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 Abu–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 −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 Abu–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 Abu–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 −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. Theof the proposed combined classes of estimatorsis 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 , = 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 ; = 1,2, …, 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:
If conditions (72) to (86) hold, then the proposed combined classes of estimators perform better than the other existing combined estimators.
4.2. Separate Estimators
On comparing the minimum of the proposed separate estimators from (69) and (70) with the minimum 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:
If the conditions (87) to (101) hold then the proposed separate classes of estimators perform better than the other existing separate estimators.
4.3. Comparison of Proposed Combined and Separate Estimators
By comparing the minimum of the proposed combined and separate classes of estimators and , , we get
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 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 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 size = 2000 units using a trivariate normal distribution in software with parameters , , , , , and different amounts of correlation coefficients , , and .(ii)Stratify the above population into 4 equal disjoint strata and quantify a sample of size = 50 units from each stratum.(iii)Tabulate all necessary statistics.(iv)Using 15,000 iterations to calculate the absolute relative bias and percent relative efficiency of various combined and separate classes of estimators. The and of different estimators are calculated regarding the classical ratio and usual mean estimators and results are reported in Table 1 and Table 2. The and are calculated using the following expressions. where and .
The simulation findings of the combined and separate classes of estimators are exposited in terms of and 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 and , , 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 . Also, the proposed combined and separate class of estimators and 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 . The mathematical expressions of bias and 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 estimators , perform better than the combined form of usual mean estimator , classical ratio and regression estimators & , Olkin [24] type estimator , Abu–Dayyeh et al. [25] type estimators & , Kadilar and Cingi [7] type estimators , Koyuncu and Kadilar [12] estimator , Singh et al. [16] estimator , Tailor and Chouhan [14] estimator , Lone et al. [18, 19] estimators and Muneer et al. [21, 22] estimators & for different values of correlation coefficients.(ii)The proposed separate classes of estimators , dominate the separate form of usual mean estimator , classical ratio and regression estimators & , Olkin [24] type estimator , Abu-Dayyeh et al. [25] type estimators & , Kadilar and Cingi [7] type estimators , Koyuncu and Kadilar [12] estimator , Singh et al. [16] estimator , Tailor and Chouhan [14] estimator , Lone et al. [18, 19] estimators and Muneer et al. [21, 22] estimators & for different values of correlation coefficients.(iii)Since, the proposed combined and separate classes of estimators and , are, respectively, superior to the combined and separate ratio and chain ratio exponential estimators , and , envisaged by Muneer et al. [22] consequently the proposed combined and separate classes of estimators and , will also dominate those 65 estimators that are the members of the combined and separate ratio and chain ratio exponential estimators , and , , respectively.(iv)The proposed combined and separate class of estimators and perform better among the proposed classes of estimators.(v)The proposed separate classes of estimators , dominate the proposed combined classes of estimators , in terms of greater for various amounts of correlation coefficients.(vi)The 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.
Appendix
A. Bias and MSE of the Existing Combined Estimators
The bias and expressions of the existing combined estimators are expressed as mentioned below:where; ; ; ; ; ; ; ; ; ; ; ; ; ; and .
The optimum values of the scalars are given as
B. Bias and MSE of the Existing Separate Estimators
The bias and expressions of the existing separate estimators are expressed as where; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .
The optimum values of scalars are tabulated as
C. Bias and MSE of the Proposed Combined and Separate Estimators
This section addresses the precis of the proof of Theorem 2 and Corollary 1 of Subsection 3.1.
Consider the estimator
Using the notations defined in the earlier section, we express the estimator in terms as
Now, squaring and taking expectations both sides of (C.2), we will get the of the estimator as
The optimum values of , , and can be obtained by minimizing (C.3) w.r.t , , and as
Putting , and in (C.3), we get the minimum as
In similar way, we can tabulate the of other estimators as
The minimum of the estimators is given by
The optimum values of the scalars involved are given hereunderwhere
Similarly, we can obtain the derivation of of the proposed separate estimators.
Data Availability
There are no data associated with this article.
Conflicts of Interest
The authors have no conflicts of interest.