New classes of difference cum-ratio-type exponential estimators for a finite population variance in stratified random sampling

The problem of estimating the variance of a finite population is an important issue in practical situations where controlling variability is difficult. During experiments conducted in the fields of agriculture and biology, researchers often face this issue, resulting in outcomes that appear uncontrollable for the desired results. Using auxiliary information effectively has the potential to enhance the precision of estimators. This article aims to introduce improved classes of efficient estimators that are specifically designed to estimate the study variable's finite population variance. When stratified random sampling is used, these estimators are particularly efficient when the minimum and maximum values of the auxiliary variable are known. The bias and mean squared error (MSE) of the proposed classes of estimators are determined by a first-order approximation. In order to evaluate their performance and verify the theoretical results, we performed simulation research. The proposed estimators show higher percent relative efficiencies (PREs) in all simulation scenarios compared to other existing estimators, according to the results. Three datasets are utilized in the application section, which are used to further validate the effectiveness of the proposed estimators.


Introduction
In order to optimize the performance of the estimators under study while minimizing costs, time, and effort, survey sampling aims to get complete details on particular characteristics of the population.In many populations, there are a few extreme observations, and estimating unknown population parameters without considering this data can be highly sensitive.In such cases, the outcomes may either be underestimated or overestimated.However, it is important to note that the efficiency of estimators usually declines in terms of mean square error when confronted with extreme values in the data set.Removing such data from the sample may be tempting.To successfully address this difficulty, it is extremely important to include this information in the process of determining population characteristics.By transforming the known largest and smallest observations of the auxiliary variable using a linear transformation, [20] introduced two estimators.After that, these concepts were not investigated any further until the ideas of [17].They used the technique of applying the largest and smallest observations of the auxiliary variable to different finite population mean estimators.[18] optimized the estimate of the finite population mean under largest and smallest observations using a two-phase sampling method.For more details, see [2][3][4]8,9,11,12,19,26] and references therein.
The problem of estimating the variance of a finite population is an important issue in practical situations where controlling variability is difficult.During experiments conducted in the fields of agriculture and biology, researchers often face this issue, resulting in outcomes that appear uncontrollable for the desired results.Using auxiliary information effectively has the potential to enhance the precision of estimators.In this paper, we propose two enhanced classes of estimators for finite population variance estimation that use the known extreme values of the auxiliary variable for further improvement, as discussed by [11][12][13].To some extent, stratified sampling can help us deal with variability in the planning, design, and estimation phases.Using stratified sampling, homogeneous groups known as strata are generated from different groups, and samples can be taken from each stratum independently.A lot of researchers have worked on different types of estimators to estimate the finite population variance, including [1,5,6,[14][15][16][21][22][23][24][27][28][29].
Extreme values of the auxiliary variable are used as auxiliary information in this article, and are retained in the data.With a stratified random sampling technique, we propose two enhanced classes of estimators for finite population variance estimation that use the known extreme values of the auxiliary variable.
This article is divided into the following sections.The notations and methods are introduced in Section 2. Section 3 covers some existing estimators.We provide comprehensive detail on our suggested estimators in Section 4. The mathematical comparison is given in Section 5.In Section 6, a simulation study is presented to produce six distinct artificial populations using various probability distributions in order to validate the theoretical outcomes covered in Section 5.This section also includes some numerical examples to further clarify our theoretical findings.Finally, some conclusions and recommendations for further research are included in Section 7.

Notations and symbols
Consider a finite population  = (  1 , respectively, where the known stratum weight is denoted by be the population variances in the ℎ ℎ stratum.Let ȳℎ and xℎ be the sample means of the study and the auxiliary variables in the ℎ ℎ stratum, and their corresponding sample variances are The following terms are defined in order to calculate the biases and mean square errors for different estimators: , where Here,  40ℎ =  2(ℎ) and  04ℎ =  2(ℎ) .

Existing estimators
This section introduces the existing estimators of finite population variances.
In stratified random sampling, the variance of the usual estimator ȳ  = ∑  ℎ=1  ℎ ȳℎ , is defined as follows The unbiased estimator Ŝ2  of  2  , is defined as The usual variance estimator of Ŝ2  =  2  for population variance is given by (3.1) [15] suggested a ratio estimator for population variance Ŝ2  , which is given by The following expressions represent the bias and and ) . (3.4) The separate linear regression estimator Ŝ2  , is defined as where , is the sample regression coefficient.
The following expression represents the where [6] introduced an exponential ratio type estimator Ŝ2   , is defined as . (3.7) The following expressions represent the bias and  of Ŝ2 ) .
[16] suggested some ratio estimators Ŝ2 are defined as where  ℎ =

𝑆 𝑥ℎ Xℎ
is the population coefficient of variation.

Proposed estimators
This section introduces some improved classes of estimators inspired by [11][12][13] that use the known largest and smallest values of the auxiliary variables to calculate the finite population variance.The suggested estimators are defined as where the scalar quantities ( 1 ,  2 ) contain the values (0, −1, 1), (  ℎ ,  = 1, 2 ) are unknown constants whose values need to be determined so as to minimize the biases and the mean squared errors, and . The parameters of the auxiliary variable are  1ℎ ,  2ℎ , and  1ℎ =  ℎ −  ℎ .The different classes of the proposed estimator (I) that we derive from (4.1) are presented in 1 where  ℎ = exp .

Table 1
Different classes of the suggested estimator-I.

Properties of the suggested estimator-I
To obtain the bias and  of the first suggested estimator Ŝ2  , we now rewrite (4.1) in terms of errors, i.e. where .
The following equation is obtained by employing the first-order Taylor series approximation ) Using (4.4), the bias of Ŝ2  up to first order of approximation is given by where and ) .
The  of Ŝ2  up to the first order of approximation is obtained by squaring both sides of (4.4) and applying expectation, which is given by where After minimizing (4.6), the optimal values of  1ℎ and  2ℎ are, respectively provided by The minimum bias and  for Ŝ2  are obtained by putting the optimum values of  1ℎ and  2ℎ into (4.5) and (4.6), which are given by and

Properties of the suggested estimator-II
Now to discuss the properties of the second suggested estimator (II), we rewrite (4.2) in terms of errors, i.e.

Numerical comparison
In this section, we compare the percent relative efficiency ( ) of different estimators using simulated and real data sets in order to assess the effectiveness of the suggested estimators.

Simulation study
In order to verify the theoretical results discussed in Section 5, we utilize the idea from [12,13] to conduct a simulation study to determine the effectiveness of the suggested estimators based on known extreme values of the auxiliary variable.The following probability distributions can be used to artificially obtain six different populations for auxiliary variable X: The variable of interest  is then computed as where  ∼ (0, 1) denotes the error term, and   = 0.80 is the correlation coefficient between (, ).
To find the (PREs) of the proposed estimators, we adopted the following procedures in R-software: Step 1: To generate a population of size 1000, we firstly employ specific types of probability distributions.This population is then divided into two different strata to compute various values of recommended and existing estimators through the use of stratified random sampling methods.
Step 2: We obtain the population total from Step 1, as well as the smallest and largest values of the auxiliary variable.Additionally, we calculate the optimal values of the suggested estimators' unknown constants.
Step 3: To obtain different sample sizes for each population, we apply simple random sampling without replacement (SRSWOR).The various sample sizes are defined as (n = 20%; 30%; 40% of N).
Step 4: For each sample size, find the   values of all the estimators discussed in this article.Step 5: Following 50000 repetitions of steps 3 and 4, the results for artificial populations are displayed in Table 2, while the results for real data sets are given in Table 3.Finally, obtain the MSEs and PREs for each estimator over all replications using the following formulas: where k is one of , ,  , , 1, 2, 3, ( = 1, 2, … , 8, ).

Numerical examples
We compared the percent relative efficiency ( ) of different estimators using real datasets in order to assess the performance of the suggested classes of estimators.The data-sets are listed below.For efficiency comparisons, we use the following formula: × 100 where k is one of , ,  , , 1, 2, 3, ( = 1, 2, … , 8, ).Table 3 summarizes the percent relative efficiency for three different data sets.

Conclusion
We proposed some classes of efficient estimators for calculating the finite population variance in this article.These estimators make use of the auxiliary variable's known extreme values.In Section 5, we discussed theoretical conditions that show the better efficiency of the proposed estimators, enabling us to compare their characteristics with those of existing estimators.We performed a simulation study and examined a number of empirical datasets in order to validate these conditions.The proposed estimators are consistently better than existing estimators in terms of  , according to the results displayed in Table 2.The empirical results in Table 3 further support this observation and validate the theoretical conclusions in Section 5.
The suggested estimators Ŝ2  ( = 1, 2, 3, … , 8, ) show higher efficiency compared to the other existing estimators, and Ŝ2 8 is especially selected among these proposed estimators because of its minimum , according to both the simulation and empirical results.However, we examined the characteristics of the proposed effective classes of estimators using a stratified random sampling approach.Our results can be useful in identifying the more efficient estimators that can provide the lowest .It is also possible to offer some novel estimators in the two-phase stratified sampling approach.It is also an interesting topic for further research.

Table 2
Percent relative efficiency (PRE) of the estimators using the artificial populations.
: Food costs associated with the work, :