get the results of mentioned estimators in this study:Step 1: the first step was to create different populations (as an auxiliary variable) of 1000 units using the different distributions, and then Y is computed using the model given in equation (53).Step 2: the unknown constants’ optimal values for the suggested estimators are obtained using the datasets computed inStep1.Step 3: we use SRS without replacement to draw a sample of sizen=250 and calculate Var/MSE for all the estimators covered in this research.Step 4: under the same environment, the variances and MSE of the mean estimators are computed by drawing 50 thousand samples from each population under SRS given inStep3. The variances/MSE of the proposed and existing estimators based on SRS are approximated by using the following formulae:(54)Vary¯≈1ψ−1∑t=1ψy¯t−Y¯2,MSEy¯Di≈1ψ−1∑t=1ψy¯Di,t−Y¯2,MSEy¯De≈1ψ−1∑t=1ψy¯De,t−Y¯2,where ψ=50,000 and i=1,2,…,8. On similar lines, the MSE of other estimators y¯R, y¯lr, y¯t1, etc., given in Section 3, are obtained. The PRE of MSEy¯Di with respect to Vary¯ is given by(55)PREMSEy¯Di,Vary¯=Vary¯MSEy¯Di×100.On similar lines, the PREs of the other estimator based on SRS may be computed. The REs of these proposed and existing estimators are reported in Table2. It can be seen that the proposed estimators are more efficient than usual unbiased estimators and existing estimators as well in terms of PRE, i.e., all values of the PREs are greater than a hundred. The effect of increasing the number of sample size n is precluded. However, generally, with an increase in the sample size, the PREs tend to increase and vice versa.Table 2
PREs of proposed class of estimators using populations 1–6.
EstimatorPopulation 1Population 2Population 3Population 4Population 5Population 6y¯100100100100100100y¯R146.7758101.3152103.5414129.8623120.5604107.7454y¯lr155.5963104.1005104.9525131.1766122.7052108.5657y¯t1154.7504195.4096116.0315134.3762121.4452106.4048y¯t2144.7504165.8892114.0315132.6784119.0782109.4256y¯hm1100.9828104.1228108.2917122.7292124.5897112.4089y¯hm2100.1832103.7872107.8894121.2048123.1645110.5284y¯D15158.9673002.2021619.323519.6661625.71291095.498y¯D2251313.9143590.043483.9124482.3844110.7247409.46y¯D33932.3871880.8681023.353605.6147710.3697910.2687y¯D4210009.3133592.932736.9210462.9129291.2835035.28y¯D54822.2393126.5541735.952673.8917695.28911201.333y¯D64815.0772162.5041343.679983.9214989.94921259.309y¯D73233.5231708.126810.6883330.1844439.2591668.9487y¯D86050.2813023.4011723.625660.7379761.13241211.036y¯De228.7653170.0789130.9133207.0891133.9556361.1317


## 6.2. Real-Life Data

We used three real datasets to compare the PREs of all these estimators to see how well they performed compared to the comparable existing estimators. These datasets’ descriptions and summary statistics are listed as follows.


### 6.2.1. Population Ι

This dataset is taken from [20] page 226 and was conducted in Pakistan during the year 2012, which comprised 33 divisions. This dataset may be downloaded from the Pakistan Bureau of Statistics web page via the link: https://www.pbs.gov.pk/content/microdata. The study variable Y corresponds to the employment level by divisions in 2012 and the number of registered factories in 2012, respectively, while R corresponds to the rank number of registered factories in 2012. Here, our objective is to estimate the finite population mean under extreme values in SRS. The population constants are N=33;n=10;Y¯=27.4909;X¯=72.5455;R¯=17;Sy=10.1308;Sx=10.5770;Sr=9.638;XM=95;Xm=58;RM=33;Rm=1.5;Cy=0.3685;Cx=0.1459;Cr=0.5669;g1=0.6608;g2=0.1151;ρyx=0.2522;ρyr=0.1950;ρxr=0.9810;andθ=0.0697


### 6.2.2. Population ΙΙ

Another dataset is taken from [20] page 135, conducted in Pakistan during the year 2012, which comprised 33 divisions. This dataset may be downloaded from the Pakistan Bureau of Statistics web page via the link: https://www.pbs.gov.pk/content/microdata. The number of pupils enrolled in each division and the total number of government primary and secondary schools for boys and girls in each division are the research variables Y and X in 2012, respectively, and R correspond to the rank of auxiliary variable X. Here, our objective is to estimate the finite population mean under extreme values in SRS. The population constants are N=36;n=10;Y¯=148718.7;X¯=1054.39;R¯=18.5;Sy=182315.1;Sx=402.6098;Sr=10.535;XM=2370;Xm=388;RM=36;Rm=1;Cy=1.2259;Cx=0.3818;Cr=0.5695;g1=0.3472;g2=0.3498;ρyx=0.1799;ρyr=0.188;ρxr=0.9378;andθ=0.0722


### 6.2.3. Population ΙΙΙ

This dataset is taken from [2], which comprises 36 units of food cost and weekly income of families. The study variable Y and auxiliary variable X are the food cost of families’ employment and weekly income of families, respectively, and R correspond to the rank of weekly income of families. For more detail, we can refer to [2] page 24. To estimate the finite population mean under extreme values, population constants are N=36;n=10;Y¯=148718.7;X¯=1054.39;R¯=18.5;Sy=182315.1;Sx=402.6098;Sr=10.535;XM=2370;Xm=388;RM=36;Rm=1;Cy=1.2259;Cx=0.3818;Cr=0.5695;g1=0.3472;g2=0.3498;ρyx=0.1799;ρyr=0.188;ρxr=0.9378;andθ=0.0722.On the abovementioned datasets, the PREs of these proposed and current estimators are provided in Table3. In terms of PRE, it can be seen that the proposed estimators are more efficient than the standard unbiased estimator and existing estimators, i.e., all values are more than one hundred.Table 3
Using real datasets, PRE of the proposed class of estimators.
EstimatorPopulation IPopulation IIPopulation IIIy¯100100100y¯R117.6365101.5271104.4765y¯lr117.651103.3431106.7879y¯t1117.5989103.1005106.6716y¯t2104.1569102.5963106.0517y¯hm1116.9265101.5758104.7442y¯hm2116.9108101.5783104.7662y¯D1812.30341519.9681604.261y¯D2263419.446122.156313.68y¯D31460.4421004.19405.2918y¯D4139601.236976.194762.583y¯D55208.6031061.294603.8153y¯D62848.7161292.207765.8887y¯D71014.97806.8052382.3507y¯D8917.26861539.1841683.153y¯De119.3596118.1962115.8183


## 6.2.1. Population Ι

This dataset is taken from [20] page 226 and was conducted in Pakistan during the year 2012, which comprised 33 divisions. This dataset may be downloaded from the Pakistan Bureau of Statistics web page via the link: https://www.pbs.gov.pk/content/microdata. The study variable Y corresponds to the employment level by divisions in 2012 and the number of registered factories in 2012, respectively, while R corresponds to the rank number of registered factories in 2012. Here, our objective is to estimate the finite population mean under extreme values in SRS. The population constants are N=33;n=10;Y¯=27.4909;X¯=72.5455;R¯=17;Sy=10.1308;Sx=10.5770;Sr=9.638;XM=95;Xm=58;RM=33;Rm=1.5;Cy=0.3685;Cx=0.1459;Cr=0.5669;g1=0.6608;g2=0.1151;ρyx=0.2522;ρyr=0.1950;ρxr=0.9810;andθ=0.0697


## 6.2.2. Population ΙΙ

Another dataset is taken from [20] page 135, conducted in Pakistan during the year 2012, which comprised 33 divisions. This dataset may be downloaded from the Pakistan Bureau of Statistics web page via the link: https://www.pbs.gov.pk/content/microdata. The number of pupils enrolled in each division and the total number of government primary and secondary schools for boys and girls in each division are the research variables Y and X in 2012, respectively, and R correspond to the rank of auxiliary variable X. Here, our objective is to estimate the finite population mean under extreme values in SRS. The population constants are N=36;n=10;Y¯=148718.7;X¯=1054.39;R¯=18.5;Sy=182315.1;Sx=402.6098;Sr=10.535;XM=2370;Xm=388;RM=36;Rm=1;Cy=1.2259;Cx=0.3818;Cr=0.5695;g1=0.3472;g2=0.3498;ρyx=0.1799;ρyr=0.188;ρxr=0.9378;andθ=0.0722


## 6.2.3. Population ΙΙΙ

This dataset is taken from [2], which comprises 36 units of food cost and weekly income of families. The study variable Y and auxiliary variable X are the food cost of families’ employment and weekly income of families, respectively, and R correspond to the rank of weekly income of families. For more detail, we can refer to [2] page 24. To estimate the finite population mean under extreme values, population constants are N=36;n=10;Y¯=148718.7;X¯=1054.39;R¯=18.5;Sy=182315.1;Sx=402.6098;Sr=10.535;XM=2370;Xm=388;RM=36;Rm=1;Cy=1.2259;Cx=0.3818;Cr=0.5695;g1=0.3472;g2=0.3498;ρyx=0.1799;ρyr=0.188;ρxr=0.9378;andθ=0.0722.On the abovementioned datasets, the PREs of these proposed and current estimators are provided in Table3. In terms of PRE, it can be seen that the proposed estimators are more efficient than the standard unbiased estimator and existing estimators, i.e., all values are more than one hundred.Table 3
Using real datasets, PRE of the proposed class of estimators.
EstimatorPopulation IPopulation IIPopulation IIIy¯100100100y¯R117.6365101.5271104.4765y¯lr117.651103.3431106.7879y¯t1117.5989103.1005106.6716y¯t2104.1569102.5963106.0517y¯hm1116.9265101.5758104.7442y¯hm2116.9108101.5783104.7662y¯D1812.30341519.9681604.261y¯D2263419.446122.156313.68y¯D31460.4421004.19405.2918y¯D4139601.236976.194762.583y¯D55208.6031061.294603.8153y¯D62848.7161292.207765.8887y¯D71014.97806.8052382.3507y¯D8917.26861539.1841683.153y¯De119.3596118.1962115.8183


## 7. Conclusion

In this paper, we present some effective estimators for estimating the finite population mean using known information about the minimum and maximum values of auxiliary data. We have identified certain theoretical situations in which the recommended estimators outperform existing estimators. Tables2 and 3 offer the PREs for all estimators over the mean per unit estimator. According to our findings, the recommended estimators y¯Di outperform the estimators evaluated in this research. They are recommended among the suggested classes of estimators because of their high PREs for all populations.



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*Source: 5866085-2022-05-11.xml*