#### 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.

#### 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 , whereas Shahzad et al. [3] suggested L-Moments and calibration-based variance estimators under double 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 , 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 . Tailor et al. [13] envisaged ratio-cum-product estimator of population mean in . 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 . Along the lines of Singh et al. [17]; Lone et al. [18] introduced a generalized ratio-cum-product type exponential estimator in , 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 . Recently, Muneer et al. [22] introduced a chain ratio exponential family of estimators based on . In the present study, we propose some improved classes of estimators for the estimation of population mean by employing bivariate auxiliary information under .Consider a finite population based on size units with study variable and two auxiliary variables and , respectively, associated with each unit of the population. Let the population be divided into disjoint strata with the stratum comprises of , units. Let a simple random sample of size be quantified without replacement from the stratum such that . Let the observed values of , and on the unit of the stratum be denoted by , . Let , and , respectively, be the sample means corresponding to the population means , and of variables , and , where is the weight in the stratum . Let , and be the sample means corresponding to the population means , and of variables , and in the stratum . Let , , , , , , respectively, be the sample variances and covariances corresponding to the population variances and covariances , , , , , in the stratum .

To derive the bias and mean square error of different combined estimators, the following notations will be used throughout the paper.

and

Following (1), we can write ; ; ; ; and , where and , and be the population coefficient of correlation with their respective subscripts in stratum .

Again, to determine the bias and of the separate estimators, the following notations will be used throughout the paper.

and ; ; ; ; and .

The aim of the present paper is to suggest some improved combined and separate classes of estimators in the presence of bivariate auxiliary information under . 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 under is given by

On the lines of Singh [23]; one may consider the classical combined ratio estimator of population mean using bivariate auxiliary information under as

The classical combined regression estimator of population mean based on bivariate auxiliary information under is given bywhere and are the regression coefficients of on and , respectively.

Following Olkin [24]; the combined ratio type estimator using bivariate auxiliary information under is given bywhere is a duly opted scalar.

Along the lines of Abuâ€“Dayyeh et al. [25]; some combined classes of ratio type estimators using bivariate auxiliary information under are given bywhere are duly opted scalars and .

Following Kadilar and Cingi [7], one may suggest some combined ratio-cum-product estimators based on bivariate auxiliary information under as (11) where and are the coefficient of kurtosis of variables and , respectively, in stratum .

Following Upadhyaya et al. [15]; Singh et al. [16] considered a combined class of ratio-cum-product estimators using bivariate auxiliary information in aswhere , are duly opted scalars and , 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 using bivariate auxiliary information under aswhere , , , and are prescribed scalars, whereas , and , are either real numbers or functions of the known parameters of auxiliary variables and , 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 aswhere 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 aswhere and are duly opted scalars.

The combined version of Lone et al. [19] estimator for estimating population mean is given by

We remark that the minimum of Abu-Dayyeh et al. [25] type estimator , Koyuncu and Kadilar [12] estimator , and Lone et al. [18, 19] estimators are equal to the minimum of the classical regression estimator .

Along the lines of Singh and Singh [20]; Muneer et al. [21] introduced a class of combined estimators in aswhere , are duly opted scalars and 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 is given by

On the lines of Searls [27]; an improved form of the above-combined estimator is given bywhere , , , , , and is a duly opted scalar, assumes values âˆ’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 and , respectively, for different values of scalars.

The bias and of the estimators considered in this section are readily discussed in Appendix A.

##### 2.2. Separate Estimators

The conventional separate mean estimator under is given by

On the lines of Singh [23]; the classical separate ratio estimator of population mean using bivariate auxiliary information under is defined as

The classical separate regression estimator of population mean under bivariate auxiliary information using iswhere and are the regression coefficients of on and , respectively, in stratum .

Motivated by Olkin [24]; the separate ratio estimator in using bivariate auxiliary information is given bywhere is a duly opted scalar in the stratum to be determined.

Following Abuâ€“Dayyeh et al. [25]; a separate class of ratio type estimators using bivariate auxiliary information under is given bywhere are duly opted scalars in stratum and .

Following Kadilar and Cingi [7]; one can suggest some separate ratio-cum-product type estimators based on bivariate auxiliary information under as

The separate version of Singh et al. [16] estimator is given bywhere , are duly opted scalars in stratum and , are scalars in stratum taking real values.

The separate version of Koyuncu and Kadilar [12] family of estimators is given bywhere , , and are some prescribed scalars whereas , and , are either real numbers or functions of the known parameters of the auxiliary variables and , respectively, in stratum .

The separate version of Tailor and Chouhan [14] estimator for population mean using bivariate auxiliary information under is defined aswhere is a duly opted scalar in stratum .

On the lines of Lone et al. [18]; a generalized separate ratio-cum-product type exponential estimator in is defined aswhere and are duly opted scalars in the stratum .

The separate version of Lone et al. [19] estimator for estimating population mean as

It is to be noted that the minimum of separate Abuâ€“Dayyeh [25] type estimator , Koyuncu and Kadilar [12] estimator and Lone et al. [18, 19] estimators & are equal to the minimum of classical separate regression estimator .The separate type of Muneer et al. [21] estimator in is given bywhere and are suitably chosen scalars in stratum and is a real constant in stratum .

Muneer et al. [22] suggested a separate chain ratio exponential family of estimators in as

On the lines of Searls [27], the modified form of the above separate estimator is given bywhere, , , and and is a duly opted scalar in stratum , assumes values âˆ’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 and for different values of scalars.The bias and of the estimators considered in this section are readily discussed in the Appendix .

#### 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 by incorporating bivariate auxiliary information under .

##### 3.1. Combined Estimators

We propose some improved combined classes of estimators based on bivariate auxiliary information under aswhere , and , are duly opted scalars to be determined.

Theorem 1. *The bias of the proposed combined classes of estimators** is given by*

*Proof. *The precis of the derivations are given in Appendix for quick review.

Theorem 2. *The**of the proposed combined classes of estimators**is given by*

*Proof. *The precis of the derivations are given in Appendix C for quick review.

Corollary 1. *The minimum of the proposed combined classes of estimators , â€‰=â€‰1,2,..., 5 is given by*

*Proof. *The precis of derivations and the definition of parametric functions and are given in Appendix C for quick review.

We note that Theorem 2 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 under aswhere , and , are duly opted scalars in stratum .

Theorem 3. *The bias of the proposed separate classes of estimators is given by*

*Proof. *The precis of the derivations are given in Appendix C for quick review

Theorem 4. *The of the proposed separate classes of estimators is given by*

*Proof. *The precis of the derivations are given in Appendix C for quick review.

Corollary 2. *The minimum of the proposed separate classes of estimators ;â€‰=â€‰1,2, â€¦, 5 is given by*

*Proof. *The precis of the derivations and definition of parametric functions and are given in Appendix

*Proof. *C for quick review.

We again note that Theorem 4 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 minimum of the proposed combined estimators from (53) and (54) with the minimum 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: