Dual Use of Auxiliary Information for Estimating the Finite Population Mean under the Stratified Random Sampling Scheme

In this paper, we proposed an improved family of estimators for finite population mean under stratified random sampling, which needed a helping variable on the sample mean and rank of the auxiliary variable.+e expression of the bias and mean square error of the proposed and existing estimators are computed up to the first-order approximation. +e estimators proposed in different situations were investigated and provided a minimum mean square error relative to all other estimators considered. Four actual data sets and simulation studies are carried out to observe the performance of the estimators. For simulation study, R software is used. +e mean square errors of all four data sets are minimum and percent relative efficiencies are more than a hundred percent higher than the other existing estimators, which indicated the importance of the newly proposed family of estimators. From the simulation study, it is concluded that the suggested family of estimators achieved better results. We demonstrate theoretically and numerically that the proposed estimator produces efficient results compared to all other contend estimators in entire situations. Overall, we conclude that the performance of the family of suggested estimators is better than all existing estimators.


Introduction
In the literature of survey sampling, the use of auxiliary variables was discussed by many researchers to improve the efficiency of their developed estimators, for estimating some usual parameters, such as mean, median, variance, and standard deviation. In such situations, traditional ratio, product, and regression estimators give efficient outcomes of the unspecified parameters. ere is a situation where the strength of linear association between the study and auxiliary variable is positive; then the rank of the auxiliary variable is also positively correlated with the study variable. In this article, we developed a family of the estimator for estimating mean that utilizes supplementary variable, not only in the shape of auxiliary variable but also on the rank of auxiliary variable.
If the population of interest is homogeneous, then simple random sampling performs reasonably well. However, when the population of interest is heterogeneous, in such a situation it is advisable to use stratified random sampling instead of simple random sampling. In stratified random sampling, we split the whole aggregate into several nonoverlapping groups or subgroups called strata. ese groups are homogeneous entirely and a sample is drawn independently from each stratum separately. To obtain the maximum benefit from stratification, the values of the N h must be known. When the strata have been determined, a sample is drawn from each stratum and the drawings are being made independently. If a simple random sample is taken from each stratum, then the entire procedure is described as stratified random sampling. Different researchers used different sample allocation techniques to distribute the sample in different strata. e use of a separate ratio estimate in each stratum is more precise if the sample in each stratum is large enough, so in this article, we use the proportional allocation method. More efforts have been pointed out about the population mean under stratified random sampling. Stratification improves efficiency when the variance between strata is much larger than the variances within strata as stated by Zaman and Kadilar [1]. More studies regarding population mean using auxiliary variables are Hussain et al. [2], Aladag and Cingi [3], Grover and Kaur [4], Shabbir and Gupta [5], Rao [6], Singh et al. [7], Kadilar and Cingi [8], Khalid [9,10], Zaman and Bulut [11], Al-Marzouki [12], and Aamir et al. [13,14].
In this paper, we propose a new family of estimators for estimating the mean using the information on the mean and ranks of the auxiliary variable based on stratified random sampling. e remaining document is set as follows.
In Section 2, notations and symbols are described. e literature review and proposed estimator are given in Sections 3 and 4. In Sections 5 and 6, theoretical and numerical identification are made, while Section 7 consists of the simulation study. In Section 8, discussion of the numerical results is provided. Concluding remarks of the article are given in Section 9.

Notations and Symbols
Let Y and X be the study and auxiliary variables which take values y h and x h , respectively, where i � 1, 2, . . . , N h and h � 1, 2, . . . , L; for estimating mean, suppose a sample of size n h is taken from the h th stratum using SRSWOR. L h�1 n h � n, where n is the sample size. Consider Y the study variable, X and Z the auxiliary and rank of the auxiliary variable, the covariance between (Y, X), (Y, Z), and (X, Z) for the hth stratum, R yxh � S yxh /(S yh S xh ), R yzh � S yzh /(S yh S zh ), and R xzh � S xzh /(S xh S zh ) the population correlation between (Y, X), (Y, Z), and (X, Z) for the hth stratum, the correlation coefficient between X and Z based on stratified random sampling, and R 2 To obtain the properties of the existing and proposed estimators of Y, we consider the following relative error terms based on stratified random sampling. Let

Existing Estimators
In this portion, we consider some estimators of population mean based on stratified sampling schemes that are available in the literature. e properties of these existing estimators are obtained up to the first degree of approximation. Different estimators with their error terms and mean square errors are as follows: (1) e usual estimator of Y is given by e variance of Y SRS st is given by (2) e existing ratio estimator of Y is given by Cochran [15].

Journal of Mathematics
e properties of Y R st are given by (3) e existing product estimator of Y is given by Murthy [16].
e properties of Y P st given as (4) e existing regression estimator of Y is given by e minimum variance of X Reg st at the optimal value Q is given by Equation (9) will become (5) e existing difference-type estimator of Y is given by where Q 1 and Q 2 are unknown constants. e bias and MSE of Y R,D st are given as e optimal values of Q 1 and Q 2 are given as e minimum MSE of Y R,D st at the optimal values of Q 1 , Q 2 is written as Equation (14) can also be written as (6) Exponential estimator [13] is given by e properties of Y S st is given by where θ � aX/(aX + b). (7) Grover and Kaur [17] are given by e properties of Y gr,kr st are given by

Journal of Mathematics
From (19), the optimal values are given by e minimum MSE of Y gr,kr st at the optimal values is given by or and Y gr,kr st is more efficient as compared to Y Reg st .

Proposed Estimator
e use of auxiliary variables may improve the efficiency of an estimator in the design or estimation stages. When there is a correlation between the study and the auxiliary variable, the rank of the auxiliary variable is correlated with the study variables as well. erefore, the rank of the auxiliary variable can be treated as a new auxiliary variable, and it is useful to increase the effectiveness of an estimator. By taking motivation from Hussain et al. [2], we developed a new family of estimators of Y which needed a helping variable on the sample mean and rank of the auxiliary variable. Moreover, some members of our proposed and existing estimates are also presented in Table 1.
where Q 5 , Q 6 , and Q 7 are unknown values. Equation (23) can be expressed as After simplification (24), we have e properties of Y Pr st are shown as Journal of Mathematics e optimal values of Q 5 , Q 6 , and Q 7 , determined by reducing (26), are given by e simplified least mean square error of Y Pr st at optimal values of Q 5 , Q 6 , and Q 7 is as follows: Pr st Here, (28) may be written as where

Efficiency Comparisons
In the following portion, we made an efficiency comparison of all estimators.

Numerical Investigation
In this portion, the mathematical outcome is illustrated to check the efficiency of all estimators. Four data sets are contemplated. e finding outcomes of these data sets are listed in Tables 2-5. e percent relative efficiency of the estimator Y i w.r.t. Y SRS st is as follows: where i � R st , P st , . . . , Pr st . e percent relative efficiency of the finite population means, measured from four data sets, is listed in Tables 6-9.
Population 1 (source: Koyuncu and Kadilar [18]): Y is the number of instructors and X is the number of the trainees in 2007 for 923 districts in six regions. Population 2 (source: Kadilar and Cingi [19]): Y is yield of apples in 1999 and X is yield of apples in 1998.

Simulation Study
A simulation study is performed to determine the efficiency of the estimators suggested for stratified sampling technique with the help of information of single auxiliary variable X and adapted rank set sampling procedure on auxiliary variable X and then made variable Z. In this section, the           Journal of Mathematics performance of the suggested exponential cum ratio type family of estimators is appraised for the artificially generated population of different stratum and sample sizes. e population elements are randomly generated from N 1 ∼ (2, 6) and N 1 ∼ (3, 9) compared to that from the values of x 1 and x 2 generated y 1 and y 2 . Applying the procedure of ranked set sampling, we obtained Z variable. e comparison is offered in the form of percentage relative efficiency (PRE). e relative efficiency in Table 10 is computed using the following formula: where i � R st , P st , . . . , Pr st . In this study, consider the generated population for summarizing the simulation procedures. e process is used to find out the efficiency of the family of suggested estimators over all the existing estimators.
From the described generated population of the normal population, the sample size is considered as n 1 � 25 and n 2 � 25. We take r � 10 and m � 50 for the ranks of the auxiliary variable x. e procedure has been repeated 10000 times and the whole aggregate is split into two strata, where N � 1000 and N 1 � 500, N 2 � 500 evaluate several available families of existing estimators under stratified random sampling technique. Using the sample, simulate 10,000 times estimators values of existing estimators class Y S st , and Y (1) gr,kr st , Y (2) gr,kr st , Y gr,kr st , Y (4) gr,kr st , Y (5) gr,kr st , Y (6) gr,kr st , Y (7) gr,kr st , Y (8) gr,kr st , Y (9) gr,kr st , Y (10) gr,kr st , and Y (1) S st , separately are obtained for the formula described in  is simulation study is evaluating the percent relative efficiencies of the suggested exponential cum ratio type family of estimators and available estimators for sample sizes in two strata. From the critical comparison, it is concluded that the suggested family of estimators perform better result. Overall, we can conclude the performance of the family of suggested estimators is better than all existing estimators.

Discussion
As mentioned above, we used two real data sets and a simulation study to determine the efficiency of the estimators suggested for stratified sampling technique with the help of information of single auxiliary variable X and rank of the auxiliary variable. We also considered different sample sizes from the populations for scenario I and scenario II. e proposed family of estimators and the adapted stratified estimators were compared to each other with respect to their mean square error and percentage relative efficiency. In Tables 2-5, we present the data description. e results of mean square error for the real data sets are presented in Tables 6-9. And the percentage relative efficiency is shown in Tables 11-14. From the numerical findings, which are put in Tables 6-9  and Tables 10-14, it is observed that the MSEs and PREs of our proposed family of estimators change with the choices of a and b. It is also observed that the proposed family of estimators are more precise than the existing estimators on both real data sets and simulation study, in terms of MSEs and PREs.

Concluding Remarks
In this manuscript, we proposed a new family of estimators for the finite population mean under the stratified sampling scheme, which needed a helping variable on the sample mean and rank of the auxiliary variable. Based on the real data sets, it is observed that the proposed class of estimators perform better theoretically and numerically compared to their existing counterparts and should be preferable over the existing estimators available in the literature. To check the robustness and generalizability of the proposed family of estimators, a simulation study is also conducted. e results of the simulation study also support the usefulness of the proposed family of estimators. Hence, it is recommended that the new suggested family of estimators should be used for better results when estimating population mean under stratified random sampling scheme.

Data Availability
All the data used for this study can be found within the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.