# Some Improved Classes of Estimators in Stratified Sampling Using Bivariate Auxiliary Information

Authors: Shashi Bhushan; Anoop Kumar; Ronald Onyango; Saurabh Singh
Journal: Journal of Probability and Statistics
 (2022)

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## Abstract

This manuscript considers some improved combined and separate classes of estimators of population mean using bivariate auxiliary information under stratified simple random sampling. The expressions of bias and mean square error of the proposed classes of estimators are determined to the first order of approximation. It is exhibited that under some particular conditions, the proposed classes of estimators dominate the existing prominent estimators. The theoretical findings are supported by a simulation study performed over a hypothetically generated population.


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## Body

## 1. Introduction

In sampling theory, the appropriate utilization of auxiliary information plays a leading role to improve the efficiency of the estimators. This information may be utilized either at the design phase (sampling design) or at the estimation phase or in both phases. It is very popular when auxiliary information is considered at the estimation phase, the ratio, product, regression, and exponential type estimators are mostly the preferred methods in different dimensions. Shabbir et al. [1] examined the performance of ratio-exponential log type class of estimators using two auxiliary variables. Shahzad et al. [2] introduced a novel family of variance estimators based on L-moments and calibration approach under stratified simple random sampling SSRS, whereas Shahzad et al. [3] suggested L-Moments and calibration-based variance estimators under double SSRS and discussed an application of COVID-19 pandemic. Shahzad et al. [4] considered the estimation of coefficient of variation using L-moments and calibration approach for nonsensitive and sensitive variables, whereas Shahzad et al. [5] developed variance estimation based on L-moments and auxiliary information. The estimation of population mean is a widely discussed approach in sample surveys and many renowned authors have utilized these auxiliary pieces of information at estimation stage and suggested various modified estimators to date. Especially, under the availability of multi-auxiliary information, the literature contains different kinds of ratio, product, regression, and exponential type estimators.In SSRS, the utilization of auxiliary information at the estimation stage has been discussed by many authors to enhance the efficiency of the estimators. In presence of univariate auxiliary information, Hansen et al. [6]; Kadilar and Cingi [7]; Shabbir and Gupta [8]; Singh and Vishwakarma [9]; Solanki and Singh [10]; Bhushan et al. [11]; etc., suggested various modified estimators of population mean whereas, in presence of multi-auxiliary information, Koyuncu and Kadilar [12] suggested a family of estimators of population mean based on SSRS. Tailor et al. [13] envisaged ratio-cum-product estimator of population mean in SSRS. Tailor and Chouhan [14] addressed ratio-cum-product type exponential estimator of finite population mean. Following Upadhyaya et al. [15]; Singh et al. [16] considered a class of ratio-cum-product estimators using information on two auxiliary variables in SSRS. Along the lines of Singh et al. [17]; Lone et al. [18] introduced a generalized ratio-cum-product type exponential estimator in SSRS, whereas Lone et al. [19] suggested efficient separate class of estimators of population mean. Following Singh and Singh [20]; Muneer et al. [21] suggested a class of combined estimators in SSRS. Recently, Muneer et al. [22] introduced a chain ratio exponential family of estimators based on SSRS. In the present study, we propose some improved classes of estimators for the estimation of population mean by employing bivariate auxiliary information under SSRS.Consider a finite population κ=κ1,κ2,…,κN based on size N units with study variable y and two auxiliary variables x and z, respectively, associated with each unit κi,i=1,2,…,N of the population. Let the population be divided into L disjoint strata with the stratum h comprises of Nh, h=1,2,…,L units. Let a simple random sample of size nh be quantified without replacement from the stratum h such that ∑h=1Lnh=n. Let the observed values of y, x and z on the ith unit of the stratum h be denoted by yhi,xhi,zhi, i=1,2,…,Nh. Let y¯st=∑h=1LWhy¯h, x¯st=∑h=1LWhx¯h and z¯st=∑h=1LWhz¯h, respectively, be the sample means corresponding to the population means Y¯=Y¯st=∑h=1LWhY¯h, X¯=X¯st=∑h=1LWhX¯h and Z¯=Z¯st=∑h=1LWhZ¯h of variables y, x and z, where Wh=Nh/N is the weight in the stratum h. Let y¯h=∑i=1nhyhi/nh, x¯h=∑i=1nhxhi/nh and z¯h=∑i=1nhzhi/nhbe the sample means corresponding to the population means Y¯h=∑i=1NhYhi/Nh, X¯h=∑i=1NhXhi/Nh and Z¯h=∑i=1NhZhi/Nh of variables y, x and z in the stratum h. Let syh2=∑h=1Lyhi−y¯h2/nh−1, sxh2=∑h=1Lxhi−x¯h2/nh−1, szh2=∑h=1Lzhi−z¯h2/nh−1, sxyh=∑h=1Lxhi−x¯hyhi−y¯h/nh−1, szyh=∑h=1Lzhi−z¯hyhi−y¯h/nh−1, sxzh=∑h=1Lxhi−x¯hzhi−z¯h/nh−1, respectively, be the sample variances and covariances corresponding to the population variances and covariances Syh2=∑h=1Lyhi−Y¯h2/Nh−1, Sxh2=∑h=1Lxhi−X¯h2/Nh−1, Szh2=∑h=1Lzhi−Z¯h2/Nh−1, Sxyh=∑h=1Lxhi−X¯hNh−1/Nh−1, Szyh=∑h=1Lzhi−Z¯hyhi−Y¯h/Nh−1, Sxzh=∑h=1Lxhi−X¯hzhi−Z¯h/Nh−1 in the stratum h.To derive the bias and mean square errorMSE of different combined estimators, the following notations will be used throughout the paper.y¯st=Y¯+ε0,x¯st=X¯+ε1,z¯st=Z¯+ε2,such thatEε0=Eε1=Eε2=0 and(1)Vrst=∑h=1LWhr+s+tEy¯h−Y¯hrx¯h−X¯hsz¯h−Z¯ht.Following (1), we can write Eε02=∑h=1LWh2γhSyh2=V200; Eε12=∑h=1LWh2γhSxh2=V020; Eε22=∑h=1LWh2γhSzh2=V002; Eε0,ε1=∑h=1LWh2γhρxyhSxhSyh=V110; Eε1,ε2=∑h=1LWh2γhρxzhSxhSzh=V011 and Eε0,ε2=∑h=1LWh2γhρyzhSyhSzh=V101, where γ=1/nh and ρxyh, ρxzh and ρyzh be the population coefficient of correlation with their respective subscripts in stratum h.Again, to determine the bias andMSE of the separate estimators, the following notations will be used throughout the paper.y¯h=Y¯h+ε0h,x¯h=X¯h+ε1h,z¯h=Z¯h+ε2hsuch thatEε0h=Eε1h=Eε2h=0 and Eε0h2=γhSyh2=U200; Eε1h2=γhSxh2=U020; Eε2h2=γhSzh2=U002; Eε0h,ε1h=γhρxyhSxhSyh=U110; Eε1h,ε2h=γhρxzhSxhSzh=U011 and Eε0h,ε2h=γhρyzhSyhSzh=U101.The aim of the present paper is to suggest some improved combined and separate classes of estimators in the presence of bivariate auxiliary information underSSRS. The remainder of the paper is drafted in the following sections. Section 2 deals with the existing combined and separate estimators, whereas Section 3 considers the proposed combined and separate classes of estimators along with their properties. The theoretical comparison of the proposed combined and separate classes of estimators with the existing combined and separate estimators is given in Section 4. The credibility of the theoretical findings is furnished with a simulation study in Section 5. Finally, a conclusion of this study is drawn in Section 6.


## 2. Existing Estimators

### 2.1. Combined Estimators

The conventional combined mean estimator underSSRS is given by(2)ymc=y¯st.On the lines of Singh [23]; one may consider the classical combined ratio estimator of population mean Y¯ using bivariate auxiliary information under SSRS as(3)y¯rc=y¯stX¯x¯stZ¯z¯st.The classical combined regression estimator of population meanY¯ based on bivariate auxiliary information under SSRS is given by(4)y¯lc=y¯st+β1X¯−x¯st+β2Z¯−z¯st,where β1 and β2 are the regression coefficients of y on x and z, respectively.Following Olkin [24]; the combined ratio type estimator using bivariate auxiliary information under SSRS is given by(5)y¯oc=y¯stwX¯x¯st+1−wZ¯z¯st,where w is a duly opted scalar.Along the lines of Abu–Dayyeh et al. [25]; some combined classes of ratio type estimators using bivariate auxiliary information under SSRS are given by(6)y˜ac=y¯stX¯x¯sta1Z¯z¯sta2,(7)y˜wc=y¯stw1X¯x¯sta1+w2Z¯z¯sta2,where a1,a2,w1,andw2 are duly opted scalars and w1+w2=1.Following Kadilar and Cingi [7], one may suggest some combined ratio-cum-product estimators based on bivariate auxiliary information under SSRS as(8)y¯kc1c=y¯st∑h=1LWhX¯h+CxhZ¯h+Czh∑h=1LWhx¯h+Cxhz¯h+Czh,(9)y¯kc2c=y¯st∑h=1LWhX¯h+β2xhZ¯h+β2zh∑h=1LWhx¯h+β2xhz¯h+β2zh,(10)y¯kc3c=y¯st∑h=1LWhX¯hβ2xh+CxhZ¯hβ2zh+Czh∑h=1LWhx¯hβ2xh+Cxhz¯hβ2zh+Czh,y¯kc4c=y¯st∑h=1LWhX¯hCxh+β2xhZ¯hCzh+β2zh∑h=1LWhx¯hCxh+β2xhz¯hCzh+β2zh, (11) where β2xh and β2zh are the coefficient of kurtosis of variables x and z, respectively, in stratum h.Following Upadhyaya et al. [15]; Singh et al. [16] considered a combined class of ratio-cum-product estimators using bivariate auxiliary information in SSRS as(12)y¯sc=η1y¯st+η2y¯stX¯x¯stαz¯stZ¯β,where η1, η2 are duly opted scalars and α, β are scalars taking real values.Motivated by Khoshnevisan et al. [26]; Koyuncu and Kadilar [12] suggested a general family of combined estimators for population mean Y¯ using bivariate auxiliary information under SSRS as(13)y¯kc=y¯staX¯+bλ1ax¯st+b+1−λ1aX¯+bg1cZ¯+dλ2cz¯st+d+1−λ2cZ¯+dg2,where λ1, λ2, g1, and g2 are prescribed scalars, whereas a≠0, b and c≠0, d are either real numbers or functions of the known parameters of auxiliary variables x and z, respectively.Along the lines of Singh et al. [17]; Tailor and Chouhan [14] suggested a combined ratio-cum-product type exponential estimator for population mean using bivariate auxiliary information under SSRS as(14)y¯tcc=ky¯stexpX¯−x¯stX¯+x¯stexpz¯st−Z¯z¯st+Z¯,where k is a duly opted scalar.Following Singh et al. [17], Lone et al. [18] suggested a combined generalized ratio-cum-product type exponential estimator in SSRS as(15)y¯l1c=y¯stexpL1X¯−x¯stX¯+x¯st+L2z¯st−Z¯z¯st+Z¯,where L1 and L2 are duly opted scalars.The combined version of Lone et al. [19] estimator for estimating population mean Y¯ is given by(16)y¯l2=y¯staX¯+bax¯st+bcz¯st+dcZ¯+d.We remark that the minimumMSE of Abu-Dayyeh et al. [25] type estimator y¯ac, Koyuncu and Kadilar [12] estimator y¯kkc , and Lone et al. [18, 19] estimators y¯l1c,i=1,2 are equal to the minimum MSE of the classical regression estimator y¯lc.Along the lines of Singh and Singh [20]; Muneer et al. [21] introduced a class of combined estimators in SSRS as(17)y¯muc=k3y¯st−k4x¯st−X¯Θ2−expz¯st−Z¯z¯st+Z¯+1−ΘexpZ¯−z¯stZ¯−z¯st,where k3, k4 are duly opted scalars and Θ is a scalar assuming values 0 and 1 to design different estimators.The combined form of Muneer et al. [22] chain ratio exponential family of estimator in SSRS is given by(18)y¯mu1c=y¯stX¯∗w1x¯st∗+1−w1X¯∗α1Z¯∗w2z¯st∗+1−w2Z¯∗α2expα3X¯−x¯stX¯+x¯stexpα4Z¯−z¯stZ¯+z¯st.On the lines of Searls [27]; an improved form of the above-combined estimator is given by(19)y¯mu2c=k1y¯stX¯∗w1x¯st∗+1−w1X¯∗α1Z¯∗w2z¯st∗+1−w2Z¯∗α2expα3X¯−x¯stX¯+x¯stexpα4Z¯−z¯stZ¯+z¯st,where X¯∗=aX¯+b, X¯∗=aX¯+b, x¯st∗=ax¯st+b, Z¯∗=cZ¯+d, z¯st∗=cz¯st+d, w1,w2=0,1 and k1 is a duly opted scalar, αj,j=1,2,3,4 assumes values −1, 0, and +1 to form different new and existing estimators. Moreover, the authors have shown that more than 65 combined classes of estimators are the members of the estimators y¯mu1c and y¯mu2c, respectively, for different values of scalars.The bias andMSE of the estimators considered in this section are readily discussed in Appendix A.


### 2.2. Separate Estimators

The conventional separate mean estimator underSSRS is given by(20)y¯ms=∑h=1LWhy¯h.On the lines of Singh [23]; the classical separate ratio estimator of population mean y¯ using bivariate auxiliary information under SSRS is defined as(21)y¯rs=∑h=1LWhy¯hX¯hx¯hZ¯hz¯h.The classical separate regression estimator of population meanY¯ under bivariate auxiliary information using SSRS is(22)y¯ls=∑h=1LWhy¯h+β1hX¯h−x¯h+β2hZ¯h−z¯h,where β1h and β2h are the regression coefficients of y on x and z, respectively, in stratum h.Motivated by Olkin [24]; the separate ratio estimator in SSRS using bivariate auxiliary information is given by(23)y¯os=∑h=1LWhy¯hwhX¯hx¯h+1−whZ¯hz¯h,where wh is a duly opted scalar in the stratum h to be determined.Following Abu–Dayyeh et al. [25]; a separate class of ratio type estimators using bivariate auxiliary information under SSRS is given by(24)y¯as=∑h=1LWhy¯hX¯hx¯ha1hZ¯hz¯ha2h,(25)y¯wh=∑h=1LWhy¯hw1hX¯hx¯ha1h+w2hZ¯hz¯ha2h,where a1h,a2h,w1handw2h are duly opted scalars in stratum h and w1h+w2h=1.Following Kadilar and Cingi [7]; one can suggest some separate ratio-cum-product type estimators based on bivariate auxiliary information under SSRS as(26)y¯kc1s=∑h=1LWhy¯hX¯h+CxhZ¯h+Czhx¯h+Cxhz¯h+Czh,(27)y¯kc2s=∑h=1LWhy¯hX¯h+β2xhZ¯h+β2zhx¯h+β2xhz¯h+β2zh,(28)y¯kc3s=∑h=1LWhy¯hX¯hβ2xh+CxhZ¯hβ2zh+Czhx¯hβ2xh+Cxhz¯hβ2zh+Czh,(29)y¯kc4c=∑h=1LWsy¯hX¯hCxh+β2xhZ¯hCzh+β2zhx¯hCxh+β2xhz¯hCzh+β2zh.The separate version of Singh et al. [16] estimator is given by(30)y¯ss=∑h=1LWhη1hy¯h+η2hy¯hX¯hx¯hαhz¯hZ¯hβh,where η1h, η2h are duly opted scalars in stratum h and αh, βh are scalars in stratum h taking real values.The separate version of Koyuncu and Kadilar [12] family of estimators is given by(31)y¯kks=∑h=1LWhy¯hahX¯h+bhλ1hahx¯h+bh+1−λ1hahX¯h+bhg1chZ¯h+dhλ2hchz¯h+dh+1−λ2hchZ¯h+dhg2,where λ1h, λ2h, g1 and g2 are some prescribed scalars whereas ah≠0, bh and ch≠0, dh are either real numbers or functions of the known parameters of the auxiliary variables x and z, respectively, in stratum h.The separate version of Tailor and Chouhan [14] estimator for population mean using bivariate auxiliary information under SSRS is defined as(32)y¯tcs=∑h=1LWkhy¯hexpX¯h−x¯hX¯h+x¯hexpz¯h−Z¯hz¯h+Z¯hh,where kh is a duly opted scalar in stratum h.On the lines of Lone et al. [18]; a generalized separate ratio-cum-product type exponential estimator in SSRS is defined as(33)y¯l1s=∑h=1LWhy¯hexpL1hX¯h−x¯hX¯h+x¯h+L2hz¯h−Z¯hz¯h+Z¯h,where L1h and L2h are duly opted scalars in the stratum h.The separate version of Lone et al. [19] estimator for estimating population mean Y¯ as(34)y¯l2s=∑h=1LWhy¯hahX¯h+bhahx¯h+bhchz¯h+dhchZ¯h+dh.It is to be noted that the minimumMSE of separate Abu–Dayyeh [25] type estimator y¯as, Koyuncu and Kadilar [12] estimator y¯kks and Lone et al. [18, 19] estimators y¯l1s & y¯l2s are equal to the minimum MSE of classical separate regression estimator y¯ls.The separate type of Muneer et al. [21] estimator in SSRS is given by(35)y¯mus=∑h=1LWhk3hy¯h−k4hx¯h−X¯hΘh2−expz¯h−Z¯hz¯h+Z¯h+1−ΘhexpZ¯h−z¯hZ¯h−z¯h,where k3h and k4h are suitably chosen scalars in stratum h and Θh is a real constant in stratum h.Muneer et al. [22] suggested a separate chain ratio exponential family of estimators in SSRS as(36)y¯mu1s=∑h=1LWhy¯hX¯h∗w1hx¯h∗+1−w1hX¯h∗α1hZ¯h∗w2hz¯h∗+1−w2hZ¯h∗α2h×expα3hX¯h−x¯hX¯h+x¯hexpα4hZ¯h−z¯hZ¯h+z¯h.On the lines of Searls [27], the modified form of the above separate estimator is given by(37)y¯mu2s=∑h=1LWhk1hy¯hX¯h∗w1hx¯h∗+1−w1hX¯h∗α1hZ¯h∗w1hz¯h∗+1−w1hZ¯h∗α2h×expα3hX¯h−x¯hX¯h+x¯hexpα4hZ¯h−z¯hZ¯h+z¯h,whereX¯∗=ahX¯h+bh, x¯h∗=ahx¯h+bh, Z¯∗=chZ¯h+dh, z¯h∗=chz¯h+dh and w1h,w2h=0,1 and k1h is a duly opted scalar in stratum h, αjh,j=1,2,3,4 assumes values −1, 0, and +1 in order to form different new and existing separate estimators. Furthermore, one can generate more than 65 separate classes of estimators from y¯mu1s and y¯mu2s for different values of scalars.The bias and MSE of the estimators considered in this section are readily discussed in the Appendix B.


## 2.1. Combined Estimators

The conventional combined mean estimator underSSRS is given by(2)ymc=y¯st.On the lines of Singh [23]; one may consider the classical combined ratio estimator of population mean Y¯ using bivariate auxiliary information under SSRS as(3)y¯rc=y¯stX¯x¯stZ¯z¯st.The classical combined regression estimator of population meanY¯ based on bivariate auxiliary information under SSRS is given by(4)y¯lc=y¯st+β1X¯−x¯st+β2Z¯−z¯st,where β1 and β2 are the regression coefficients of y on x and z, respectively.Following Olkin [24]; the combined ratio type estimator using bivariate auxiliary information under SSRS is given by(5)y¯oc=y¯stwX¯x¯st+1−wZ¯z¯st,where w is a duly opted scalar.Along the lines of Abu–Dayyeh et al. [25]; some combined classes of ratio type estimators using bivariate auxiliary information under SSRS are given by(6)y˜ac=y¯stX¯x¯sta1Z¯z¯sta2,(7)y˜wc=y¯stw1X¯x¯sta1+w2Z¯z¯sta2,where a1,a2,w1,andw2 are duly opted scalars and w1+w2=1.Following Kadilar and Cingi [7], one may suggest some combined ratio-cum-product estimators based on bivariate auxiliary information under SSRS as(8)y¯kc1c=y¯st∑h=1LWhX¯h+CxhZ¯h+Czh∑h=1LWhx¯h+Cxhz¯h+Czh,(9)y¯kc2c=y¯st∑h=1LWhX¯h+β2xhZ¯h+β2zh∑h=1LWhx¯h+β2xhz¯h+β2zh,(10)y¯kc3c=y¯st∑h=1LWhX¯hβ2xh+CxhZ¯hβ2zh+Czh∑h=1LWhx¯hβ2xh+Cxhz¯hβ2zh+Czh,y¯kc4c=y¯st∑h=1LWhX¯hCxh+β2xhZ¯hCzh+β2zh∑h=1LWhx¯hCxh+β2xhz¯hCzh+β2zh, (11) where β2xh and β2zh are the coefficient of kurtosis of variables x and z, respectively, in stratum h.Following Upadhyaya et al. [15]; Singh et al. [16] considered a combined class of ratio-cum-product estimators using bivariate auxiliary information in SSRS as(12)y¯sc=η1y¯st+η2y¯stX¯x¯stαz¯stZ¯β,where η1, η2 are duly opted scalars and α, β are scalars taking real values.Motivated by Khoshnevisan et al. [26]; Koyuncu and Kadilar [12] suggested a general family of combined estimators for population mean Y¯ using bivariate auxiliary information under SSRS as(13)y¯kc=y¯staX¯+bλ1ax¯st+b+1−λ1aX¯+bg1cZ¯+dλ2cz¯st+d+1−λ2cZ¯+dg2,where λ1, λ2, g1, and g2 are prescribed scalars, whereas a≠0, b and c≠0, d are either real numbers or functions of the known parameters of auxiliary variables x and z, respectively.Along the lines of Singh et al. [17]; Tailor and Chouhan [14] suggested a combined ratio-cum-product type exponential estimator for population mean using bivariate auxiliary information under SSRS as(14)y¯tcc=ky¯stexpX¯−x¯stX¯+x¯stexpz¯st−Z¯z¯st+Z¯,where k is a duly opted scalar.Following Singh et al. [17], Lone et al. [18] suggested a combined generalized ratio-cum-product type exponential estimator in SSRS as(15)y¯l1c=y¯stexpL1X¯−x¯stX¯+x¯st+L2z¯st−Z¯z¯st+Z¯,where L1 and L2 are duly opted scalars.The combined version of Lone et al. [19] estimator for estimating population mean Y¯ is given by(16)y¯l2=y¯staX¯+bax¯st+bcz¯st+dcZ¯+d.We remark that the minimumMSE of Abu-Dayyeh et al. [25] type estimator y¯ac, Koyuncu and Kadilar [12] estimator y¯kkc , and Lone et al. [18, 19] estimators y¯l1c,i=1,2 are equal to the minimum MSE of the classical regression estimator y¯lc.Along the lines of Singh and Singh [20]; Muneer et al. [21] introduced a class of combined estimators in SSRS as(17)y¯muc=k3y¯st−k4x¯st−X¯Θ2−expz¯st−Z¯z¯st+Z¯+1−ΘexpZ¯−z¯stZ¯−z¯st,where k3, k4 are duly opted scalars and Θ is a scalar assuming values 0 and 1 to design different estimators.The combined form of Muneer et al. [22] chain ratio exponential family of estimator in SSRS is given by(18)y¯mu1c=y¯stX¯∗w1x¯st∗+1−w1X¯∗α1Z¯∗w2z¯st∗+1−w2Z¯∗α2expα3X¯−x¯stX¯+x¯stexpα4Z¯−z¯stZ¯+z¯st.On the lines of Searls [27]; an improved form of the above-combined estimator is given by(19)y¯mu2c=k1y¯stX¯∗w1x¯st∗+1−w1X¯∗α1Z¯∗w2z¯st∗+1−w2Z¯∗α2expα3X¯−x¯stX¯+x¯stexpα4Z¯−z¯stZ¯+z¯st,where X¯∗=aX¯+b, X¯∗=aX¯+b, x¯st∗=ax¯st+b, Z¯∗=cZ¯+d, z¯st∗=cz¯st+d, w1,w2=0,1 and k1 is a duly opted scalar, αj,j=1,2,3,4 assumes values −1, 0, and +1 to form different new and existing estimators. Moreover, the authors have shown that more than 65 combined classes of estimators are the members of the estimators y¯mu1c and y¯mu2c, respectively, for different values of scalars.The bias andMSE of the estimators considered in this section are readily discussed in Appendix A.


## 2.2. Separate Estimators

The conventional separate mean estimator underSSRS is given by(20)y¯ms=∑h=1LWhy¯h.On the lines of Singh [23]; the classical separate ratio estimator of population mean y¯ using bivariate auxiliary information under SSRS is defined as(21)y¯rs=∑h=1LWhy¯hX¯hx¯hZ¯hz¯h.The classical separate regression estimator of population meanY¯ under bivariate auxiliary information using SSRS is(22)y¯ls=∑h=1LWhy¯h+β1hX¯h−x¯h+β2hZ¯h−z¯h,where β1h and β2h are the regression coefficients of y on x and z, respectively, in stratum h.Motivated by Olkin [24]; the separate ratio estimator in SSRS using bivariate auxiliary information is given by(23)y¯os=∑h=1LWhy¯hwhX¯hx¯h+1−whZ¯hz¯h,where wh is a duly opted scalar in the stratum h to be determined.Following Abu–Dayyeh et al. [25]; a separate class of ratio type estimators using bivariate auxiliary information under SSRS is given by(24)y¯as=∑h=1LWhy¯hX¯hx¯ha1hZ¯hz¯ha2h,(25)y¯wh=∑h=1LWhy¯hw1hX¯hx¯ha1h+w2hZ¯hz¯ha2h,where a1h,a2h,w1handw2h are duly opted scalars in stratum h and w1h+w2h=1.Following Kadilar and Cingi [7]; one can suggest some separate ratio-cum-product type estimators based on bivariate auxiliary information under SSRS as(26)y¯kc1s=∑h=1LWhy¯hX¯h+CxhZ¯h+Czhx¯h+Cxhz¯h+Czh,(27)y¯kc2s=∑h=1LWhy¯hX¯h+β2xhZ¯h+β2zhx¯h+β2xhz¯h+β2zh,(28)y¯kc3s=∑h=1LWhy¯hX¯hβ2xh+CxhZ¯hβ2zh+Czhx¯hβ2xh+Cxhz¯hβ2zh+Czh,(29)y¯kc4c=∑h=1LWsy¯hX¯hCxh+β2xhZ¯hCzh+β2zhx¯hCxh+β2xhz¯hCzh+β2zh.The separate version of Singh et al. [16] estimator is given by(30)y¯ss=∑h=1LWhη1hy¯h+η2hy¯hX¯hx¯hαhz¯hZ¯hβh,where η1h, η2h are duly opted scalars in stratum h and αh, βh are scalars in stratum h taking real values.The separate version of Koyuncu and Kadilar [12] family of estimators is given by(31)y¯kks=∑h=1LWhy¯hahX¯h+bhλ1hahx¯h+bh+1−λ1hahX¯h+bhg1chZ¯h+dhλ2hchz¯h+dh+1−λ2hchZ¯h+dhg2,where λ1h, λ2h, g1 and g2 are some prescribed scalars whereas ah≠0, bh and ch≠0, dh are either real numbers or functions of the known parameters of the auxiliary variables x and z, respectively, in stratum h.The separate version of Tailor and Chouhan [14] estimator for population mean using bivariate auxiliary information under SSRS is defined as(32)y¯tcs=∑h=1LWkhy¯hexpX¯h−x¯hX¯h+x¯hexpz¯h−Z¯hz¯h+Z¯hh,where kh is a duly opted scalar in stratum h.On the lines of Lone et al. [18]; a generalized separate ratio-cum-product type exponential estimator in SSRS is defined as(33)y¯l1s=∑h=1LWhy¯hexpL1hX¯h−x¯hX¯h+x¯h+L2hz¯h−Z¯hz¯h+Z¯h,where L1h and L2h are duly opted scalars in the stratum h.The separate version of Lone et al. [19] estimator for estimating population mean Y¯ as(34)y¯l2s=∑h=1LWhy¯hahX¯h+bhahx¯h+bhchz¯h+dhchZ¯h+dh.It is to be noted that the minimumMSE of separate Abu–Dayyeh [25] type estimator y¯as, Koyuncu and Kadilar [12] estimator y¯kks and Lone et al. [18, 19] estimators y¯l1s & y¯l2s are equal to the minimum MSE of classical separate regression estimator y¯ls.The separate type of Muneer et al. [21] estimator in SSRS is given by(35)y¯mus=∑h=1LWhk3hy¯h−k4hx¯h−X¯hΘh2−expz¯h−Z¯hz¯h+Z¯h+1−ΘhexpZ¯h−z¯hZ¯h−z¯h,where k3h and k4h are suitably chosen scalars in stratum h and Θh is a real constant in stratum h.Muneer et al. [22] suggested a separate chain ratio exponential family of estimators in SSRS as(36)y¯mu1s=∑h=1LWhy¯hX¯h∗w1hx¯h∗+1−w1hX¯h∗α1hZ¯h∗w2hz¯h∗+1−w2hZ¯h∗α2h×expα3hX¯h−x¯hX¯h+x¯hexpα4hZ¯h−z¯hZ¯h+z¯h.On the lines of Searls [27], the modified form of the above separate estimator is given by(37)y¯mu2s=∑h=1LWhk1hy¯hX¯h∗w1hx¯h∗+1−w1hX¯h∗α1hZ¯h∗w1hz¯h∗+1−w1hZ¯h∗α2h×expα3hX¯h−x¯hX¯h+x¯hexpα4hZ¯h−z¯hZ¯h+z¯h,whereX¯∗=ahX¯h+bh, x¯h∗=ahx¯h+bh, Z¯∗=chZ¯h+dh, z¯h∗=chz¯h+dh and w1h,w2h=0,1 and k1h is a duly opted scalar in stratum h, αjh,j=1,2,3,4 assumes values −1, 0, and +1 in order to form different new and existing separate estimators. Furthermore, one can generate more than 65 separate classes of estimators from y¯mu1s and y¯mu2s for different values of scalars.The bias and MSE of the estimators considered in this section are readily discussed in the Appendix B.


## 3. Proposed Estimators

The objective of this paper is to suggest some improved combined and separate classes of estimators over the existing combined and separate estimators discussed in the previous section. We have extended the work of Bhushan et al. [28] for the estimation of population mean Y¯ by incorporating bivariate auxiliary information under SSRS.


### 3.1. Combined Estimators

We propose some improved combined classes of estimators based on bivariate auxiliary information underSSRS as(38)y¯s1c=ξ1y¯st1+logx¯stX¯θ11+logz¯stZ¯δ1,(39)y¯s2c=ξ2y¯st1+θ2logx¯stX¯1+δ2logz¯stZ¯,(40)y¯s3c=ξ3y¯st+θ3x¯st−X¯+δ3z¯st−Z¯,(41)y¯s4c=ξ4y¯stX¯x¯stθ4Z¯z¯stδ4,(42)y¯s5c=ξ5y¯stX¯X¯+θ5x¯st−X¯Z¯Z¯+δ5z¯st−Z¯,where ξi, θi and δi, i=1,2,…,5 are duly opted scalars to be determined.Theorem 1.
The bias of the proposed combined classes of estimatorsy¯sic,i=1,2,…,5is given by(43)Biasy¯s1c=Y¯ξ11+θ122−θ1V020+δ122−δ1V002+θ1V110+δ1V101+θ1δ1V011−1,(44)Biasy¯s2c=Y¯ξ21−θ22V020−δ22V002+θ2V110+δ2V101+θ2δ2V011−1,(45)Biasy¯s3c=Y¯ξ3−1,(46)Biasy¯s4c=Y¯ξ41+θ4θ4+12V020+δ4δ4+12V002−θ4V110−δ4V101+θ4δ4V011−1,(47)Biasy¯s5c=Y¯ξ51+θ52V020+δ52V002−θ5V110−δ5V101+θ5δ5V011−1.Proof.
The precis of the derivations are given in AppendixC for quick review.Theorem 2.
TheMSEof the proposed combined classes of estimatorsy¯sic,i=1,2,…,5is given by(48)MSEy¯s1c=Y¯21+ξ121+V200+2θ12V020+2δ12V002−2θ1V020−2δ1V002+4θ1V110+4δ1V101+4θ1δ1V011−2ξ11+θ122V020+δ122V002−θ1V020−δ1V002+θ1V110+δ1V101+θ1δ1V011,(49)MSEy¯s2c=Y¯21+ξ221+V200+θ22V020+δ22V002−θ2V020−δ2V002+4θ2V110+4δ2V101+4θ2δ2V011−2ξ21−θ22V020−δ22V002+θ2V110+δ2V101+θ2δ2V011,(50)MSEy¯s3c=Y¯2ξ3−12Y¯2+ξ32Y¯2V200+θ32X¯2V020+δ32Z¯2V002+2ξ3θ3X¯Y¯V110+2ξ3δ3Z¯Y¯V101+2θ3δ3X¯Z¯V011,(51)MSEy¯s4c=Y¯21+ξ421+V200+θ4V020+δ4V002+2θ42V020+2δ42V002−4θ4V110−4δ4V101+4θ4δ4V011−2ξ41+θ4θ4+12V020+δ4δ4+12V002−θ4V110−δ4V101+θ4δ4V011,(52)MSEy¯s5c=Y¯21+ξ521+V200+3θ52V020+3δ52V002−4θ5V110−4δ5V101+4θ5δ5V011−2ξ51+θ52V020−θ5V110+δ52V002−δ5V101+θ5δ5V011.Proof.
The precis of the derivations are given in AppendixC for quick review.Corollary 1.
The minimumMSE of the proposed combined classes of estimators y¯sic, i = 1,2,..., 5 is given by(53)minMSEy¯sic=Y¯21−Bi2Ai;i=1,2,4,5,(54)minMSEy¯s3c=Y¯21−ξ3opt=Y¯21−B32A3.Proof.
The precis of derivations and the definition of parametric functionsAi and Bi are given in Appendix C for quick review.
We note that Theorem2 and Corollary 1 are important to derive the efficiency conditions given in Subsection 4.1.


### 3.2. Separate Estimators

We propose some improved separate classes of estimators based on bivariate auxiliary information underSSRS as(55)y¯s1s=∑h=1LWhξ1hy¯h1+logx¯hX¯hθ1h1+logz¯hZ¯hδ1h,(56)y¯s2s=∑h=1LWhξ2hy¯h1+θ2hlogx¯hX¯h1+δ2hlogz¯hZ¯h,(57)y¯s3s=∑h=1LWhξ3hy¯h+θ3hx¯h−X¯h+δ3hz¯h−Z¯h,(58)y¯s4s=∑h=1LWhξ4hy¯hX¯hx¯hθ4hZ¯hz¯hδ4h,(59)y¯s5s=∑h=1LWhξ5hy¯hX¯hX¯h+θ5hx¯h−X¯hZ¯hZ¯h+δ5hz¯h−Z¯h,where ξih, θih and δih, i=1,2,…,5 are duly opted scalars in stratum h.Theorem 3.
The bias of the proposed separate classes of estimatorsy¯sis,i=1,2,…,5is given by(60)Biasy¯s1s=∑h=1LWhY¯hξ1h1+θ1h22−θ1hU020+δ1h22−δ1hU002+θ1hU110+δ1hU101+θ1hδ1hU011−1,(61)Biasy¯s2s=∑h=1LWhY¯hξ2h1−θ2h2U020−δ2h2U002+θ2hU110+δ2hU101+θ2hδ2hU011−1,(62)Biasy¯s3s=∑h=1LWhY¯hξ3h−1,(63)Biasy¯s4s=∑h=1LWhY¯hξ4h1+θ4hθ4h+12U020+δ4hδ4h+12U002−θ4hU110−δ4hU101+θ4hδ4hU011−1,(64)Biasy¯s5s=∑h=1LWhY¯hξ5h1+θ5h2U020+δ5h2U002−θ5hU110−δ5hU101+θ5hδ5hU011−1.Proof.
The precis of the derivations are given in Appendix C for quick reviewTheorem 4.
TheMSE of the proposed separate classes of estimators y¯sis,i=1,2,…,5 is given by(65)MSEy¯s1s=∑h=1LWh2Y¯h21+ξ1h21+U200+2θ1h2U020+2δ1h2U002−2δ1hU020−2δ1hU002+4θ1hU110+4δ1hU101+4θ1hδ1hU011−2ξ1h1+θ1h22U020+δ1h22U002−θ1U020−δ1hU002+θ1hU110+δ1hU101+θ1hδ1hU011,(66)MSEy¯s2s=∑h=1LWh2Y¯h21+ξ2h21+U200+θ2h2U020+δ2h2U002−θ2hU020−δ2hU002+4θ2hU110+4δ2hU101+4θ2hδ2hU011−2ξ2h1−θ2h2U020−δ2h2U002+θ2hU110+δ2hU101+θ2hδ2hU011,(67)MSEy¯s3s=∑h=1LWh2Y¯h2ξ3h−12Y¯h2+ξ3h2Y¯h2U200+θ3h2X¯h2U020+δ3h2Z¯h2U002+2ξ3hθ3hX¯hY¯hU110+2ξ3hδ3hZ¯hY¯hU101+2θ3hδ3hX¯hZ¯hU011,(68)MSEy¯s4s=∑h=1LWh2Y¯h21+ξ4h21+U200+θ4hU020+δ4hU002+2θ4h2U020+2δ42U002−4θ4hU110−4δ4hU101+4θ4hδ4hU011−2ξ4h1+θ4hθ4h+12U020+δ4hδ4h+12U002−θ4hU110−δ4hU101+θ4hδ4hU011,(69)MSEy¯s5s=∑h=1LWh2Y¯h21+ξ5h21+U200+3θ5h2U020+3δ5h2U002−4θ5hU110−4δ5hU101+4θ5hδ5hU011−2ξ5h1+θ5h2U020−θ5hU110+δ5h2U002−δ5hU101+θ5hδ5hU011.Proof.
The precis of the derivations are given in AppendixC for quick review.Corollary 2.
The minimumMSE of the proposed separate classes of estimators y¯sis;i = 1,2, …, 5 is given by(70)minMSEy¯sis=∑h=1LWh2Y¯h21−Bih2Aih;i=1,2,4,5,(71)minMSEy¯s3s=∑h=1LWh2Y¯h21−ξ3hopt=∑h=1LWh2Y¯h21−B3h2A3h.Proof.
The precis of the derivations and definition of parametric functionsAih and Bih are given in AppendixProof.
C for quick review.
We again note that Theorem4 and Corollary 2 are important in order to derive the efficiency conditions given in Subsection 4.2.


## 3.1. Combined Estimators

We propose some improved combined classes of estimators based on bivariate auxiliary information underSSRS as(38)y¯s1c=ξ1y¯st1+logx¯stX¯θ11+logz¯stZ¯δ1,(39)y¯s2c=ξ2y¯st1+θ2logx¯stX¯1+δ2logz¯stZ¯,(40)y¯s3c=ξ3y¯st+θ3x¯st−X¯+δ3z¯st−Z¯,(41)y¯s4c=ξ4y¯stX¯x¯stθ4Z¯z¯stδ4,(42)y¯s5c=ξ5y¯stX¯X¯+θ5x¯st−X¯Z¯Z¯+δ5z¯st−Z¯,where ξi, θi and δi, i=1,2,…,5 are duly opted scalars to be determined.Theorem 1.
The bias of the proposed combined classes of estimatorsy¯sic,i=1,2,…,5is given by(43)Biasy¯s1c=Y¯ξ11+θ122−θ1V020+δ122−δ1V002+θ1V110+δ1V101+θ1δ1V011−1,(44)Biasy¯s2c=Y¯ξ21−θ22V020−δ22V002+θ2V110+δ2V101+θ2δ2V011−1,(45)Biasy¯s3c=Y¯ξ3−1,(46)Biasy¯s4c=Y¯ξ41+θ4θ4+12V020+δ4δ4+12V002−θ4V110−δ4V101+θ4δ4V011−1,(47)Biasy¯s5c=Y¯ξ51+θ52V020+δ52V002−θ5V110−δ5V101+θ5δ5V011−1.Proof.
The precis of the derivations are given in AppendixC for quick review.Theorem 2.
TheMSEof the proposed combined classes of estimatorsy¯sic,i=1,2,…,5is given by(48)MSEy¯s1c=Y¯21+ξ121+V200+2θ12V020+2δ12V002−2θ1V020−2δ1V002+4θ1V110+4δ1V101+4θ1δ1V011−2ξ11+θ122V020+δ122V002−θ1V020−δ1V002+θ1V110+δ1V101+θ1δ1V011,(49)MSEy¯s2c=Y¯21+ξ221+V200+θ22V020+δ22V002−θ2V020−δ2V002+4θ2V110+4δ2V101+4θ2δ2V011−2ξ21−θ22V020−δ22V002+θ2V110+δ2V101+θ2δ2V011,(50)MSEy¯s3c=Y¯2ξ3−12Y¯2+ξ32Y¯2V200+θ32X¯2V020+δ32Z¯2V002+2ξ3θ3X¯Y¯V110+2ξ3δ3Z¯Y¯V101+2θ3δ3X¯Z¯V011,(51)MSEy¯s4c=Y¯21+ξ421+V200+θ4V020+δ4V002+2θ42V020+2δ42V002−4θ4V110−4δ4V101+4θ4δ4V011−2ξ41+θ4θ4+12V020+δ4δ4+12V002−θ4V110−δ4V101+θ4δ4V011,(52)MSEy¯s5c=Y¯21+ξ521+V200+3θ52V020+3δ52V002−4θ5V110−4δ5V101+4θ5δ5V011−2ξ51+θ52V020−θ5V110+δ52V002−δ5V101+θ5δ5V011.Proof.
The precis of the derivations are given in AppendixC for quick review.Corollary 1.
The minimumMSE of the proposed combined classes of estimators y¯sic, i = 1,2,..., 5 is given by(53)minMSEy¯sic=Y¯21−Bi2Ai;i=1,2,4,5,(54)minMSEy¯s3c=Y¯21−ξ3opt=Y¯21−B32A3.Proof.
The precis of derivations and the definition of parametric functionsAi and Bi are given in Appendix C for quick review.
We note that Theorem2 and Corollary 1 are important to derive the efficiency conditions given in Subsection 4.1.


## 3.2. Separate Estimators

We propose some improved separate classes of estimators based on bivariate auxiliary information underSSRS as(55)y¯s1s=∑h=1LWhξ1hy¯h1+logx¯hX¯hθ1h1+logz¯hZ¯hδ1h,(56)y¯s2s=∑h=1LWhξ2hy¯h1+θ2hlogx¯hX¯h1+δ2hlogz¯hZ¯h,(57)y¯s3s=∑h=1LWhξ3hy¯h+θ3hx¯h−X¯h+δ3hz¯h−Z¯h,(58)y¯s4s=∑h=1LWhξ4hy¯hX¯hx¯hθ4hZ¯hz¯hδ4h,(59)y¯s5s=∑h=1LWhξ5hy¯hX¯hX¯h+θ5hx¯h−X¯hZ¯hZ¯h+δ5hz¯h−Z¯h,where ξih, θih and δih, i=1,2,…,5 are duly opted scalars in stratum h.Theorem 3.
The bias of the proposed separate classes of estimatorsy¯sis,i=1,2,…,5is given by(60)Biasy¯s1s=∑h=1LWhY¯hξ1h1+θ1h22−θ1hU020+δ1h22−δ1hU002+θ1hU110+δ1hU101+θ1hδ1hU011−1,(61)Biasy¯s2s=∑h=1LWhY¯hξ2h1−θ2h2U020−δ2h2U002+θ2hU110+δ2hU101+θ2hδ2hU011−1,(62)Biasy¯s3s=∑h=1LWhY¯hξ3h−1,(63)Biasy¯s4s=∑h=1LWhY¯hξ4h1+θ4hθ4h+12U020+δ4hδ4h+12U002−θ4hU110−δ4hU101+θ4hδ4hU011−1,(64)Biasy¯s5s=∑h=1LWhY¯hξ5h1+θ5h2U020+δ5h2U002−θ5hU110−δ5hU101+θ5hδ5hU011−1.Proof.
The precis of the derivations are given in Appendix C for quick reviewTheorem 4.
TheMSE of the proposed separate classes of estimators y¯sis,i=1,2,…,5 is given by(65)MSEy¯s1s=∑h=1LWh2Y¯h21+ξ1h21+U200+2θ1h2U020+2δ1h2U002−2δ1hU020−2δ1hU002+4θ1hU110+4δ1hU101+4θ1hδ1hU011−2ξ1h1+θ1h22U020+δ1h22U002−θ1U020−δ1hU002+θ1hU110+δ1hU101+θ1hδ1hU011,(66)MSEy¯s2s=∑h=1LWh2Y¯h21+ξ2h21+U200+θ2h2U020+δ2h2U002−θ2hU020−δ2hU002+4θ2hU110+4δ2hU101+4θ2hδ2hU011−2ξ2h1−θ2h2U020−δ2h2U002+θ2hU110+δ2hU101+θ2hδ2hU011,(67)MSEy¯s3s=∑h=1LWh2Y¯h2ξ3h−12Y¯h2+ξ3h2Y¯h2U200+θ3h2X¯h2U020+δ3h2Z¯h2U002+2ξ3hθ3hX¯hY¯hU110+2ξ3hδ3hZ¯hY¯hU101+2θ3hδ3hX¯hZ¯hU011,(68)MSEy¯s4s=∑h=1LWh2Y¯h21+ξ4h21+U200+θ4hU020+δ4hU002+2θ4h2U020+2δ42U002−4θ4hU110−4δ4hU101+4θ4hδ4hU011−2ξ4h1+θ4hθ4h+12U020+δ4hδ4h+12U002−θ4hU110−δ4hU101+θ4hδ4hU011,(69)MSEy¯s5s=∑h=1LWh2Y¯h21+ξ5h21+U200+3θ5h2U020+3δ5h2U002−4θ5hU110−4δ5hU101+4θ5hδ5hU011−2ξ5h1+θ5h2U020−θ5hU110+δ5h2U002−δ5hU101+θ5hδ5hU011.Proof.
The precis of the derivations are given in AppendixC for quick review.Corollary 2.
The minimumMSE of the proposed separate classes of estimators y¯sis;i = 1,2, …, 5 is given by(70)minMSEy¯sis=∑h=1LWh2Y¯h21−Bih2Aih;i=1,2,4,5,(71)minMSEy¯s3s=∑h=1LWh2Y¯h21−ξ3hopt=∑h=1LWh2Y¯h21−B3h2A3h.Proof.
The precis of the derivations and definition of parametric functionsAih and Bih are given in AppendixProof.
C for quick review.
We again note that Theorem4 and Corollary 2 are important in order to derive the efficiency conditions given in Subsection 4.2.


## 4. Efficiency Conditions

In this section, we derive the efficiency conditions under which the proposed combined and separate classes of estimators dominate the existing combined and separate estimators.


### 4.1. Combined Estimators

On comparing the minimumMSE of the proposed combined estimators y¯sic,i=1,2,…,5 from (53) and (54) with the minimum MSE of existing combined estimators from (A.1), (A.3), (A.4), (A.6), (A.8), (A.10), (A.12), (A.14), (A.16), (A.20), (A.22), (A.24), (A.26), and (A.28), we get the following conditions:(72)MSEy¯mc>MSEy¯sic⇒Bi2Ai>1−V200,(73)MSEy¯rc>MSEtsic⇒Bi2Ai>1−V200−V020−V002+2V110+2V101−2V011,(74)MSEy¯lc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(75)MSEy¯oc>MSEy¯sic⇒Bi2Ai>1−V200−V002−2V101+V101−V002−V110+V0112V020+V002−2V011,(76)MSEy¯ac>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(77)MSEy¯wc>MSEy¯sic⇒Bi2Ai>1−V200−a22V002+2V002V101+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(78)MSEy¯kcic>MSEy¯sic⇒Bi2Ai>1−V200−λi2V020−Δi2V002+2λiV110+2ΔiV101−2λiΔiV011,(79)MSEy¯sc>MSEy¯sic⇒Bi2Ai>A2−A3A4A1A2−A32−A4A1A4−A3A1A2−A32,(80)MSEy¯kkc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(81)MSEy¯tcc>MSEy¯sic⇒Bi2Ai>1−k∗2V200+14V020+V002−2V002+V101−V110−2k∗k∗−1183V020−V002−2V011−k∗−12,(82)MSEy¯l1c>MSEy¯sic⇒Bi2Ai>1−V200+V1012V020+V1102V002−2V011V101V110V020V0021−ρxzh2,(83)MSEy¯l2c>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(84)MSEy¯muc>MSEy¯sic⇒Bi2Ai>V01124V020+Am2Bm,(85)MSEy¯mu1c>MSEy¯sic⇒Bi2Ai>2O−P,(86)MSEy¯mu2c>MSEy¯sic⇒Bi2Ai>O2P.If conditions (72) to (86) hold, then the proposed combined classes of estimators y¯sic,i=1,2,…,5 perform better than the other existing combined estimators.


### 4.2. Separate Estimators

On comparing the minimumMSE of the proposed separate estimators y¯sis,i=1,2,…,5 from (69) and (70) with the minimum MSE of the existing separate estimators from (B.1), (B.3), (B.4), (B.6), (B.8), (B.10), (B.12), (B.14), (B.16), (B.18), (B.20), (B.22), (B.24), and (B.26), we get the following efficiency conditions:(87)MSEy¯ms>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200,(88)MSEy¯rs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+U020+U002−2U110−2U101+2U011,(89)MSEy¯ls>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(90)MSEy¯os>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+U002+2U101−U101−U002−U110+U0112U020+U002−2U011,(91)MSEy¯as>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(92)MSEy¯ws>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+a2h2U002−2U002U101−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(93)MSEy¯kcis>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+λih2U020+Δih2U002−2λihU110−2ΔihU101+2λihΔihU011,(94)MSEy¯ss>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−A2h−A3hA4hA1hA2h−A3h2−A4hA1hA4h−A3hA1hA2h−A3h2,(95)MSEy¯kks>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(96)MSEy¯tcs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2kh∗2U200+14U020+U002−2U002+U101−U110+2kh∗kh∗−1183U020−U002−2U011+12U101−U110+kh∗−12,(97)MSEy¯l1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U1012U020+U1102U002−2U011U101U110U020U0021−ρxzh2,(98)MSEy¯l2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(99)MSEy¯mus>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−U01124U020−Amh2Bmh,(100)MSEy¯mu1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21+Ph−2Oh,(101)MSEy¯mu2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−Oh2Ph.If the conditions (87) to (101) hold then the proposed separate classes of estimators y¯sis,i=1,2,…,5 perform better than the other existing separate estimators.


### 4.3. Comparison of Proposed Combined and Separate Estimators

By comparing the minimumMSE of the proposed combined and separate classes of estimators y¯sic and y¯sis, i=1,2,…,5, we get(102)minMSEy¯sic−minMSEy¯sis=∑h=1LY¯2−Wh2Y¯h2−Y¯2Bi2Ai−Wh2Y¯h2Bih2Aih.If the ratio estimate is veritable and the relationship between auxiliary and study variables within each stratum is a straight line passing through origin then the last term of (102) is broadly small and it vanished.Furthermore, unlessRh is invariant from stratum to stratum, separate estimators probably become more efficient in each stratum if the sample in each stratum is large enough so that the approximate formula for MSEy¯sis,i=1,2,…,5 is valid and the cumulative bias that can affect the proposed estimators is negligible, whereas the proposed combined estimators are to be preferably recommended with only a small sample in each stratum ([29]).Furthermore, the conditions of Subsection 4.1, Subsection 4.2, and Subsection 4.3 are held in practice by being verified through a simulation study.


## 4.1. Combined Estimators

On comparing the minimumMSE of the proposed combined estimators y¯sic,i=1,2,…,5 from (53) and (54) with the minimum MSE of existing combined estimators from (A.1), (A.3), (A.4), (A.6), (A.8), (A.10), (A.12), (A.14), (A.16), (A.20), (A.22), (A.24), (A.26), and (A.28), we get the following conditions:(72)MSEy¯mc>MSEy¯sic⇒Bi2Ai>1−V200,(73)MSEy¯rc>MSEtsic⇒Bi2Ai>1−V200−V020−V002+2V110+2V101−2V011,(74)MSEy¯lc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(75)MSEy¯oc>MSEy¯sic⇒Bi2Ai>1−V200−V002−2V101+V101−V002−V110+V0112V020+V002−2V011,(76)MSEy¯ac>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(77)MSEy¯wc>MSEy¯sic⇒Bi2Ai>1−V200−a22V002+2V002V101+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(78)MSEy¯kcic>MSEy¯sic⇒Bi2Ai>1−V200−λi2V020−Δi2V002+2λiV110+2ΔiV101−2λiΔiV011,(79)MSEy¯sc>MSEy¯sic⇒Bi2Ai>A2−A3A4A1A2−A32−A4A1A4−A3A1A2−A32,(80)MSEy¯kkc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(81)MSEy¯tcc>MSEy¯sic⇒Bi2Ai>1−k∗2V200+14V020+V002−2V002+V101−V110−2k∗k∗−1183V020−V002−2V011−k∗−12,(82)MSEy¯l1c>MSEy¯sic⇒Bi2Ai>1−V200+V1012V020+V1102V002−2V011V101V110V020V0021−ρxzh2,(83)MSEy¯l2c>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,(84)MSEy¯muc>MSEy¯sic⇒Bi2Ai>V01124V020+Am2Bm,(85)MSEy¯mu1c>MSEy¯sic⇒Bi2Ai>2O−P,(86)MSEy¯mu2c>MSEy¯sic⇒Bi2Ai>O2P.If conditions (72) to (86) hold, then the proposed combined classes of estimators y¯sic,i=1,2,…,5 perform better than the other existing combined estimators.


## 4.2. Separate Estimators

On comparing the minimumMSE of the proposed separate estimators y¯sis,i=1,2,…,5 from (69) and (70) with the minimum MSE of the existing separate estimators from (B.1), (B.3), (B.4), (B.6), (B.8), (B.10), (B.12), (B.14), (B.16), (B.18), (B.20), (B.22), (B.24), and (B.26), we get the following efficiency conditions:(87)MSEy¯ms>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200,(88)MSEy¯rs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+U020+U002−2U110−2U101+2U011,(89)MSEy¯ls>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(90)MSEy¯os>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+U002+2U101−U101−U002−U110+U0112U020+U002−2U011,(91)MSEy¯as>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(92)MSEy¯ws>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+a2h2U002−2U002U101−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(93)MSEy¯kcis>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200+λih2U020+Δih2U002−2λihU110−2ΔihU101+2λihΔihU011,(94)MSEy¯ss>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−A2h−A3hA4hA1hA2h−A3h2−A4hA1hA4h−A3hA1hA2h−A3h2,(95)MSEy¯kks>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(96)MSEy¯tcs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2kh∗2U200+14U020+U002−2U002+U101−U110+2kh∗kh∗−1183U020−U002−2U011+12U101−U110+kh∗−12,(97)MSEy¯l1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U1012U020+U1102U002−2U011U101U110U020U0021−ρxzh2,(98)MSEy¯l2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(99)MSEy¯mus>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−U01124U020−Amh2Bmh,(100)MSEy¯mu1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21+Ph−2Oh,(101)MSEy¯mu2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−Oh2Ph.If the conditions (87) to (101) hold then the proposed separate classes of estimators y¯sis,i=1,2,…,5 perform better than the other existing separate estimators.


## 4.3. Comparison of Proposed Combined and Separate Estimators

By comparing the minimumMSE of the proposed combined and separate classes of estimators y¯sic and y¯sis, i=1,2,…,5, we get(102)minMSEy¯sic−minMSEy¯sis=∑h=1LY¯2−Wh2Y¯h2−Y¯2Bi2Ai−Wh2Y¯h2Bih2Aih.If the ratio estimate is veritable and the relationship between auxiliary and study variables within each stratum is a straight line passing through origin then the last term of (102) is broadly small and it vanished.Furthermore, unlessRh is invariant from stratum to stratum, separate estimators probably become more efficient in each stratum if the sample in each stratum is large enough so that the approximate formula for MSEy¯sis,i=1,2,…,5 is valid and the cumulative bias that can affect the proposed estimators is negligible, whereas the proposed combined estimators are to be preferably recommended with only a small sample in each stratum ([29]).Furthermore, the conditions of Subsection 4.1, Subsection 4.2, and Subsection 4.3 are held in practice by being verified through a simulation study.


## 5. Simulation Study

To enhance the credibility of the theoretical development of the proposed combined and separate classes of estimators, we have conducted a simulation study. In the procedure, the following steps are considered:(i)
Generate trivariate random observations of sizeN = 2000 units using a trivariate normal distribution in R software with parameters Y¯=20, X¯=25, Z¯=30, σy=5, σx=6, σz=7 and different amounts of correlation coefficients ρxy, ρyz, and ρxz.(ii)
Stratify the above population into 4 equal disjoint strata and quantify a sample of sizen = 50 units from each stratum.(iii)
Tabulate all necessary statistics.(iv)
Using 15,000 iterations to calculate the absolute relative biasARB and percent relative efficiency PRE of various combined and separate classes of estimators. The ARB and PRE of different estimators T are calculated regarding the classical ratio and usual mean estimators and results are reported in Table 1 and Table 2. The ARB and PRE are calculated using the following expressions.(103)ARB=∑i=115,000y¯r−Y¯∑i=115,000T−Y¯,(104)PRE=∑i=115,000y¯m−Y¯2∑i=115,000T−Y¯2×100,whereT=y¯mc,y¯rc,y¯lc,y¯oc,y¯mc,y¯ac,y¯wc,y¯kcic,i=1,2,3,4,y¯sc,y¯tcc,y¯muc,y¯mu1c,y¯mu2c,y¯sic,i=1,2,…,5,y¯ms,y¯rs,y¯ls,y¯os,y¯ms,y¯as,y¯ws,y¯kcis,i=1,2,3,4,y¯ss,y¯tcs,y¯mus,y¯mu1s,y¯mu2s and y¯sis,i=1,2,…,5.Table 1
ARB and PRE of different combined estimators.
ρxy0.60.70.80.90.90.80.70.6ρyz0.60.70.80.90.80.70.60.5ρxz0.60.70.80.90.70.60.50.4EstimatorsARBPREARBPREARBPREARBPREARBPREARBPREARBPREARBPREy¯mc01000100010001000100010001000100y¯rc196.001117.561154.491211.781236.591168.391127.971102.94y¯lc0178.810230.110329.150564.570550.000321.640230.370175.72y¯oc3.87178.773.46230.433.14328.862.92562.992.95541.453.19320.363.53229.214.02175.53y¯ac18.22178.8125.21230.1144.60329.15207.86564.57103.66550.0033.90321.6420.58230.3715.25175.72y¯wc18.98130.6313.23139.1610.43143.779.14119.346.40406.719.89230.5314.63172.5524.54143.63y¯kc1c2.0497.442.33119.642.67157.843.08218.063.32243.372.86171.812.45130.032.11104.34y¯kc2c2.77116.003.29147.343.99202.004.88307.315.25338.914.24217.453.46155.732.86122.12y¯kc3c9.67164.2016.15219.6645.50327.6781.99646.1772.23594.8048.80314.1417.00213.079.63161.47y¯kc4c2.0196.452.29118.202.62155.573.01213.843.25238.742.81169.442.41128.612.08103.36y¯sc3.29156.264.33196.096.39277.7812.81526.3112.29526.316.10277.784.09196.093.15156.26y¯kkc9.36178.817.67230.116.47329.155.71564.575.39550.006.14321.647.28230.379.17175.72y¯tcc1.7485.041.8986.932.0388.402.2389.992.1690.631.9689.921.8088.571.6987.28y¯l1c9.90178.818.00230.116.73329.155.87564.574.07550.004.79321.645.76230.377.18175.72y¯l2c6.76178.819.24230.1114.17329.1528.50564.5714.63550.009.47321.646.91230.375.35175.72y¯muc3.78180.615.19236.107.96345.3916.52674.0916.29731.607.47348.514.74230.253.47176.36y¯mu1c3.56180.895.99236.778.21346.1117.11674.8717.09732.417.76349.125.16231.584.37176.72y¯mu2c3.77180.965.87236.978.01346.7817.27675.1717.17733.017.94349.245.24232.194.20176.87y¯s1c21.07182.8414.45239.565.45349.673.45678.583.18750.015.09351.6814.23234.4016.98178.75y¯s2c1.27182.781.38238.111.98348.072.61678.001.22748.161.08351.241.06234.2312178.67y¯s3c3.83181.985.22238.547.97347.8816.52678.1717.52749.217.73351.454.89234.383.61178.73y¯s4c3.83182.835.22239.277.97349.3616.52678.4417.52749.917.73351.614.89234.393.61178.74y¯s5c3.82181.925.21238.547.97347.8816.52678.1717.52749.207.73351.454.89234.233.60178.73Table 2
ARB and PRE of different separate estimators.
ρxy0.60.70.80.90.90.80.70.6ρyz0.60.70.80.90.80.70.60.5ρxz0.60.70.80.90.70.60.50.4EstimatorsARBPREARBPREARBPREARBPREARBPREARBPREARBPREARBPREy¯ms01000100010001000100010001000100y¯rs192.121119.541154.961212.281236.661167.431128.431103.45y¯ls0179.220230.970329.790565.160550.800322.110230.060179.23y¯os3.86179.563.45230.953.15328.242.95563.352.97542.163.20320.773.51230.004.00177.21y¯as18.19179.2225.24230.9744.63329.79207.92565.16103.86550.8034.33322.1121.76230.0615.23179.23y¯ws19.00131013.29139.8910.88144.869.15120.856.44408.549.93231114.77171.7324.64142.28y¯kc1s2.1098.002.31120.202.77158.883.12218.993.37244.062.89172.442.49130.882.14103.48y¯kc2s2.75116.923.40148.734.00202.914.90308.005.31342.154.30219.353.36151.532.70120.77y¯kc3s9.60167.4516.19221.3446.00327.8982.08648.0172.45596.6048.89313.3117.11210.989.43160.64y¯kc4s2.1296.872.30117.992.59155.023.09214.473.30242.002.88173.072.45127.762.09104.55y¯ss3.31156.864.35196.976.47279.0012.94527.2112.37528.216.12277.874.15196.833.19156.99y¯kks9.35179.227.66230.976.46329.795.73565.165.42550.806.15322.117.31230.069.19179.23y¯tcs1.7685.911.9186.992.0988.882.2690.432.1090.981.9989.981.8489.251.7388.03y¯l1s9.93179.228.11230.976.77329.795.89565.164.13550.804.82322.115.70230.067.20179.23y¯l2s6.80179.229.29230.9715.17329.7929.00565.1615.66550.809.44322.116.96230.065.39178.23y¯mus3.84181.095.23236.977.87346.6516.71674.8516.38732.287.51348.814.83232.033.52178.24y¯mu1s3.62181.246.09237.848.24347.1517.13675.5117.13732.767.80349.955.20232.134.50177.89y¯mu2s3.84181.855.90237.578.20349.1717.76676.7417.10732.997.98351.535.32234.114.23176.97y¯s1s20.76183.6414.4724155.44352.993.46679.743.20751.325.15353.8214.67236.7817.00182.52y¯s2s1.28182.851.39240.071.30350.996.61678.681.25750.7310352.4611235.5214180.76y¯s3s3.88182.175.25239.478.07350.5917.00678.9617.59749.837.80352.694.95235.753.63180.77y¯s4s3.88182.915.25239.988.07351.8716.90679.0317.60750.797.81352.964.91236.043.601817y¯s5s3.86182.165.22239.478.06350.5917.01678.9617.59749.827.880352.694.95235.753.62180.77The simulation findings of the combined and separate classes of estimators are exposited in terms ofARB and PRE in Table 1 and Table 2 for different values of correlation coefficients. The results exhibit the dominance of the proposed combined and separate classes of estimators y¯sic and y¯sis, i=1,2,…,5, respectively, over the combined and separate usual mean estimators, classical ratio and regression estimators, Olkin [24] type estimator, Abu–Dayyeh et al. [25] type estimators, Kadilar and Cingi [7] type estimator, Singh et al. [16] estimator, Koyuncu and Kadilar [12] estimator, Tailor and Chouhan [14] estimator, Lone et al. [18, 19] estimators and Muneer et al. [21, 22] estimators in terms of PRE. Also, the proposed combined and separate class of estimators y¯s1c and y¯s1s are found to be most efficient among the proposed combined and separate classes of estimators for each passably chosen values of correlation coefficients.


## 6. Conclusion

In this article, we propose some improved classes of estimators for population mean by extending the work of Bhushan et al. [28] using bivariate auxiliary information under SSRS. The mathematical expressions of bias and MSE of the proposed classes of estimators are obtained up to the first order of approximation. The efficiency conditions are derived under which the suggested estimators perform better than the other existing estimators. In support of the theoretical results, a simulation study is carried out using an artificially generated population with various amounts of correlation coefficients. From the perusal of the theoretical and simulation results reported in Table 1 and Table 2, we conclude that:(i)
The proposed combined classes of estimatorsy¯sic, i=1,2,…,5 perform better than the combined form of usual mean estimator y¯mc, classical ratio and regression estimators y¯rc & y¯lc, Olkin [24] type estimator y¯oc, Abu–Dayyeh et al. [25] type estimators y¯ac & y¯wc, Kadilar and Cingi [7] type estimators y¯kcic,i=1,2,3,4, Koyuncu and Kadilar [12] estimator y¯kkc, Singh et al. [16] estimator y¯sc, Tailor and Chouhan [14] estimator y¯tcc, Lone et al. [18, 19] estimators y¯lic,i=1,2 and Muneer et al. [21, 22] estimators y¯muc & y¯muic,i=1,2 for different values of correlation coefficients.(ii)
The proposed separate classes of estimatorsy¯sis, i=1,2,…,5 dominate the separate form of usual mean estimator y¯ms, classical ratio and regression estimators y¯rs & y¯ls, Olkin [24] type estimator y¯os, Abu-Dayyeh et al. [25] type estimators y¯as & y¯ws, Kadilar and Cingi [7] type estimators y¯kcis,i=1,2,3,4, Koyuncu and Kadilar [12] estimator y¯kks, Singh et al. [16] estimator y¯ss, Tailor and Chouhan [14] estimator y¯tcs, Lone et al. [18, 19] estimators y¯lis,i=1,2 and Muneer et al. [21, 22] estimators y¯mus & y¯muis,i=1,2 for different values of correlation coefficients.(iii)
Since, the proposed combined and separate classes of estimatorsy¯sic and y¯sis, i=1,2,…,5 are, respectively, superior to the combined and separate ratio and chain ratio exponential estimators y¯mu1c, y¯mu1s and y¯mu2c, y¯mu2s envisaged by Muneer et al. [22] consequently the proposed combined and separate classes of estimators y¯sic and y¯sis, i=1,2,…,5 will also dominate those 65 estimators that are the members of the combined and separate ratio and chain ratio exponential estimators y¯mu1c, y¯mu2c and y¯mu1s, y¯mu2s, respectively.(iv)
The proposed combined and separate class of estimatorsy¯s1c and y¯s1s perform better among the proposed classes of estimators.(v)
The proposed separate classes of estimatorsy¯sis, i=1,2,…,5 dominate the proposed combined classes of estimators y¯sic, i=1,2,…,5 in terms of greater PRE for various amounts of correlation coefficients.(vi)
ThePRE of the proposed combined and separate classes of estimators increases as the values of correlation coefficients increase and vice versa.Thus, the proposed combined and separate classes of estimators can be preferably used by the survey professionals in practice.



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*Source: 2660114-2022-08-31.xml*