A Simulation-Based Study for Progressive Estimation of Population Mean through Traditional and Nontraditional Measures in Stratified Random Sampling

'is study suggests a new optimal family of exponential-type estimators for estimating population mean in stratified random sampling. 'ese estimators are based on the traditional and nontraditional measures of auxiliary information. Expressions for the bias, mean square error, and minimum mean square error of the proposed estimators are derived up to first order of approximation. It is observed that proposed estimators perform better than the traditional estimators (unbiased, combined ratio, and combined regression) and other recent estimators. A real dataset is used to highlight the applicability of proposed estimators. In addition, a simulation study is carried out to assess the performance of new family as compared to other estimators.


Introduction
Nowadays, it is common practice to use the auxiliary/ancillary information to boost the efficiency of estimators in survey sampling. Most of the researchers only deal with the traditional information of auxiliary variable(s) such as standard deviation, coefficient of variation, coefficient of skewness, coefficient of kurtosis, and coefficient of correlation. Having edge of this traditional information, many authors have been trying to explore new optimal estimators and families of estimators for estimating population mean under stratified random sampling. Stratified random sampling has often proved needful in improving the precision of estimators over simple random sampling, for instance, see works of Kadilar and Cingi [1,2], Koyuncu and Kadilar [3,4], Singh and Vishwakarma [5][6][7], Shabbir and Gupta [8], Haq and Shabbir [9], Singh and Solanki [10], Yadav et al. [11], Solanki and Singh [10,12], Javed et al. [13], and Javed and Irfan [14]. e motivation behind this article is to utilize the nontraditional information as well as the traditional information of the auxiliary variable to progress the estimation of population mean in stratified random sampling. is idea is initiated first time in this article under stratified random sampling.
Nontraditional information includes quartile deviation, midrange, interquartile range, quartile average, decile mean, tri-mean, Hodges-Lehmann estimator, and L-moments of an auxiliary variable. L-moments are determined by linear combinations of the expected values of the order statistics (for detail, check the works of Hosking [15] and Shahzad et al. [16]). Furthermore, efficiency of the estimators is uncertain in the occurrence of the extreme values in the dataset. Some of the above nontraditional measures such as decile mean, Hodges-Lehmann estimator, and tri-mean are robust measures. Utilizing these measures, we can well cope with the extreme values/outliers in the dataset. In addition, L-moments also are used to reduce the negative effect of outliers on the estimators.
Rest of the article is organized in the following way. Section 2 presents the useful notations. Section 3 gives comprehensive detail of existing families of estimators. Section 4 suggests a new optimal family of estimators for estimating population mean using traditional and nontraditional measures of auxiliary variable. Expressions for the bias, mean squared error (MSE) and minimum MSE of this family are derived up to first degree of approximation in the same section. A real dataset is used in Section 5 to check the potential of new estimators as compared to existing ones. In Section 6, the performance of suggested family is evaluated by carrying out a simulation study using the same dataset used in Section 5. Section 7 contains the final discussion.

Useful Notations
Let us consider a finite population U � U 1 , U 2 , U 3 , ..., U N of size N, and it can be stratified into L homogenous strata with h th stratum containing N h , ( To derive the expressions for the bias, mean square error (MSE), and minimum mean square error of the existing and proposed estimators, we consider the following relative error terms along with their expectations as such that (2) From (2), we can write as below: where Some other formulas for h th stratum, under stratified random sampling are listed below: , where Q 1 , Q 2 , and Q 3 are the first, second, and third quartiles, respectively, x (1) is the minimum value, and x (N) is the maximum value of the data.

Some Existing Estimators/Classes of Estimators
is section gives a brief introduction of some well-known estimators/classes of estimators from the literature.

Usual Estimators.
In stratified random sampling usual unbiased Y st , combined ratio Y CR and combined regression Y CReg estimators and their MSEs are detailed below: where Bahl and Tuteja [17] suggested ratio and product exponential-type estimators for population mean under stratified random sampling as Average of (7) and (8) can be written as [3]. To estimate the population mean under stratified random sampling, a family of ratio estimators was introduced by Koyuncu and Kadilar [3] as below: (10) where c and g are suitable constants and a st ( ≠ 0) and b st are either real numbers or functions of known parameters of the auxiliary variable such as coefficient of skewness, coefficient of kurtosis, coefficient of variation, and coefficient of correlation.

Koyuncu and Kadilar
Up to the first order of approximation, the bias and MSE of Y K are given by , the minimum MSE Y K is given as, [4]. Koyuncu and Kadilar [4] considered the ratio estimator of Gupta and Shabbir [18] and suggested an improved estimator defined as below:

Koyuncu and Kadilar
where θ 1 and θ 2 are suitably chosen weights. Given below are the expressions, up to first degree of approximation, for the bias and MSE of Y KK , respectively: e suitable weights of θ 1 and θ 2 are given by Journal of Mathematics Inserting the above weights of θ 1 and θ 2 in (14), we get the minimum MSE of Y KK as [8]. Given below is a ratio-type estimator suggested by Shabbir and Gupta [8] in stratified random sampling:

Shabbir and Gupta
where θ 3 and θ 4 are the constants to be determined. Also, we consider that Expressions for the bias and the MSE of Y SG , respectively, are given below: e suitable weights of θ 3 and θ 4 are given as Putting weights of θ 3 and θ 4 in (20), we have minimum MSE of Y SG as 3.5. Haq and Shabbir [9]. Haq and Shabbir [9] proposed two exponential ratio-type families of estimators detailed below: where θ 5 , θ 6 , θ 7 , and θ 8 are the suitable constants. Given below are the expressions for bias and MSE of Y HS1 and Y HS2 , respectively: where ψ � cη and η is defined earlier. e weights of θ 5 , θ 6 , θ 7 , and θ 8 are determined as below:

Journal of Mathematics
By substituting values of θ 5 and θ 6 in (25) and θ 7 and θ 8 in (26), we get minimum MSE of Y HS1 and Y HS2 , respectively: 3.6. Singh and Solanki [19]. Singh and Solanki [19] proposed a family of estimators as given below: where δ 1 and δ 2 are suitable scalars and θ 9 and θ 10 are the constants to be determined to make the MSE minimum.
Assuming different values of δ 1 , δ 2 , and c proposed family Y SS1 reduces to the ratio-type Y SS1R , product-type Y SS1P , and ratio-cum-product-type Y SS1RP estimators.
For ratio-type, product-type, and ratio-cum-producttype estimators suitable values are ( For bias and MSE of Y SS1 , we consider the expressions given below: where e weights of θ 9 and θ 10 are determined as below: (33) Substituting the above weights in (31), we get the minimum MSE as given by [10]. Given below is the class of estimators suggested by Solanki and Singh [10]:

Solanki and Singh
where θ 11 and θ 12 are the feasible weights to be found such that the MSE is minimal. Here, c � 1; δ 3 and δ 4 being the constants take values (0, 1, −1) for obtaining different estimators like For bias and MSE of Y SS2 , we consider the expressions given below: where Journal of Mathematics 7 e suitable weights of θ 11 and θ 12 are as below: Substituting these suitable weights in (37), we have the minimum MSE as given by 3.8. Solanki and Singh [12]. Recently, Solanki and Singh [12] defined an improved estimation given as where θ 13 and θ 14 are the suitably chosen weights to get minimum MSE.
are real number to parameters related to auxiliary variate x. Here, δ 5 , δ 6 , δ 7 , and δ 8 being the constants take values (−1, 0, 1) for obtaining different classes of estimators Y SS3i : For bias and MSE of Y SS3 , we consider the expressions given below: where Given below are the weights of θ 13 and θ 14 for minimizing the MSE: us, the minimum MSE by putting the above values of θ 13 and θ 14 in (43) is given by

Suggested Family of Estimators
Following the lines of Shabbir et al. [20], a generalized estimator for the estimation of population mean is proposed using some traditional and nontraditional measures of an auxiliary variable. For more details of these nontraditional measures, see the works of Hettmansperger and McKean [21], Wang et al. [22], and Irfan et al. [23][24][25][26]: where θ 15 Table 1.
Using (1), the suggested class of estimators Y P can be rewritten as As defined earlier, η � a st X/a st X + b st . Expanding the right-hand side of (48), up to first order of approximation and subtracting Y from both sides, we obtain Journal of Mathematics Up to first order of approximation, the bias and the MSE are given by By minimizing (51), suitable weights of θ 15 and θ 16 are obtained as below:

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Putting the optimal weights of θ 15 and θ 16 in (51), we have the minimum MSE given by

Application to a Dataset
To examine the performance of the proposed class of estimators, we considered a real data of Turkey (2007) used by Koyuncu and Kadilar [3] given in Table 2. In this data, let y be the number of teachers (study variable) and x be the number of students (auxiliary variable) which are recorded for primary and secondary schools at 6 regions for N � 923 districts. A total sample of size n � 180 is selected through Neyman allocation from 6 strata. (source: Ministry of Education, Republic of Turkey). We , and MSE min (Y P ) for the population dataset given in Table 2 and are reported in Tables 3 and 4.

Important Findings
(ii) It is obvious from

Simulation Study
In this section, we carried out a simulation study using R statistical software to evaluate the behavior of proposed Table 3 is used for the simulation study. ree different sample sizes n � 180, 250, and 350 are taken from this population on the basis of proportional allocation. e following steps summarize the procedure of finding the average MSE of an estimator.
Step 1: select a bivariate stratified sample of size n using simple random sampling without replacement from the bivariate stratified normal population Step 2: use sample data from Step 1 to find the MSE of all the estimators under the study  Table 5.
MSEs of the other estimators under the study are presented in Tables 6-11, and the following important considerations are made from them.

Discussion
In this study, we proposed a new optimal family of estimators for estimating population mean under stratified random sampling. Bias, MSE, and minimum MSE of this family of estimators are derived up to first degree of approximation. e proposed family is compared with some well-known estimators/classes of estimators under stratified random sampling such as the works of Koyuncu and Kadilar [3,4], Shabbir and Gupta [8], Haq and Shabbir [9], Singh and Solanki [19], and Solanki and Singh [10,12]. It is numerically inferred that the proposed family behaves optimal

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Journal of Mathematics             14 Journal of Mathematics    as compared to other estimators. A simulation study is also carried out in support of efficient proposed estimators. So, to get more enhanced results in practice under stratified random sampling, our suggested family of estimators is recommended. e possible extensions of this work are to estimate the (1) finite population mean using robust quantile regression and L-moments characteristics of an auxiliary information under stratified ranked set sampling, (2) finite population parameters including median, variance, and proportions using L-moments under different sampling designs, and (3) population mean in the presence of nonsampling errors using L-moments and calibration approach.
Data Availability e data used to support the findings of this study are included within the article.

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