# On Efficient Estimation of the Population Mean under Stratified Ranked Set Sampling

Authors: Shashi Bhushan; Anoop Kumar; Sana Shahab; Showkat Ahmad Lone; Md Tanwir Akhtar
Journal: Journal of Mathematics
 (2022)

---

## Abstract

This paper considers some efficient combined and separate classes of estimators of the population mean in the presence of bivariate auxiliary information under stratified ranked set sampling. The mean square errorMSE expressions of the proffered combined and separate classes of estimators are derived to the first order of approximation. The theoretical conditions are obtained under which the proffered combined and separate classes of estimators perform better than the existing combined and separate class of estimators. Subsequently, numerical and simulation studies are performed using real and artificially generated populations. The numerical and simulation results are found to be rewarding, showing the superiority of the proffered estimators over the existing estimators.


---

## Body

## 1. Introduction

It is evident from sample surveys that the convenient utilization of auxiliary information may always boost the efficiency of the estimator. This information may be used either at the design phase (sampling design) or at the estimation phase or at both phases. It is a well-known fact that when this information is employed for the estimation phase, the ratio, exponential, product, and regression-type estimators are often used in distinct aspects. Various esteemed authors have considered this auxiliary information at the estimation stage and envisaged a wide range of modified estimators till date. However, the literature encompasses a variety of estimators in the presence of multiple auxiliary information. Koyuncu and Kadilar [1] introduced a family of estimators of the population mean using stratified simple random sampling SSRS. Tailor et al. [2] suggested the ratio-cum-product estimator in SSRS for estimating the population mean. Tailor and Chouhan [3] envisaged the ratio-cum-product-type exponential estimator of the finite population mean. Lone et al. [4] employed a generalized ratio-cum-product-type exponential estimator in SSRS, whereas Lone et al. [5] mooted enhanced separate classes of estimators of the population mean. Muneer et al. [6] investigated a class of combined estimators in SSRS. Recently, Muneer et al. [7] investigated the apparent family of chain exponential estimators in SSRS.The concept of ranked set samplingRSS was initiated by Mclntyre [8] but did not furnish any mathematical support. Takahasi and Wakimoto [9] improved the lacuna of the RSS method by furnishing the necessary mathematical theory. Dell and Clutter [10] demonstrated that the mean under RSS is an unbiased estimator of the population mean under the condition of perfect and imperfect ranking. Muttlak [11] suggested the estimation of parameters in simple linear regression under RSS. Samawi and Muttlak [12] shown that ranking of the denominator variable of the ratio estimator improves the efficiency. The interested readers may refer to some recent works such as Bhushan and Kumar [13–16] and Bhushan et al. [17, 18] for a comprehensive study about RSS. In extensive surveys, when the information on more than one auxiliary variable is available, then, Abu-Dayyeh et al. [19] introduced two estimators under RSS for estimating the population mean. Following Olkin [20], Mehta and Mandowara [21] suggested an improved ratio estimator under RSS. Khan and Shabbir [22] introduced a generalized exponential-type ratio-cum-ratio estimators using RSS and SRSS for estimating the population mean. This study is aimed at proffering some efficient classes of combined and separate estimators in the presence of bivariate auxiliary information using SRSS.Neutrosophic statistics is an extension of classical statistics and is applied when the data is coming from a complex process or from an uncertain environment. Smarandache [23] considered a neutrosophic set as a generalization of the intuitionistic fuzzy set. Smarandache [24] discussed the neutrosophic measure, neutrosophic integral, and neutrosophic probability. Alhabib et al. [25] discussed some neutrosophic probability distributions. Aslam et al. [26] proposed a new diagnosis test under the neutrosophic statistics with an application to diabetic patients, whereas Aslam et al. [27] devised a vague data analysis using neutrosophic the Jarque-Bera test. Aslam [27] suggested the neutrosophic statistical test for counts in climatology. Tahir et al. [28] investigated neutrosophic ratio-type estimators for estimating the population mean. Recently, Vishwakarma and Singh [29] introduced a generalized estimator for computation of the population mean under neutrosophic RSS with an application to solar energy data. In future, we intend to extend the current study using neutrosophic statistics.


### 1.1. Sampling Methodology

The procedure of ranked set sampling is based on selectingm independent random samples of size m units with equal probability and with replacement from a population of size N units. The units of each random sample are now ranked with respect to the variable of choice. Let the study variable be denoted by Y and the two auxiliary variables by X and Z. Then, m2 trivariate samples are selected randomly from the population and distributed into m sets, each of size m units. Let the ranking be performed over each unit of the auxiliary variable X. Now, from the first sample, the unit with the smallest rank of X, together with the variables Y and Z associated with the smallest rank of X, is measured. From the second sample of the same size, the variables Y and Z associated with the second smallest rank of X are measured. This procedure is continued until the variables Y and Z associated with the highest rank of X are measured from the mth sample. This process completes a single cycle. The whole procedure is repeated r times to get a desired sample of size n=mr units.The procedure of stratified ranked set samplingSRSS is consisting of quantifying mh2 trivariate random samples from the hth stratum of the population. These quantified random samples are fixed up into mh sets, each of size mh units. The procedure of RSS is now employed on each set to get mh sets of ranked set samples, each of size mh units. These ranked set samples are jointly formed mh sets, each of size mh units. Iterate this procedure r times for each stratum to get the desired stratified ranked set sample of size nh=mhr units.Consider a finite populationκ=κ1,κ2,⋯,κN based on N identifiable units with a study variable Y and two auxiliary variables X and Z associated with each unit κi,i=1,2,⋯,N of the population. Let the population be divided into L disjoint strata with stratum h based on Nh, h=1,2,,⋯,L units. Let (Yh1j, Xh1j, Zh1j), (Yh2j, Xh2j, Zh2j),…,(Yhmhj, Xhmhj, Zhmhj) be the stratified ranked set sample for jth,j=1,2,⋯,r cycle in the hth stratum. Here, Yhij and Zhij are the ith judgement order for the variables Y and Z and Xhij is the ith order statistics for variable X in the ith sample at the jth cycle of the hth stratum. Let y¯SRSS=∑h=1LWhy¯hRSS, x¯SRSS=∑h=1LWhx¯hRSS, and z¯SRSS=∑h=1LWhz¯hRSS be the stratified ranked set sample means corresponding to the population means Y¯=∑h=1LWhY¯h, X¯=∑h=1LWhX¯h, and Z¯=∑h=1LWhZ¯h of variables Y, X, and Z, where Wh=Nh/N is the weight in stratum h. Let y¯hRSS=∑i=1mh∑j=1rYhij/mhr, x¯hRSS=∑i=1mh∑j=1rXhij/mhr, and z¯hRSS=∑i=1mh∑j=1rZhij/mhr be the stratified ranked set sample means corresponding to the population means Y¯h=∑j=1NhYhij/Nh, X¯h=∑j=1NhXhij/Nh, and Z¯h=∑j=1NhZhij/Nh of variables Y, X, and Z in stratum h. Let syh2=∑h=1LYhi−y¯hRSS2/nh−1, sxh2=∑h=1LXhi−x¯hRSS2/nh−1, szh2=∑h=1LZhi−z¯hRSS2/nh−1, sxyh=∑h=1LXhi−x¯hRSSYhi−y¯hRSS/nh−1, syzh=∑h=1LYhi−y¯hRSSZhi−z¯hRSS/nh−1, and sxzh=∑h=1LXhi−x¯hRSSZhi−z¯hRSS/nh−1 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¯hYhi−Y¯h/Nh−1, Syzh=∑h=1LYhi−Y¯hZhi−Z¯h/Nh−1, and Sxzh=∑h=1LXhi−X¯hZhi−Z¯h/Nh−1 in the stratum h. Let Cyh, Cxh, and Czh be the population coefficient of variation of variables Y, X, and Z, respectively.To derive theMSE of the proffered combined estimators, the following notations will be utilized throughout the paper.
(1)y¯SRSS=Y¯1+ε0,x¯SRSS=X¯1+ε1,z¯SRSS=Z¯1+ε2,such that Eε0=Eε1=Eε2=0 and
(2)Vrst=∑h=1LWhr+s+tEy¯SRSS−Y¯rx¯SRSS−X¯sz¯SRSS−Z¯tY¯rX¯sZ¯t.On the basis of (2), it can be written as follows:
(3)Eε02=∑h=1LWh2γhCyh2−Dyhi2=V200,Eε12=∑h=1LWh2γhCxh2−Dxhi2=V020,Eε22=∑h=1LWh2γhCzh2−Dzhi2=V002,Eε0,ε1=∑h=1LWh2γhρxyhCxhCyh−Dxyhi=V110,Eε1,ε2=∑h=1LWh2γhρxzhCxhCzh−Dxzhi=V011,Eε0,ε2=∑h=1LWh2γhρyzhCyhCzh−Dyzhi=V101,where γh=1/mhr,Cxh=Sxh/X¯,Czh=Szh/Z¯,Cyh=Syh/Y¯, Dxhi2=∑i=1mhτxhi2/mh2rX¯2, Dzhi2=∑i=1mhτzhi2/mh2rZ¯2, Dyhi2=∑i=1mhτyhi2/mh2rY¯2, Dyxhi=∑i=1mhτyxhi/mh2rY¯X¯, Dyzhi=∑i=1mhτyzhi/mh2rY¯Z¯, Dxzhi=∑i=1mhτxzhi/mh2rX¯Z¯,τxhi=Xhi−X¯h,τyhi=Yhi−Y¯h,τxyhi=Xhi−X¯hYhi−Y¯h,τyzhi=Yhi−Y¯hZhi−Z¯h,andτxzhi=Xhi−X¯hZhi−Z¯h.Again, to determine the properties of the separate estimators, the following notations will be used throughout the paper.(4)Lety¯hRSS=Y¯h1+ε0h,x¯hRSS=X¯h1+ε1h,z¯hRSS=Z¯h1+ε2h,such that Eε0h=Eε1h=Eε2h=0,Eε0h2=γhCyh2−Myhi2=U200,Eε1h2=γhCxh2−Mxhi2=U020,Eε2h2=γhCzh2−Mzhi2=U002,Eε0h,ε1h=γhρxyhCxhCyh−Mxyhi=U110,Eε1h,ε2h=γhρxhzhCxhCzh−Mxzhi=U011,andEε0h,ε2h=γhρyhzhCyhCzh−Myzhi=U101, where Cxh=Sxh/X¯h,Czh=Szh/Z¯h,Cyh=Syh/Y¯h, Mxhi2=∑i=1mhτxhi2/mh2rX¯h2, Mzhi2=∑i=1mhτzhi2/mh2rZ¯h2, Myhi2=,Mxyhi=,Myzhi=∑i=1mhτyzhi/mh2rY¯hZ¯h,andMxzhi=∑i=1mhτxzhi/mh2rX¯hZ¯h.The paper is divided into some sections. Section2 is devoted to the existing combined and separate estimators, whereas Section 3 deals with the proffered combined classes of estimators. The suggested combined and separate estimators are given in Section 3 along with their properties. The theoretical comparisons of the proffered and existing combined and separate estimators are given in Section 4. The theoretical results are illustrated through a simulation study in Section 5. Lastly, the study is concluded in Section 6.


## 1.1. Sampling Methodology

The procedure of ranked set sampling is based on selectingm independent random samples of size m units with equal probability and with replacement from a population of size N units. The units of each random sample are now ranked with respect to the variable of choice. Let the study variable be denoted by Y and the two auxiliary variables by X and Z. Then, m2 trivariate samples are selected randomly from the population and distributed into m sets, each of size m units. Let the ranking be performed over each unit of the auxiliary variable X. Now, from the first sample, the unit with the smallest rank of X, together with the variables Y and Z associated with the smallest rank of X, is measured. From the second sample of the same size, the variables Y and Z associated with the second smallest rank of X are measured. This procedure is continued until the variables Y and Z associated with the highest rank of X are measured from the mth sample. This process completes a single cycle. The whole procedure is repeated r times to get a desired sample of size n=mr units.The procedure of stratified ranked set samplingSRSS is consisting of quantifying mh2 trivariate random samples from the hth stratum of the population. These quantified random samples are fixed up into mh sets, each of size mh units. The procedure of RSS is now employed on each set to get mh sets of ranked set samples, each of size mh units. These ranked set samples are jointly formed mh sets, each of size mh units. Iterate this procedure r times for each stratum to get the desired stratified ranked set sample of size nh=mhr units.Consider a finite populationκ=κ1,κ2,⋯,κN based on N identifiable units with a study variable Y and two auxiliary variables X and Z associated with each unit κi,i=1,2,⋯,N of the population. Let the population be divided into L disjoint strata with stratum h based on Nh, h=1,2,,⋯,L units. Let (Yh1j, Xh1j, Zh1j), (Yh2j, Xh2j, Zh2j),…,(Yhmhj, Xhmhj, Zhmhj) be the stratified ranked set sample for jth,j=1,2,⋯,r cycle in the hth stratum. Here, Yhij and Zhij are the ith judgement order for the variables Y and Z and Xhij is the ith order statistics for variable X in the ith sample at the jth cycle of the hth stratum. Let y¯SRSS=∑h=1LWhy¯hRSS, x¯SRSS=∑h=1LWhx¯hRSS, and z¯SRSS=∑h=1LWhz¯hRSS be the stratified ranked set sample means corresponding to the population means Y¯=∑h=1LWhY¯h, X¯=∑h=1LWhX¯h, and Z¯=∑h=1LWhZ¯h of variables Y, X, and Z, where Wh=Nh/N is the weight in stratum h. Let y¯hRSS=∑i=1mh∑j=1rYhij/mhr, x¯hRSS=∑i=1mh∑j=1rXhij/mhr, and z¯hRSS=∑i=1mh∑j=1rZhij/mhr be the stratified ranked set sample means corresponding to the population means Y¯h=∑j=1NhYhij/Nh, X¯h=∑j=1NhXhij/Nh, and Z¯h=∑j=1NhZhij/Nh of variables Y, X, and Z in stratum h. Let syh2=∑h=1LYhi−y¯hRSS2/nh−1, sxh2=∑h=1LXhi−x¯hRSS2/nh−1, szh2=∑h=1LZhi−z¯hRSS2/nh−1, sxyh=∑h=1LXhi−x¯hRSSYhi−y¯hRSS/nh−1, syzh=∑h=1LYhi−y¯hRSSZhi−z¯hRSS/nh−1, and sxzh=∑h=1LXhi−x¯hRSSZhi−z¯hRSS/nh−1 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¯hYhi−Y¯h/Nh−1, Syzh=∑h=1LYhi−Y¯hZhi−Z¯h/Nh−1, and Sxzh=∑h=1LXhi−X¯hZhi−Z¯h/Nh−1 in the stratum h. Let Cyh, Cxh, and Czh be the population coefficient of variation of variables Y, X, and Z, respectively.To derive theMSE of the proffered combined estimators, the following notations will be utilized throughout the paper.
(1)y¯SRSS=Y¯1+ε0,x¯SRSS=X¯1+ε1,z¯SRSS=Z¯1+ε2,such that Eε0=Eε1=Eε2=0 and
(2)Vrst=∑h=1LWhr+s+tEy¯SRSS−Y¯rx¯SRSS−X¯sz¯SRSS−Z¯tY¯rX¯sZ¯t.On the basis of (2), it can be written as follows:
(3)Eε02=∑h=1LWh2γhCyh2−Dyhi2=V200,Eε12=∑h=1LWh2γhCxh2−Dxhi2=V020,Eε22=∑h=1LWh2γhCzh2−Dzhi2=V002,Eε0,ε1=∑h=1LWh2γhρxyhCxhCyh−Dxyhi=V110,Eε1,ε2=∑h=1LWh2γhρxzhCxhCzh−Dxzhi=V011,Eε0,ε2=∑h=1LWh2γhρyzhCyhCzh−Dyzhi=V101,where γh=1/mhr,Cxh=Sxh/X¯,Czh=Szh/Z¯,Cyh=Syh/Y¯, Dxhi2=∑i=1mhτxhi2/mh2rX¯2, Dzhi2=∑i=1mhτzhi2/mh2rZ¯2, Dyhi2=∑i=1mhτyhi2/mh2rY¯2, Dyxhi=∑i=1mhτyxhi/mh2rY¯X¯, Dyzhi=∑i=1mhτyzhi/mh2rY¯Z¯, Dxzhi=∑i=1mhτxzhi/mh2rX¯Z¯,τxhi=Xhi−X¯h,τyhi=Yhi−Y¯h,τxyhi=Xhi−X¯hYhi−Y¯h,τyzhi=Yhi−Y¯hZhi−Z¯h,andτxzhi=Xhi−X¯hZhi−Z¯h.Again, to determine the properties of the separate estimators, the following notations will be used throughout the paper.(4)Lety¯hRSS=Y¯h1+ε0h,x¯hRSS=X¯h1+ε1h,z¯hRSS=Z¯h1+ε2h,such that Eε0h=Eε1h=Eε2h=0,Eε0h2=γhCyh2−Myhi2=U200,Eε1h2=γhCxh2−Mxhi2=U020,Eε2h2=γhCzh2−Mzhi2=U002,Eε0h,ε1h=γhρxyhCxhCyh−Mxyhi=U110,Eε1h,ε2h=γhρxhzhCxhCzh−Mxzhi=U011,andEε0h,ε2h=γhρyhzhCyhCzh−Myzhi=U101, where Cxh=Sxh/X¯h,Czh=Szh/Z¯h,Cyh=Syh/Y¯h, Mxhi2=∑i=1mhτxhi2/mh2rX¯h2, Mzhi2=∑i=1mhτzhi2/mh2rZ¯h2, Myhi2=,Mxyhi=,Myzhi=∑i=1mhτyzhi/mh2rY¯hZ¯h,andMxzhi=∑i=1mhτxzhi/mh2rX¯hZ¯h.The paper is divided into some sections. Section2 is devoted to the existing combined and separate estimators, whereas Section 3 deals with the proffered combined classes of estimators. The suggested combined and separate estimators are given in Section 3 along with their properties. The theoretical comparisons of the proffered and existing combined and separate estimators are given in Section 4. The theoretical results are illustrated through a simulation study in Section 5. Lastly, the study is concluded in Section 6.


## 2. Review of Literature

This section considers the review of existing combined and separate classes of estimators.


### 2.1. Combined Estimators

The classical combined ratio estimator of population meanY¯ using bivariate auxiliary information under SRSS is defined as follows:
(5)y¯rc=y¯SRSSX¯x¯SRSSZ¯z¯SRSS.The classical combined regression estimator of population meanY¯ based on bivariate auxiliary information under SRSS is defined as follows:
(6)y¯lrc=y¯SRSS+β1X¯−x¯SRSS+β2Z¯−z¯SRSS,where β1 and β2 are the regression coefficients of Y on X and Z, respectively.Motivated by Khoshnevisan et al. [30] and Koyuncu and Kadilar [1], we introduce a general family of combined estimators for population mean Y¯ under SRSS as follows:
(7)y¯kkc=y¯SRSSaX¯+bλ1ax¯SRSS+b+1−λ1aX¯+bg1cZ¯+dλ2cz¯SRSS+d+1−λ2cZ¯+dg2,where λ1, λ2, g1, and g2 are suitably chosen scalars, whereas a≠0, b and c≠0, d are either real numbers or the function of known parameters of variables X and Z such as standard deviation, coefficients of variation, coefficient of skewness, coefficient of kurtosis, and correlation coefficient.On the lines of Tailor and Chouhan [3], one may suggest the following combined class of the estimator of population mean under SRSS as follows:
(8)y¯tcc=ky¯SRSSexpX¯−x¯SRSSX¯+x¯SRSSexpz¯SRSS−Z¯z¯SRSS+Z¯,where k is a duly opted scalar.Following Lone et al. [4], we define a generalized combined ratio-cum-product-type exponential estimator in SRSS as follows:
(9)y¯l1c=y¯SRSSexpL1X¯−x¯SRSSX¯+x¯SRSS+L2z¯SRSS−Z¯z¯SRSS+Z¯,where L1 and L2 are suitably chosen scalars.Following Lone et al. [5], one may introduce a combined estimator for estimating population mean Y¯ under SRSS as follows:
(10)y¯l2=y¯SRSSaX¯+bax¯SRSS+bcz¯SRSS+dcZ¯+d.On the lines of Muneer et al. [6], one may investigate a combined class of estimators under SRSS as follows:
(11)y¯muc=k3y¯SRSS−k4x¯SRSS−X¯Θ2−expz¯SRSS−Z¯z¯SRSS+Z¯+1−ΘexpZ¯−z¯SRSSZ¯−z¯SRSS,where k3 and k4 are duly opted scalars.Khan and Shabbir [22] suggested a generalized combined exponential-type ratio-cum-ratio estimators under SRSS as follows:
(12)y¯ksc=y¯SRSSexpX¯−x¯SRSSX¯+η1−1x¯SRSSexpZ¯−z¯SRSSZ¯+η2−1z¯SRSS,where η1 and η2 are duly opted scalars.Following Muneer et al. [7], one can introduce a combined chain ratio exponential family of estimators in SRSS as follows:
(13)y¯mu1c=y¯SRSSX¯∗w1x¯SRSS∗+1−w1X¯∗α1Z¯∗w2z¯SRSS∗+1−w2Z¯∗α2×expα3X¯−x¯SRSSX¯+x¯SRSSexpα4Z¯−z¯SRSSZ¯+z¯SRSS.On the lines of Searls [31], the above-combined chain ratio exponential family of estimators becomes
(14)y¯mu2c=k1y¯SRSSX¯∗w1x¯SRSS∗+1−w1X¯∗α1Z¯∗w2z¯SRSS∗+1−w2Z¯∗α2×expα3X¯−x¯SRSSX¯+x¯SRSSexpα4Z¯−z¯SRSSZ¯+z¯SRSS,where X¯∗=aX¯+b, x¯SRSS∗=ax¯SRSS+b, Z¯∗=cZ¯+d, z¯SRSS∗=cz¯SRSS+d, w1,w2=0,1, and k1 are duly opted scalars; αj,j=1,2,3,4 assumes values −1, 0, and +1 in order to form different new and existing estimators.It is noticed that the minimumMSE of the Koyuncu and Kadilar- [1] type estimator y¯kkc, [4, 5] estimator y¯l1c and y¯l2c and Khan and Shabbir- [22] estimator y¯ksc are equal to the minimum MSE of classical regression estimator y¯lrc.TheMSE of the estimators considered in this section is discussed in Appendix A.


### 2.2. Separate Estimators

The classical separate ratio estimator of population meanY¯ using bivariate auxiliary information under SRSS can be defined as follows:
(15)y¯rs=∑h=1LWhy¯hRSSX¯hx¯hRSSZ¯hz¯hRSS.The classical separate regression estimator of population meanY¯ under bivariate auxiliary information using SRSS is as follows:
(16)y¯lrs=∑h=1LWhy¯hRSS+β1hX¯h−x¯hRSS+β2hZ¯h−z¯hRSS,where β1h and β2h are the regression coefficients of Y on X and Z in stratum h, respectively.Following Koyuncu and Kadilar [1], one may consider a general family of separate estimators for population mean Y¯ under SRSS as follows:
(17)y¯kks=∑h=1LWhy¯hRSSahX¯h+bhλ1hahx¯hRSS+bh+1−λ1hahX¯h+bhg1chZ¯h+dhλ2hchz¯hRSS+dh+1−λ2hchZ¯h+dhg2,where λ1h, λ2h, g1, and g2 are suitably chosen constants whereas ah≠0, bh and ch≠0, dh are either real numbers or the function of known parameters of variables X and Z, respectively, in stratum h.Following Tailor and Chouhan [3], one may suggest the following separate class of the estimator of the population mean using bivariate auxiliary information under SRSS as follows:
(18)y¯tcs=∑h=1LWhkhy¯hRSSexpX¯h−x¯hRSSX¯h+x¯hRSSexpz¯hRSS−Z¯hz¯hRSS+Z¯h,where kh is a duly opted scalar.On the lines of Lone et al. [4], we consider a generalized separate ratio-cum-product-type exponential estimator in SRSS as follows:
(19)y¯l1s=∑h=1LWhy¯hRSSexpL1hX¯h−x¯hRSSX¯h+x¯hRSS+L2hz¯hRSS−Z¯hz¯hRSS+Z¯h,where L1h and L2h are suitably chosen scalars.The separate version of the Lone et al. [5] estimator for estimating population mean Y¯ under SRSS is defined as follows:
(20)y¯l2s=∑h=1LWhy¯hRSSahX¯h+bhahx¯hRSS+bhchz¯hRSS+dhchZ¯h+dh.Following Muneer et al. [6], we introduce a separate class of estimators in SRSS as follows:
(21)y¯mus=∑h=1LWhk3hy¯hRSS−k4hx¯hRSS−X¯hΘh2−expz¯hRSS−Z¯hz¯hRSS+Z¯h+1−ΘhexpZ¯h−z¯hRSSZ¯h−z¯hRSS,where k3h and k4h are duly opted scalars.Khan and Shabbir [22] suggested a generalized separate exponential-type ratio-cum-ratio estimator under SRSS as follows:
(22)y¯kss=∑h=1LWhy¯hRSSexpX¯h−x¯hRSSX¯h+η1h−1x¯hRSSexpZ¯h−z¯hRSSZ¯h+η2h−1z¯hRSS,where η1h and η2h are duly opted scalars.Motivated by Muneer et al. [7], a separate chain ratio exponential family of estimators is defined in SRSS as follows:
(23)y¯mu1s=∑h=1LWhy¯hRSSX¯h∗w1hx¯hRSS∗+1−w1hX¯h∗α1hZ¯h∗w2hz¯hRSS∗+1−w2hZ¯h∗α2h×expα3hX¯h−x¯hRSSX¯h+x¯hRSSexpα4hZ¯h−z¯hRSSZ¯h+z¯hRSS.On the lines of Searls [31], the above separate chain ratio exponential family of estimators becomes
(24)y¯mu2s=∑h=1LWhk1hy¯hRSSX¯h∗w1hx¯hRSS∗+1−w1hX¯h∗α1hZ¯h∗w1hz¯hRSS∗+1−w1hZ¯h∗α2h×expα3hX¯h−x¯hRSSX¯h+x¯hRSSexpα4hZ¯h−z¯hRSSZ¯h+z¯hRSS,where X¯∗=ahX¯h+bh, x¯hRSS∗=ahx¯hRSS+bh, Z¯∗=chZ¯hRSS+dh, z¯hRSS∗=chz¯hRSS+dh, and w1h,w2h=0,1 and k1h is a duly opted scalar; αjh,j=1,2,3,4 assumes values −1, 0, and +1 in order to form different new and existing estimators.It is to be noted that the minimumMSE of Koyuncu and Kadilar- [1] type estimator y¯kks, Lone et al.- [4, 5] type estimator y¯l1s, y¯l2s, and Khan and Shabbir- [22] estimator y¯kss are similar to the minimum MSE of classical regression estimator y¯lrs.TheMSE of the estimators discussed in this section is given in Appendix B for ready reference.


## 2.1. Combined Estimators

The classical combined ratio estimator of population meanY¯ using bivariate auxiliary information under SRSS is defined as follows:
(5)y¯rc=y¯SRSSX¯x¯SRSSZ¯z¯SRSS.The classical combined regression estimator of population meanY¯ based on bivariate auxiliary information under SRSS is defined as follows:
(6)y¯lrc=y¯SRSS+β1X¯−x¯SRSS+β2Z¯−z¯SRSS,where β1 and β2 are the regression coefficients of Y on X and Z, respectively.Motivated by Khoshnevisan et al. [30] and Koyuncu and Kadilar [1], we introduce a general family of combined estimators for population mean Y¯ under SRSS as follows:
(7)y¯kkc=y¯SRSSaX¯+bλ1ax¯SRSS+b+1−λ1aX¯+bg1cZ¯+dλ2cz¯SRSS+d+1−λ2cZ¯+dg2,where λ1, λ2, g1, and g2 are suitably chosen scalars, whereas a≠0, b and c≠0, d are either real numbers or the function of known parameters of variables X and Z such as standard deviation, coefficients of variation, coefficient of skewness, coefficient of kurtosis, and correlation coefficient.On the lines of Tailor and Chouhan [3], one may suggest the following combined class of the estimator of population mean under SRSS as follows:
(8)y¯tcc=ky¯SRSSexpX¯−x¯SRSSX¯+x¯SRSSexpz¯SRSS−Z¯z¯SRSS+Z¯,where k is a duly opted scalar.Following Lone et al. [4], we define a generalized combined ratio-cum-product-type exponential estimator in SRSS as follows:
(9)y¯l1c=y¯SRSSexpL1X¯−x¯SRSSX¯+x¯SRSS+L2z¯SRSS−Z¯z¯SRSS+Z¯,where L1 and L2 are suitably chosen scalars.Following Lone et al. [5], one may introduce a combined estimator for estimating population mean Y¯ under SRSS as follows:
(10)y¯l2=y¯SRSSaX¯+bax¯SRSS+bcz¯SRSS+dcZ¯+d.On the lines of Muneer et al. [6], one may investigate a combined class of estimators under SRSS as follows:
(11)y¯muc=k3y¯SRSS−k4x¯SRSS−X¯Θ2−expz¯SRSS−Z¯z¯SRSS+Z¯+1−ΘexpZ¯−z¯SRSSZ¯−z¯SRSS,where k3 and k4 are duly opted scalars.Khan and Shabbir [22] suggested a generalized combined exponential-type ratio-cum-ratio estimators under SRSS as follows:
(12)y¯ksc=y¯SRSSexpX¯−x¯SRSSX¯+η1−1x¯SRSSexpZ¯−z¯SRSSZ¯+η2−1z¯SRSS,where η1 and η2 are duly opted scalars.Following Muneer et al. [7], one can introduce a combined chain ratio exponential family of estimators in SRSS as follows:
(13)y¯mu1c=y¯SRSSX¯∗w1x¯SRSS∗+1−w1X¯∗α1Z¯∗w2z¯SRSS∗+1−w2Z¯∗α2×expα3X¯−x¯SRSSX¯+x¯SRSSexpα4Z¯−z¯SRSSZ¯+z¯SRSS.On the lines of Searls [31], the above-combined chain ratio exponential family of estimators becomes
(14)y¯mu2c=k1y¯SRSSX¯∗w1x¯SRSS∗+1−w1X¯∗α1Z¯∗w2z¯SRSS∗+1−w2Z¯∗α2×expα3X¯−x¯SRSSX¯+x¯SRSSexpα4Z¯−z¯SRSSZ¯+z¯SRSS,where X¯∗=aX¯+b, x¯SRSS∗=ax¯SRSS+b, Z¯∗=cZ¯+d, z¯SRSS∗=cz¯SRSS+d, w1,w2=0,1, and k1 are duly opted scalars; αj,j=1,2,3,4 assumes values −1, 0, and +1 in order to form different new and existing estimators.It is noticed that the minimumMSE of the Koyuncu and Kadilar- [1] type estimator y¯kkc, [4, 5] estimator y¯l1c and y¯l2c and Khan and Shabbir- [22] estimator y¯ksc are equal to the minimum MSE of classical regression estimator y¯lrc.TheMSE of the estimators considered in this section is discussed in Appendix A.


## 2.2. Separate Estimators

The classical separate ratio estimator of population meanY¯ using bivariate auxiliary information under SRSS can be defined as follows:
(15)y¯rs=∑h=1LWhy¯hRSSX¯hx¯hRSSZ¯hz¯hRSS.The classical separate regression estimator of population meanY¯ under bivariate auxiliary information using SRSS is as follows:
(16)y¯lrs=∑h=1LWhy¯hRSS+β1hX¯h−x¯hRSS+β2hZ¯h−z¯hRSS,where β1h and β2h are the regression coefficients of Y on X and Z in stratum h, respectively.Following Koyuncu and Kadilar [1], one may consider a general family of separate estimators for population mean Y¯ under SRSS as follows:
(17)y¯kks=∑h=1LWhy¯hRSSahX¯h+bhλ1hahx¯hRSS+bh+1−λ1hahX¯h+bhg1chZ¯h+dhλ2hchz¯hRSS+dh+1−λ2hchZ¯h+dhg2,where λ1h, λ2h, g1, and g2 are suitably chosen constants whereas ah≠0, bh and ch≠0, dh are either real numbers or the function of known parameters of variables X and Z, respectively, in stratum h.Following Tailor and Chouhan [3], one may suggest the following separate class of the estimator of the population mean using bivariate auxiliary information under SRSS as follows:
(18)y¯tcs=∑h=1LWhkhy¯hRSSexpX¯h−x¯hRSSX¯h+x¯hRSSexpz¯hRSS−Z¯hz¯hRSS+Z¯h,where kh is a duly opted scalar.On the lines of Lone et al. [4], we consider a generalized separate ratio-cum-product-type exponential estimator in SRSS as follows:
(19)y¯l1s=∑h=1LWhy¯hRSSexpL1hX¯h−x¯hRSSX¯h+x¯hRSS+L2hz¯hRSS−Z¯hz¯hRSS+Z¯h,where L1h and L2h are suitably chosen scalars.The separate version of the Lone et al. [5] estimator for estimating population mean Y¯ under SRSS is defined as follows:
(20)y¯l2s=∑h=1LWhy¯hRSSahX¯h+bhahx¯hRSS+bhchz¯hRSS+dhchZ¯h+dh.Following Muneer et al. [6], we introduce a separate class of estimators in SRSS as follows:
(21)y¯mus=∑h=1LWhk3hy¯hRSS−k4hx¯hRSS−X¯hΘh2−expz¯hRSS−Z¯hz¯hRSS+Z¯h+1−ΘhexpZ¯h−z¯hRSSZ¯h−z¯hRSS,where k3h and k4h are duly opted scalars.Khan and Shabbir [22] suggested a generalized separate exponential-type ratio-cum-ratio estimator under SRSS as follows:
(22)y¯kss=∑h=1LWhy¯hRSSexpX¯h−x¯hRSSX¯h+η1h−1x¯hRSSexpZ¯h−z¯hRSSZ¯h+η2h−1z¯hRSS,where η1h and η2h are duly opted scalars.Motivated by Muneer et al. [7], a separate chain ratio exponential family of estimators is defined in SRSS as follows:
(23)y¯mu1s=∑h=1LWhy¯hRSSX¯h∗w1hx¯hRSS∗+1−w1hX¯h∗α1hZ¯h∗w2hz¯hRSS∗+1−w2hZ¯h∗α2h×expα3hX¯h−x¯hRSSX¯h+x¯hRSSexpα4hZ¯h−z¯hRSSZ¯h+z¯hRSS.On the lines of Searls [31], the above separate chain ratio exponential family of estimators becomes
(24)y¯mu2s=∑h=1LWhk1hy¯hRSSX¯h∗w1hx¯hRSS∗+1−w1hX¯h∗α1hZ¯h∗w1hz¯hRSS∗+1−w1hZ¯h∗α2h×expα3hX¯h−x¯hRSSX¯h+x¯hRSSexpα4hZ¯h−z¯hRSSZ¯h+z¯hRSS,where X¯∗=ahX¯h+bh, x¯hRSS∗=ahx¯hRSS+bh, Z¯∗=chZ¯hRSS+dh, z¯hRSS∗=chz¯hRSS+dh, and w1h,w2h=0,1 and k1h is a duly opted scalar; αjh,j=1,2,3,4 assumes values −1, 0, and +1 in order to form different new and existing estimators.It is to be noted that the minimumMSE of Koyuncu and Kadilar- [1] type estimator y¯kks, Lone et al.- [4, 5] type estimator y¯l1s, y¯l2s, and Khan and Shabbir- [22] estimator y¯kss are similar to the minimum MSE of classical regression estimator y¯lrs.TheMSE of the estimators discussed in this section is given in Appendix B for ready reference.


## 3. Proffered Estimators

Motivated by the works of Bhushan et al. [32, 33], we have extended the work of Bhushan et al. [17] using bivariate auxiliary information under SRSS.


### 3.1. Combined Estimators

We propose some improved combined classes of estimators based on bivariate auxiliary information underSRSS as follows:
(25)y¯s1c=ξ1y¯SRSS1+logx¯SRSSX¯θ11+logz¯SRSSZ¯δ1,y¯s2c=ξ2y¯SRSS1+θ2logx¯SRSSX¯1+δ2logz¯SRSSZ¯,y¯s3c=ξ3y¯SRSS+θ3x¯SRSS−X¯+δ3z¯SRSS−Z¯,y¯s4c=ξ4y¯SRSSX¯x¯SRSSθ4Z¯z¯SRSSδ4,y¯s5c=ξ5y¯SRSSX¯X¯+θ5x¯SRSS−X¯Z¯Z¯+δ5z¯SRSS−Z¯,where ξi, θi, δi, i=1,2,⋯,5 are suitably chosen scalars.Theorem 1.
TheMSE of the proffered combined class estimators to the first order of approximation is given by
(26)MSEy¯s1c=Y¯21+ξ121+V200+2θ12V020+2δ12V002−2θ1V020−2δ1V002+4θ1V110+4δ1V101+4θ1δ1V011−2ξ11+θ122V020+δ122V002−θ1V020−δ1V002+θ1V110+δ1V101+θ1δ1V011,MSEy¯s2c=Y¯21+ξ221+V200+θ22V020+δ22V002−θ2V020−δ2V002+4θ2V110+4δ2V101+4θ2δ2V011−2ξ21−θ22V020−δ22V002+θ2V110+δ2V101+θ2δ2V011,MSEy¯s3c=Y¯2ξ3−12Y¯2+ξ32Y¯2V200+θ32X¯2V020+δ32Z¯2V002+2ξ3θ3X¯Y¯V110+2ξ3δ3Z¯Y¯V101+2θ3δ3X¯Z¯V011,MSEy¯s4c=Y¯21+ξ421+V200+θ4V020+δ4V002+2θ42V020+2δ42V002−4θ4V110−4δ4V101+4θ4δ4V011−2ξ41+θ4θ4+12V020+δ4δ4+12V002−θ4V110−δ4V101+θ4δ4V011,MSEy¯s5c=Y¯21+ξ521+V200+3θ52V020+3δ52V002−4θ5V110−4δ5V101+4θ5δ5V011−2ξ51+θ52V020−θ5V110+δ52V002−δ5V101+θ5δ5V011.Proof.
The outline of the derivation is given in AppendixC for ready reference.Corollary 2.
The minimumMSE at the optimum values of ξi, θi, and δi,i=1,2,⋯,5, is given by
(27)minMSEy¯sic=Y¯21−Bi2Ai,i=1,2,4,5,minMSEy¯s3c=Y¯21−ξ3opt=Y¯21−B32A3.Proof.
The outline of the derivation is given in AppendixC for ready reference.


### 3.2. Separate Estimators

We propose some improved separate classes of estimators based on bivariate auxiliary information underSRSS as follows:
(28)y¯s1s=∑h=1LWhξ1hy¯hRSS1+logx¯hRSSX¯hθ1h1+logz¯hRSSZ¯hδ1h,y¯s2s=∑h=1LWhξ2hy¯hRSS1+θ2hlogx¯hRSSX¯h1+δ2hlogz¯hRSSZ¯h,y¯s3s=∑h=1LWhξ3hy¯hRSS+θ3hx¯hRSS−X¯h+δ3hz¯hRSS−Z¯h,y¯s4s=∑h=1LWhξ4hy¯hRSSX¯hx¯hRSSθ4hZ¯hz¯hRSSδ4h,y¯s5s=∑h=1LWhξ5hy¯hRSSX¯hX¯h+θ5hx¯hRSS−X¯hZ¯hZ¯h+δ5hz¯hRSS−Z¯h,where ξih, θih, and δih, i=1,2,⋯,5, are suitably chosen scalars.Theorem 3.
TheMSE of the proffered separate classes of estimators to the first order of approximation is given by
(29)MSEy¯s1s=∑h=1LWh2Y¯h21+ξ1h21+U200+2θ1h2U020+2δ1h2U002−2δ1hU020−2δ1hU002+4θ1hU110+4δ1hU101+4θ1hδ1hU011−2ξ1h1+θ1h22U020+δ1h22U002−θ1U020−δ1hU002+θ1hU110+δ1hU101+θ1hδ1hU011,MSEy¯s2s=∑h=1LWh2Y¯h21+ξ2h21+U200+θ2h2U020+δ2h2U002−θ2hU020−δ2hU002+4θ2hU110+4δ2hU101+4θ2hδ2hU011−2ξ2h1−θ2h2U020−δ2h2U002+θ2hU110+δ2hU101+θ2hδ2hU011,MSEy¯s3s=∑h=1LWh2Y¯h2ξ3h−12Y¯h2+ξ3h2Y¯h2U200+θ3h2X¯h2U020+δ3h2Z¯h2U002+2ξ3hθ3hX¯hY¯hU110+2ξ3hδ3hZ¯hY¯hU101+2θ3hδ3hX¯hZ¯hU011,MSEy¯s4s=∑h=1LWh2Y¯h21+ξ4h21+U200+θ4hU020+δ4hU002+2θ4h2U020+2δ42U002−4θ4hU110−4δ4hU101+4θ4hδ4hU011−2ξ4h1+θ4hθ4h+12U020+δ4hδ4h+12U002−θ4hU110−δ4hU101+θ4hδ4hU011,MSEy¯s5s=∑h=1LWh2Y¯h21+ξ5h21+U200+3θ5h2U020+3δ5h2U002−4θ5hU110−4δ5hU101+4θ5hδ5hU011−2ξ5h1+θ5h2U020−θ5hU110+δ5h2U002−δ5hU101+θ5hδ5hU011.Proof.
The outline of the derivation is given in AppendixC for ready reference.Corollary 4.
The minimumMSE at the optimum values of ξih, θih, and δih, i=1,2,⋯,5, is given by
(30)minMSEy¯sis=∑h=1LWh2Y¯h21−Bih2Aih,i=1,2,4,5,minMSEy¯s3s=∑h=1LWh2Y¯h21−ξ3hopt=∑h=1LWh2Y¯h21−B3h2A3h.Proof.
The outline of the derivation is given in AppendixD for ready reference.


## 3.1. Combined Estimators

We propose some improved combined classes of estimators based on bivariate auxiliary information underSRSS as follows:
(25)y¯s1c=ξ1y¯SRSS1+logx¯SRSSX¯θ11+logz¯SRSSZ¯δ1,y¯s2c=ξ2y¯SRSS1+θ2logx¯SRSSX¯1+δ2logz¯SRSSZ¯,y¯s3c=ξ3y¯SRSS+θ3x¯SRSS−X¯+δ3z¯SRSS−Z¯,y¯s4c=ξ4y¯SRSSX¯x¯SRSSθ4Z¯z¯SRSSδ4,y¯s5c=ξ5y¯SRSSX¯X¯+θ5x¯SRSS−X¯Z¯Z¯+δ5z¯SRSS−Z¯,where ξi, θi, δi, i=1,2,⋯,5 are suitably chosen scalars.Theorem 1.
TheMSE of the proffered combined class estimators to the first order of approximation is given by
(26)MSEy¯s1c=Y¯21+ξ121+V200+2θ12V020+2δ12V002−2θ1V020−2δ1V002+4θ1V110+4δ1V101+4θ1δ1V011−2ξ11+θ122V020+δ122V002−θ1V020−δ1V002+θ1V110+δ1V101+θ1δ1V011,MSEy¯s2c=Y¯21+ξ221+V200+θ22V020+δ22V002−θ2V020−δ2V002+4θ2V110+4δ2V101+4θ2δ2V011−2ξ21−θ22V020−δ22V002+θ2V110+δ2V101+θ2δ2V011,MSEy¯s3c=Y¯2ξ3−12Y¯2+ξ32Y¯2V200+θ32X¯2V020+δ32Z¯2V002+2ξ3θ3X¯Y¯V110+2ξ3δ3Z¯Y¯V101+2θ3δ3X¯Z¯V011,MSEy¯s4c=Y¯21+ξ421+V200+θ4V020+δ4V002+2θ42V020+2δ42V002−4θ4V110−4δ4V101+4θ4δ4V011−2ξ41+θ4θ4+12V020+δ4δ4+12V002−θ4V110−δ4V101+θ4δ4V011,MSEy¯s5c=Y¯21+ξ521+V200+3θ52V020+3δ52V002−4θ5V110−4δ5V101+4θ5δ5V011−2ξ51+θ52V020−θ5V110+δ52V002−δ5V101+θ5δ5V011.Proof.
The outline of the derivation is given in AppendixC for ready reference.Corollary 2.
The minimumMSE at the optimum values of ξi, θi, and δi,i=1,2,⋯,5, is given by
(27)minMSEy¯sic=Y¯21−Bi2Ai,i=1,2,4,5,minMSEy¯s3c=Y¯21−ξ3opt=Y¯21−B32A3.Proof.
The outline of the derivation is given in AppendixC for ready reference.


## 3.2. Separate Estimators

We propose some improved separate classes of estimators based on bivariate auxiliary information underSRSS as follows:
(28)y¯s1s=∑h=1LWhξ1hy¯hRSS1+logx¯hRSSX¯hθ1h1+logz¯hRSSZ¯hδ1h,y¯s2s=∑h=1LWhξ2hy¯hRSS1+θ2hlogx¯hRSSX¯h1+δ2hlogz¯hRSSZ¯h,y¯s3s=∑h=1LWhξ3hy¯hRSS+θ3hx¯hRSS−X¯h+δ3hz¯hRSS−Z¯h,y¯s4s=∑h=1LWhξ4hy¯hRSSX¯hx¯hRSSθ4hZ¯hz¯hRSSδ4h,y¯s5s=∑h=1LWhξ5hy¯hRSSX¯hX¯h+θ5hx¯hRSS−X¯hZ¯hZ¯h+δ5hz¯hRSS−Z¯h,where ξih, θih, and δih, i=1,2,⋯,5, are suitably chosen scalars.Theorem 3.
TheMSE of the proffered separate classes of estimators to the first order of approximation is given by
(29)MSEy¯s1s=∑h=1LWh2Y¯h21+ξ1h21+U200+2θ1h2U020+2δ1h2U002−2δ1hU020−2δ1hU002+4θ1hU110+4δ1hU101+4θ1hδ1hU011−2ξ1h1+θ1h22U020+δ1h22U002−θ1U020−δ1hU002+θ1hU110+δ1hU101+θ1hδ1hU011,MSEy¯s2s=∑h=1LWh2Y¯h21+ξ2h21+U200+θ2h2U020+δ2h2U002−θ2hU020−δ2hU002+4θ2hU110+4δ2hU101+4θ2hδ2hU011−2ξ2h1−θ2h2U020−δ2h2U002+θ2hU110+δ2hU101+θ2hδ2hU011,MSEy¯s3s=∑h=1LWh2Y¯h2ξ3h−12Y¯h2+ξ3h2Y¯h2U200+θ3h2X¯h2U020+δ3h2Z¯h2U002+2ξ3hθ3hX¯hY¯hU110+2ξ3hδ3hZ¯hY¯hU101+2θ3hδ3hX¯hZ¯hU011,MSEy¯s4s=∑h=1LWh2Y¯h21+ξ4h21+U200+θ4hU020+δ4hU002+2θ4h2U020+2δ42U002−4θ4hU110−4δ4hU101+4θ4hδ4hU011−2ξ4h1+θ4hθ4h+12U020+δ4hδ4h+12U002−θ4hU110−δ4hU101+θ4hδ4hU011,MSEy¯s5s=∑h=1LWh2Y¯h21+ξ5h21+U200+3θ5h2U020+3δ5h2U002−4θ5hU110−4δ5hU101+4θ5hδ5hU011−2ξ5h1+θ5h2U020−θ5hU110+δ5h2U002−δ5hU101+θ5hδ5hU011.Proof.
The outline of the derivation is given in AppendixC for ready reference.Corollary 4.
The minimumMSE at the optimum values of ξih, θih, and δih, i=1,2,⋯,5, is given by
(30)minMSEy¯sis=∑h=1LWh2Y¯h21−Bih2Aih,i=1,2,4,5,minMSEy¯s3s=∑h=1LWh2Y¯h21−ξ3hopt=∑h=1LWh2Y¯h21−B3h2A3h.Proof.
The outline of the derivation is given in AppendixD for ready reference.


## 4. Theoretical Conditions

### 4.1. Combined Estimators

On comparing the minimumMSE of the proffered combined classes of estimators y¯sic,i=1,2,⋯,5 and the existing combined estimators from (27) with (47), (48), (50), (52), (56), (58), (60), (62), (63), and (65), we obtain the theoretical conditions given as follows:
(31)MSEy¯mc>MSEy¯sic⇒Bi2Ai>1−V200,MSEy¯rc>MSEtsic⇒Bi2Ai>1−V200−V020−V002+2V110+2V101−2V011,MSEy¯lrc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯kkc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯tcc>MSEy¯sic⇒Bi2Ai>1−k∗2V200+14V020+V002−2V002+V101−V110−2k∗k∗−1183V020−V002−2V011+12V101−V110−k∗−12,MSEy¯l1c>MSEy¯sic⇒Bi2Ai>1−V200+V1012V020+V1102V002−2V011V101V110V020V0021−ρxhzh2,MSEy¯l2c>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯muc>MSEy¯sic⇒Bi2Ai>V01124V020+Am2Bm,MSEy¯ksc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯mu1c>MSEy¯sic⇒Bi2Ai>2O−P,MSEy¯mu2c>MSEy¯sic⇒Bi2Ai>O2P.If condition (31) holds, then, the proffered 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 proffered separate classes of estimator y¯sis,i=1,2,⋯,5, with the existing separate estimators from (30) with (81), (82), (84), (86), (90), (92), (94), (96), (97), and (99), we get the following theoretical conditions.
(32)MSEy¯ms>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200,(33)MSEy¯rs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<U200+U020+U002−2U110−2U101+2U011,(34)MSEy¯lrs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(35)MSEy¯kks>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(36)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,(37)MSEy¯l1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U1012U020+U1102U002−2U011U101U110U020U0021−ρxhzh2,(38)MSEy¯l2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(39)MSEy¯mus>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−U01124U020−Amh2Bmh,(40)MSEy¯kss>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(41)MSEy¯mu1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21+Ph−2Oh,(42)MSEy¯mu2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−Oh2Ph.If conditions (32)–(42) hold, then, the proffered separate classes of estimators y¯sis,i=1,2,⋯,5, become superior than the other existing separate estimators.


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

On comparing the minimumMSE of proffered combined and separate classes of estimators y¯sic and y¯sis, i=1,2,⋯,5, we get
(43)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 the origin, then, the last term of (43) is broadly small and it vanishes.Moreover, 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 proffered estimators is negligible, whereas the proffered combined estimators are to be preferably recommended with only a small sample in each stratum (see [34]).


## 4.1. Combined Estimators

On comparing the minimumMSE of the proffered combined classes of estimators y¯sic,i=1,2,⋯,5 and the existing combined estimators from (27) with (47), (48), (50), (52), (56), (58), (60), (62), (63), and (65), we obtain the theoretical conditions given as follows:
(31)MSEy¯mc>MSEy¯sic⇒Bi2Ai>1−V200,MSEy¯rc>MSEtsic⇒Bi2Ai>1−V200−V020−V002+2V110+2V101−2V011,MSEy¯lrc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯kkc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯tcc>MSEy¯sic⇒Bi2Ai>1−k∗2V200+14V020+V002−2V002+V101−V110−2k∗k∗−1183V020−V002−2V011+12V101−V110−k∗−12,MSEy¯l1c>MSEy¯sic⇒Bi2Ai>1−V200+V1012V020+V1102V002−2V011V101V110V020V0021−ρxhzh2,MSEy¯l2c>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯muc>MSEy¯sic⇒Bi2Ai>V01124V020+Am2Bm,MSEy¯ksc>MSEy¯sic⇒Bi2Ai>1−V200+V002V1102+V020V1012−2V110V101V011V020V002−V0112,MSEy¯mu1c>MSEy¯sic⇒Bi2Ai>2O−P,MSEy¯mu2c>MSEy¯sic⇒Bi2Ai>O2P.If condition (31) holds, then, the proffered 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 proffered separate classes of estimator y¯sis,i=1,2,⋯,5, with the existing separate estimators from (30) with (81), (82), (84), (86), (90), (92), (94), (96), (97), and (99), we get the following theoretical conditions.
(32)MSEy¯ms>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200,(33)MSEy¯rs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<U200+U020+U002−2U110−2U101+2U011,(34)MSEy¯lrs>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(35)MSEy¯kks>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(36)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,(37)MSEy¯l1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U1012U020+U1102U002−2U011U101U110U020U0021−ρxhzh2,(38)MSEy¯l2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(39)MSEy¯mus>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−U01124U020−Amh2Bmh,(40)MSEy¯kss>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h2U200−U002U1102+U020U1012−2U110U101U011U020U002−U0112,(41)MSEy¯mu1s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21+Ph−2Oh,(42)MSEy¯mu2s>MSEy¯sis,∑h=1LWh2Y¯h21−Bih2Aih<∑h=1LWh2Y¯h21−Oh2Ph.If conditions (32)–(42) hold, then, the proffered separate classes of estimators y¯sis,i=1,2,⋯,5, become superior than the other existing separate estimators.


## 4.3. Comparison of Proffered Combined and Separate Estimators

On comparing the minimumMSE of proffered combined and separate classes of estimators y¯sic and y¯sis, i=1,2,⋯,5, we get
(43)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 the origin, then, the last term of (43) is broadly small and it vanishes.Moreover, 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 proffered estimators is negligible, whereas the proffered combined estimators are to be preferably recommended with only a small sample in each stratum (see [34]).


## 5. Numerical Study

To enhance the credibility of the theoretical development of the proposed combined and separate classes of estimators, we have conducted a numerical study using two real populations which are described as follows.Population 5 (Sarndal et al. [35], p. 529).
y is the population in thousands during 1985, x is the population in thousands during 1975, z is the total number of seats in the municipal council, N=199, n=47, N1=25, N2=48, N3=32, N4=38, N5=56, n1=6, n2=11, n3=8, n4=9, n5=13, Y¯1=62.4400, Y¯2=29.6042, Y¯3=24.0625, Y¯4=31.0000, Y¯5=29.41071, X¯1=59.5200, X¯2=29.1667, X¯3=23.9375, X¯4=30.6316, X¯5=28.7143, Z¯1=51.1600, Z¯2=47.6667, Z¯3=50.2500, Z¯4=48.4737, Z¯5=46.3572, Sy12=15521.5900, Sy22=1320.2442, Sy32=445.3508, Sy42=1536.3784, Sy52=3219.8464, Sx12=16564.7600, Sx22=1228.2270, Sx32=437.1573, Sx42=1721.1579, Sx52=3565.1169, Sz12=197.9734, Sz22=106.3546, Sz32=106.7742, Sz42=82.0939, Sz52=97.9065, Syx1=10125.1783, Syx2=1270.2376, Syx3=440.4234, Syx4=1622.6216, Syx5=3385.5013, Sxz1=383.6633, Sxz2=409.8440, Sxz3=189.1129, Sxz4=232.3954, Sxz5=417.5584, Syz1=150.0933, Syz2=422.9929, Syz3=189.1129, Syz4=229.8378, and Syz5=399.9052.Population 6 (National Horticulture Baard [36]).
y is productivity (MT/hectare), x is production (000 tons), z is the area (000 hectares), N=10, n=7, N1=5, N2=5, n1=3, n2=4, Y¯1=1.70, Y¯2=3.67, X¯1=10.41, X¯2=309.14, Z¯1=6.20, Z¯2=80.67, Sy12=0.2916, Sy22=1.9881, Sx12=1.4116, Sx22=3486.6916, Sz12=1.4116, Sz22=116.8561, Syx1=1.6000, Syx2=83.47, Sxz1=1.7500, Sxz2=64.9700, Syz1=0.2000, and Syz2=5.5800.
We have calculated theMSE and PRE of different estimators T by using the above populations. The PRE is calculated with respect to the usual mean estimator using the following expression.(44)PRE=MSEy¯srssMSET×100,whereMSET is the MSE of existing and proposed combined and separate estimators. The results of the numerical study for the above populations are reported in Table 1 by MSE and PRE. The numerical results show the dominance of the proposed combined and separate classes of estimators y¯sic and y¯sis, i=1,2,⋯,5, respectively, in terms of lesser MSE and greater PRE over the combined and separate usual mean estimator, classical ratio, and regression estimators, Koyuncu and Kadilar- [1] type estimator, Tailor and Chouhan- [3] type estimator, [4, 5] type estimators, Khan and Shabbir- [22] type estimator, and Muneer et al.- [6, 7] type estimators. Also, the proposed combined and separate class of estimators y¯s1c and y¯s1s attain the lesser MSE and greater PRE among the proposed classes of estimators in both the populations.Table 1
MSE and PRE of different combined and separate estimators using real populations.
Combined estimatorsPopulation5Population6SeparatePopulation5Population6MSEPREMSEPREEstimatorsMSEPREMSEPREy¯mc127.24100.000.1485100.00y¯ms129.42100.000.1617100.00y¯rc85.53148.760.0406365.76y¯rs86.11150.290.0439368.33y¯lrc69.78182.340.0263563.72y¯lrs70.13184.540.0285567.36y¯kkc69.78182.340.0263563.72y¯kks70.13184.540.0285567.36y¯tcc156.7581.170.0985150.78y¯tcs155.6583.140.0989163.49y¯l1c69.78182.340.0263563.72y¯l1s70.13184.540.0285567.36y¯l2c69.78182.340.0263563.72y¯l2s70.13184.540.0285567.36y¯ksc69.78182.340.0263563.72y¯kss70.13184.540.0285567.36y¯muc88.03144.520.0193769.43y¯mus88.78145.770.0210770.00y¯mu1c99.64127.700.0195758.29y¯mu1s100.33128.990.0212762.73y¯mu2c98.56129.090.0195758.60y¯mu2s99.51130.050.0213759.15y¯s1c61.44207.070.0189785.71y¯s1s62.18208.130.0201804.47y¯s2c65.79193.400.0192773.43y¯s2s66.21195.460.0208777.40y¯s3c66.01192.750.0192774.24y¯s3s66.97193.250.0208778.15y¯s4c64.69196.660.0190781.57y¯s4s65.29198.220.0206784.95y¯s5c65.13195.360.0191777.48y¯s5s65.98196.150.0207779.27


## 6. Simulation Study

To generalize the results of the numerical study, we have conducted a simulation study over a hypothetically generated normal population. The simulation procedure is explained in the following points:(i)
Trivariate random observations of size 600 units are drawn from a trivariate normal distribution with parametersY¯=20, X¯=15, Z¯=10, σy=15, σx=10, and σz=5 and different amounts of correlation coefficients ρxy, ρyz and ρxz(ii)
The population generated above is divided into 3 equal strata, and a stratified ranked set sample of size 12 units with a number of cycles 4 and set size 3 is drawn from each stratum(iii)
Compute the required statistics(iv)
Iterate the above steps 10000 times to calculate theMSE and PRE of various combined and separate classes of estimators using the following expression:(45)MSET=110,000∑i=110,000Ti−Y¯2,(46)PRE=MSEy¯SRSSMSET×100TheMSE and PRE of the combined and separate classes of estimators are calculated using (45) and (46), respectively, and the results are reported for various values of correlation coefficients in Tables 2 and 3 which exhibit the ascendancy of the proposed combined and separate classes of estimators over the existing combined and separate classes of estimators.Table 2
MSE and PRE of different combined estimators.
ρxy0.60.70.80.90.90.80.70.6ρyz0.60.70.80.90.80.70.60.5ρxz0.60.70.80.90.70.60.50.4EstimatorsMSEPREMSEPREMSEPREMSEPREMSEPREMSEPREMSEPREMSEPREy¯mc0.6839100.000.6801100.000.6753100.000.5721100.000.6639100.000.6737100.000.6860100.000.6950100.00y¯rc1.512845.201.487845.711.446146.691.412240.511.449045.271.466445.941.498345.781.505346.17y¯lrc0.5858116.740.5804117.170.5719118.080.4442128.790.5622118.080.5742117.320.5923115.810.5946116.88y¯kkc0.5858116.740.5804117.170.5719118.080.4442128.790.5622118.080.5742117.320.5923115.810.5946116.88y¯tcc1.154859.221.138659.731.129059.811.060953.921.145257.971.154958.331.162259.021.182158.79y¯l1c0.5858116.740.5804117.170.5719118.080.4442128.790.5622118.080.5742117.320.5923115.810.5946116.88y¯l2c0.5858116.740.5804117.170.5719118.080.4442128.790.5622118.080.5742117.320.5923115.810.5946116.88y¯ksc0.5858116.740.5804117.170.5719118.080.4442128.790.5622118.080.5742117.320.5923115.810.5946116.88y¯muc0.5775118.420.5735118.580.5633119.880.4381130.580.5584118.890.5682118.560.5855117.160.5893117.93y¯mu1c0.780487.630.776187.630.774487.200.656587.140.755487.880.764288.150.782987.620.795187.41y¯mu2c0.734093.170.728893.310.726292.990.613293.290.712393.200.722193.290.737792.990.745493.23y¯s1c0.5667120.680.5616121.100.5532122.070.4184136.730.5443121.970.5545121.490.5726119.800.5858118.64y¯s2c0.5732119.310.5681119.710.5602120.540.4287133.440.5508120.530.5618119.910.5792118.430.5896117.87y¯s3c0.5728119.390.5678119.780.5597120.650.4254134.480.5505120.590.5621119.850.5791118.450.5899117.81y¯s4c0.5710119.770.5660120.150.5578121.060.4210135.890.5487120.990.5596120.380.5761119.070.5888118.03y¯s5c0.5720119.560.5678119.770.5601120.560.4259134.320.5505120.590.5616119.960.5771118.870.5891117.97Table 3
MSE and PRE of different separate estimators.
ρxy0.60.70.80.90.90.80.70.6ρyz0.60.70.80.90.80.70.60.5ρxz0.60.70.80.90.70.60.50.4EstimatorsMSEPREMSEPREMSEPREMSEPREMSEPREMSEPREMSEPREMSEPREy¯ms0.7347100.000.6643100.000.6584100.000.6510100.000.6503100.000.6571100.000.6594100.000.6653100.00y¯rs1.679743.731.515543.961.474644.641.421345.801.450544.831.508243.561.548342.581.638840.59y¯lrs0.6297116.670.5560117.720.5390122.150.5282123.240.5278123.200.5386122.000.5425121.540.5607118.65y¯kks0.6297116.670.5560117.720.5390122.150.5282123.240.5278123.200.5286122.000.5425121.540.5607118.65y¯tcs1.307856.171.142758.301.123358.611.110758.611.099059.171.128558.221.143257.681.113459.75y¯l1s0.6297116.670.5560117.720.5390122.150.5282123.240.5278123.200.5286122.000.5425121.540.5607118.65y¯l2s0.6297116.670.5560117.720.5390122.150.5282123.240.5278123.200.5286122.000.5425121.540.5607118.65y¯kss0.6297116.670.5560117.720.5390122.150.5282123.240.5278123.200.5286122.000.5425121.540.5607118.65y¯mus0.6127119.910.5470121.800.5304124.130.5186125.530.5176125.630.5289124.230.5328123.760.5536120.17y¯mu1s0.890182.540.792984.030.753987.330.736688.370.740487.830.756786.830.785683.930.815681.57y¯mu2s0.825890.050.731691.070.699594.120.686694.810.688694.430.702293.570.725790.860.748488.89y¯s1s0.5965123.160.5348124.580.5175127.220.5065128.520.5076128.110.5145127.710.5223126.240.5405123.09y¯s2s0.6101120.420.5421122.910.5256125.260.5157126.230.5157126.100.5244125.300.5301124.390.5469121.64y¯s3s0.6115120.140.5398123.430.5238125.690.5137126.720.5133126.690.5247125.230.5274125.020.5447122.14y¯s4s0.5978122.900.5390123.610.5224126.030.5122127.090.5123126.930.5110126.120.5267125.190.5441122.27y¯s5s0.6100120.440.5397123.450.5237125.720.5136126.750.5131126.730.5239125.420.5272125.070.5459121.87


## 7. Conclusion

In this paper, we proffer some improved classes of estimators along with their properties using bivariate auxiliary information inSRSS. The proffered estimators dominate the other existing estimators under the conditions stated in Section 4. The numerical and simulation studies are performed using real and artificially generated populations with various amounts of correlation coefficients. The results of numerical and simulation studies are reported in terms of MSE and PRE from Tables 1 to 3. From the perusal of the results of Tables 1–3, the following conclusions are drawn:
(i)
In Table1, the proffered combined classes of estimators y¯sic,i=1,2,⋯,5, perform better than the existing combined estimators, namely, conventional mean estimator y¯mc, classical ratio and regression estimators y¯rc & y¯lrc, Koyuncu and Kadilar- [1] type estimator y¯kkc, Tailor and Chouhan- [3] type estimator y¯tcc, Lone et al.- [4, 5] type estimators y¯l1c & y¯l2c, Khan and Shabbir [22] estimator y¯ksc, and Muneer et al.- [6, 7] type estimators y¯muc, y¯mu1c, & y¯mu2c in both the populations(ii)
In Table1, the proffered separate classes of estimators y¯sis,i=1,2,⋯,5, dominate the existing separate estimators, namely, conventional mean estimator y¯ms, classical ratio and regression estimators y¯rs & y¯lrs, Koyuncu and Kadilar- [1] type estimator y¯kkc, Tailor and Chouhan- [3] type estimator y¯tcs, Lone et al.- [4, 5] type estimators y¯l1s & y¯l2s, Khan and Shabbir [22] estimator y¯ksc, and Muneer et al.- [6, 7] type estimators y¯mus, y¯mu1s, & y¯mu2s(iii)
In Table2, the proffered combined classes of estimators y¯sic,i=1,2,⋯,5, perform better than the existing combined estimators. The similar inclination in the results of separate estimators can also be observed in Table 3(iv)
In Tables1, 2, and 3, the proffered combined and separate classes of estimators y¯s1c and y¯s1s, respectively, become superior among the proffered combined and separate classes of estimators(v)
In Tables1, 2, and 3, the proffered separate classes of estimators y¯sis,i=1,2,⋯,5, perform better than the proffered combined classes of estimators y¯sic,i=1,2,⋯,5, in terms of lesser MSE and greater PRE(vi)
From the simulation results of Tables2 and 3, it has been observed that the PRE increases as the correlation coefficients decreaseThus, the proffered classes of estimators are preferred for the computation of the population mean when bivariate auxiliary information is available.



---

*Source: 6196142-2022-10-06.xml*