Some ImprovedClasses of Estimators in Stratified SamplingUsing Bivariate Auxiliary Information

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


Introduction
In sampling theory, the appropriate utilization of auxiliary information plays a leading role to improve the e ciency of the estimators. is 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 di erent 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 strati ed 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 coe cient 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. e 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 modi ed estimators to date. Especially, under the availability of multi-auxiliary information, the literature contains di erent 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 e ciency 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 modi ed 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-cumproduct estimator of population mean in SSRS. Tailor and Chouhan [14] addressed ratio-cum-product type exponential estimator of nite population mean. Following Upadhyaya et al. [15]; Singh et al. [16] considered a class of ratiocum-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 � (κ 1 , κ 2 , . . . , κ N ) based on size N units with study variable y and two auxiliary variables x and z, respectively, associated with each unit κ i , i � 1, 2, . . . , N of the population. Let the population be divided into L disjoint strata with the stratum h comprises of N h , h � 1, 2, . . . , L units. Let a simple random sample of size n h be quantified without replacement from the stratum h such that L h�1 n h � n. Let the observed values of y, x and z on the i th unit of the stratum h be denoted by  ((x hi − x h ))(y hi − y h ))/(n h − 1), s zy h � L h�1 ((z hi − z h )) (y hi − y h ))/(n h − 1), s xz h � L h�1 ((x hi − x h ))(z hi − z h ))/ (n h − 1), respectively, be the sample variances and covariances corresponding to the population variances and covariances S 2 To derive the bias and mean square error (MSE) of different combined estimators, the following notations will be used throughout the paper.
where c � 1/n h and ρ xy h , ρ xz h and ρ yz h be the population coefficient of correlation with their respective subscripts in stratum h.
Again, to determine the bias and MSE of the separate estimators, the following notations will be used throughout the paper.
e aim of the present paper is to suggest some improved combined and separate classes of estimators in the presence of bivariate auxiliary information under SSRS. e 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. e theoretical comparison of the proposed combined and separate classes of estimators with the existing combined and separate estimators is given in Section 4. e 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.

Combined Estimators.
e conventional combined mean estimator under SSRS is given by 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 e classical combined regression estimator of population mean Y based on bivariate auxiliary information under SSRS is given by where β 1 and β 2 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 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 where a 1 , a 2 , w 1 , and w 2 are duly opted scalars and w 1 + w 2 � 1. Following Kadilar and Cingi [7], one may suggest some combined ratio-cum-product estimators based on bivariate auxiliary information under SSRS as (11) where β 2 (x h ) and β 2 (z h ) 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 where η 1 , η 2 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 where λ 1 , λ 2 , g 1 , and g 2 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 y c tc � ky st exp 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 where L 1 and L 2 are duly opted scalars. e combined version of Lone et al. [19] estimator for estimating population mean Y is given by We remark that the minimum MSE of Abu-Dayyeh et al. [25] type estimator y c a , Koyuncu and Kadilar [12] estimator y c kk , and Lone et al. [18,19] estimators y c l 1 , i � 1, 2 are equal to the minimum MSE of the classical regression estimator y c l . Along the lines of Singh and Singh [20]; Muneer et al. [21] introduced a class of combined estimators in SSRS as where k 3 , k 4 are duly opted scalars and Θ is a scalar assuming values 0 and 1 to design different estimators.
e combined form of Muneer et al. [22] chain ratio exponential family of estimator in SSRS is given by On the lines of Searls [27]; an improved form of the above-combined estimator is given by where and k 1 is a duly opted scalar, α j , 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 c mu 1 and y c mu 2 , respectively, for different values of scalars. e bias and MSE of the estimators considered in this section are readily discussed in Appendix A.

Separate Estimators.
e conventional separate mean estimator under SSRS is given by On the lines of Singh [23]; the classical separate ratio estimator of population mean y using bivariate auxiliary information under SSRS is defined as e classical separate regression estimator of population mean Y under bivariate auxiliary information using SSRS is where β 1 h and β 2 h 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 where w h 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 where a 1 h , a 2 h , w 1 h and w 2 h are duly opted scalars in stratum h and w 1 h + w 2 h � 1. Following Kadilar and Cingi [7]; one can suggest some separate ratio-cum-product type estimators based on bivariate auxiliary information under SSRS as e separate version of Singh et al. [16] estimator is given by where η 1 h , η 2 h are duly opted scalars in stratum h and α h , β h are scalars in stratum h taking real values. e separate version of Koyuncu and Kadilar [12] family of estimators is given by where λ 1 h , λ 2 h , g 1 and g 2 are some prescribed scalars whereas a h ( ≠ 0), b h and c h ( ≠ 0), d h are either real numbers or functions of the known parameters of the auxiliary variables x and z, respectively, in stratum h. e separate version of Tailor and Chouhan [14] estimator for population mean using bivariate auxiliary information under SSRS is defined as where k h 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 where L 1 h and L 2 h are duly opted scalars in the stratum h. e separate version of Lone et al. [19] estimator for estimating population mean Y as It is to be noted that the minimum MSE of separate Abu-Dayyeh [25] where k 3 h and k 4 h are suitably chosen scalars in stratum h and Θ h is a real constant in stratum h.
Muneer et al. [22] suggested a separate chain ratio exponential family of estimators in SSRS as and k 1 h is a duly opted scalar in stratum h, α j h , 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 s mu 1 and y s mu 2 for different values of scalars. e bias and MSE of the estimators considered in this section are readily discussed in the Appendix B.

Proposed Estimators
e 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.

Combined Estimators.
We propose some improved combined classes of estimators based on bivariate auxiliary information under SSRS as

Journal of Probability and Statistics
where ξ i , θ i and δ i , i � 1, 2, . . . , 5 are duly opted scalars to be determined.

Theorem 1.
e bias of the proposed combined classes of estimators y c s i , i � 1, 2, . . . , 5 is given by Bias y c Bias y c Proof. e precis of the derivations are given in Appendix C for quick review.
Proof. e precis of the derivations are given in Appendix C for quick review.
Proof. e precis of derivations and the definition of parametric functions A i and B i are given in Appendix C for quick review.
We note that eorem 2 and Corollary 1 are important to derive the efficiency conditions given in Subsection 4.1.
Bias y s Bias y s Bias y s Bias y s Proof.
Proof. e precis of the derivations are given in Appendix C for quick review.
min MSE y s

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.

Combined Estimators. On comparing the minimum
Journal of Probability and Statistics 9 MSE y c mu > MSE y c If conditions (72) to (86) hold, then the proposed combined classes of estimators y c s i , i � 1, 2, . . . , 5 perform better than the other existing combined estimators.

MSE y s mu
If the conditions (87) to (101) hold then the proposed separate classes of estimators y s s i , i � 1, 2, . . . , 5 perform better than the other existing separate estimators.

Comparison of Proposed Combined and Separate
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, unless R h 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 MSE(y s s i ), 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.

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:  Table 1 and Table 2. e ARB and PRE are calculated using the following expressions.   Table 1 and Table 2 for different values of correlation coefficients. e results exhibit the dominance of the proposed combined and separate classes of estimators y c s i and y s s i , 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 c s 1 and y s s 1 are found to be most efficient among the proposed combined and separate classes of estimators for each passably chosen values of correlation coefficients.

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. e mathematical expressions of bias and MSE of the proposed classes of estimators are obtained up to the first order of approximation. e 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) e proposed combined classes of estimators y c s i , i � 1, 2, . . . , 5 perform better than the combined form of usual mean estimator y c m , classical ratio and regression estimators y c r & y c l , Olkin [24]   (iii) Since, the proposed combined and separate classes of estimators y c s i and y s s i , i � 1, 2, . . . , 5 are, respectively, superior to the combined and separate ratio and chain ratio exponential estimators y c mu 1 , y s mu 1 and y c mu 2 , y s mu 2 envisaged by Muneer et al. [22] consequently the proposed combined and separate classes of estimators y c s i and y s s i , 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 c mu 1 , y c mu 2 and y s mu 1 , y s mu 2 , respectively.

Data Availability
ere are no data associated with this article.

Conflicts of Interest
e authors have no conflicts of interest.