Flexible Robust Regression-Ratio Type Estimators and Its Applications

In real-world situations, the data set under examinationmay contain uncommon noisy measurements that unreasonably aect the data’s outcome and produce incorrect model estimates. Practitioners employed robust-type estimators to reduce the weight of the noisy measurements in a data set in such a scenario. Using auxiliary information that will produce reliable estimates, we have looked at a few exible robust-type estimators in this study. In order to estimate the population mean, this study presents unique exible robust regression type ratio estimators that take into account the data from the midrange and interdecile range of the auxiliary variables. Up to the rst order of approximate computation, the bias and mean square were calculated. In order to compare the exibility of the proposed estimator to those of the existing estimators, theoretical conditions were also obtained. We took into account data sets containing outliers for empirical computation, and it was found that the suggested estimators produce results with higher precision than the existing estimators.


Introduction
In practice, collecting all of the information on an object under investigation is challenging; therefore, predictions and decision-making studies are based on samples. Using probability theory, sampling is an art form for determining the dependability of available data. Simple random sampling (SRS) is the most common and simplest approach for selecting samples with equal probability at each selection while avoiding the concentration of auxiliary information. We collect some additional information (X) that is positively or negatively connected to the variable of interest (Y) in real-life situations with the variable of interest (Y). If we incorporate new information into classical estimators, we will get exible results. Many researchers are presently striving to increase the exibility of existing estimators by incorporating additional data. For example, Kadilar and Cingi [1] worked on the regression type estimators, Yan and Tian [2], Ijaz et al. [3][4][5].
e usual estimator of the population mean is de ned by e bias and mean square error of t j up to the rst-order approximation are given as Kadilar and Cingi [2006] introduced a classical ratio and regression estimator.
e bias of t j is given as e mean square error of t j up to the first-order approximation is given as where Yan and Tian [2010] suggested the efficient ratio-type estimators e mean square error of t k where R 6 � X/X + β 1 ,R 7 � Xβ 1 /Xβ 1 + β 2 . Ijaz et al. [3] proposed ratio and regression type estimators e mean square error is, respectively, given by Other estimators of Ijaz et al. [4,5] are defined by e mean square error of proposed ratio type estimators is l � 10, 11, 12, 13, 14, 15, 16, where θ 10 � δX/δX + QD × C X ,θ 11 � δX/δX + XC X , θ 12 � δX/δX + X, where recommended ratio estimator is as follows: e mean square error of the above estimator is defined by where θ 23 � Xβ 1 /Xβ 1 + QD.

Research Problem
In actual, some data sets have a broad range of values known as outliers. e classical estimators will result in an incorrect conclusion and overfitting of the model in such a case. e primary goal of the current work is to create an estimate that will not be significantly impacted by an outlier. is paper used the midrange and interdecile range to investigate novel robust type ratio type estimators.

Methodology of the Proposed Estimators
e study is motivated by Kadilar and Cingi [1] where the authors proposed some regression type estimators. e study of Kadilar and Cingi [1] was not taken into account the data sets with an outlier. e current study focused to cover this gap and developed some robust type estimators that are not much effective against outliers. is paper presents new estimators for estimating the population means using the auxiliary information in the forms of midrange (MR) and interdecile range (IDR). e proposed estimators are defined by where W i � 1,MR,β 2(x) V i � MR,IDR, To derive the estimator bias, and mean square error, we consider where applying expectations on both sides, we get the bias of P i which is given by Squaring and applying expectations on both sides of equation (26), we get e mean square error of p i up to the first order approximation is given as

3.1.
eoretical Conditions. In this section, theoretical conditions are derived so that to assess the performance of the proposed estimators as compared to the existing estimators.
e MSE of the proposed estimators is given in equation (29) with the usual mean estimator given in equation (2) can be compared in the following way.
Similarly, the Mse of the proposed estimator given in equation (29) can be compared with that of the Mse given in equation (29), we have the following.
e proposed estimator leads to a better performance as compared to others iff the above conditions are satisfied. Table 1 defines the result of theoretical conditions using population data sets 1 and 2.
e results of Table 1 clearly demonstrate that the aforementioned theoretical requirements are met for both data sets; hence, it is anticipated that the suggested estimators will perform better for these two data sets than for others.

Applications.
e paper proposed the robust type estimators, and hence, we considered two data sets with outliers. e data sets were obtained from the Italian Bureau of the Environment Protection [7] and recently cited by Abid et al. [8].
e data statistics are given in Tables 2 and 3. e percentage relative efficiency (PRE) is shown in Tables 4 and 5computed with the following mathematical formula: Tables 4 and 5 define the Bias, Mse, and PRE of the proposed and other existing estimators. e results

Discussion and Conclusion
e traditional or common estimators are consistently inadequate for estimating the population parameter in practice. e traditional estimators overestimate the sets of data that contain an outlier(s). e current work examines various innovative estimators that are less susceptible to these outliers as a result. In this paper, novel regressionratio type estimators are proposed by using the midrange (MR) and interdecile range (IDR). To evaluate the model performance, we derived theoretical conditions and used real-world data sets to back up our findings. Furthermore, the proposed as well as alternative estimators' results for the bias, mean square error, and percentage relative efficiency (PRF) are computed. e proposed estimators are superior to others in terms of PRE, but their Mse is the lowest of all. It is obvious that the suggested estimators outperform other methods in terms of results

Data Availability
e data set is used to evaluate the real significance of the proposed estimator and is given in the manuscript.

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