Robust Location Retrieval Strategy Under Systematic Sampling

Abstract

Systematic sampling design is an ordered observation scheme that is popular in data collection because it has the property of uniform coverage and is operationally simple. This scheme is, however, susceptible to extreme observations, which may severely compromise the accuracy of traditional location estimators. To overcome this weakness, this research proposes a robust location retrieval or estimation method that regulates the impact of unusual observations in the ordered selection framework. In the suggested strategy, a set of twenty influence-adjusted estimators is built with a variety of re-descending weighting functions, which is then extended with another family of five generalized ones. Large-scale derivations of mean squared error (MSE) and percentage relative efficiency (PRE) are used to explore the theoretical properties of the proposed estimators. Significant improvements in stability and efficiency over existing methods are demonstrated by extensive empirical analyses, both with real data (e.g., mtcars and Trees) and on a wide variety of synthetic problems containing embedded outliers. The findings suggest that location retrieval based on influence-controlled processes is much more robust when using an ordered observation scheme and can be an efficient and scalable tool implemented in the modern data-intensive and computationally demanding environment.

1. Introduction

To analyze a subset of a given population, a sampling method is used, which can be classified as probabilistic or non-probabilistic. Specifically, in this context, we are concerned with systematic sampling, which involves selecting elements from the population at regular intervals. For a positive integer m, the process starts with an initial element g chosen at random and then selects the following sequence of elements $g, g+m, g+2m, \dots , g+(t-1)m$, where $m\ge g$.

Systematic sampling is especially helpful when the population is uniformly allotted or logically listed. For example, in agricultural studies, the researcher should systematically sample the plants in a field to determine crop health. In industrial quality control, every k-th product from an assembly line is inspected periodically to measure defects. This method is favored because it is simple, easy to implement, and offers evenly distributed samples across the entire population. The cost-effectiveness of systematic sampling is one of the main advantages of the approach because it takes less time and requires less effort than simple random sampling. Systematic sampling is an appropriate method for studying natural populations, such as estimating timber volume in forests (Zinger, [1]). This technique is widely used by research institutions, such as the United Nations Food and Agriculture Organization (FAO) in its 2010 Global Forest Resources Assessment survey.

The uniform systematic sampling approach has received considerable attention, and numerous modifications and improvements have been made to it. Further details on the theoretical foundations and practical application of systematic sampling are provided in existing literature on sampling theory. The primary aim of this approach is to estimate various population characteristics or trends. This study focuses on the estimation of the mean for a particular variable. Moreover, the accuracy of these estimates is improved by incorporating an additional auxiliary variable, particularly when the sample size for the principal variable is limited.

Many researchers have calculated the average using data from one or more additional variables under systematic sampling. Initially, Swain [2] provided a ratio-type mean estimator in this context. Kushwaha and Singh [3] proposed nearly unbiased product and ratio estimators aimed at estimating the true mean of a research variable. Their methods involve the use of the jackknife technique as first proposed by Ref. [4] or a refined form of the jackknife as proposed by Singh and Singh [5]. Kushwaha and Kushwaha’s [6] research introduced a group of estimators for estimating the population average in systematic sampling design by utilizing supporting information. The study extended a previously established estimator, proposed by Swain [2], by examining conditions that enhance its efficiency. Additionally, it considered difference estimators as particular cases and identified scenarios where the proposed approach outperforms traditional estimation methods. Singh et al. [7,8] developed a generalized group of estimators for the true mean of the population in systematic sampling with a linear transformation of a supplementary variable. They derived bias and MSE expressions and an asymptotically optimal estimator. Shahzad [9] extended this concept by including the auxiliary attributes. In the study by Ali et al. [10], two classes of estimators were introduced for cases where data affecting the systematic random sampling setting includes an outlier in one or both of the critical region parameters. The first class considers ratio-type robust regression-based estimators, and the second class considers a regression-based estimator. Ref. [11] presented a regression-ratio technique for average estimation from a population with high-breakdown regression to mitigate the outlier-induced effect. The latter were extended to create these estimators using quantile regression coefficients, minimizing absolute deviations instead of squared deviations. Their estimator efficiency was then evaluated in a forestry case study and found to exceed 30 percent compared with the standard procedure for timber volume. In Audu et al.’s [12] recent work, calibrated estimators were developed to estimate population proportions from diagonal systematic samples.

Outliers in the accuracy of population parameter estimates have serious effects, but efforts have been frequently made to minimize such impact with weighted methods. Such extreme values may sometimes be ignored, but having such extreme values tends to result in significant overestimation or underestimation, impeding the reliability of traditional estimators and, in particular, increasing MSE. This has been carried out previously in [13], where an array of different estimation techniques were employed, including ratio, product, and regression-based methods, accompanied by both regular and robust estimators. On the other hand, systematic sampling methods using re-descending M-estimators to deal with outlying observations have not been explored yet. This paper addresses this gap by proposing M-type average approximations with re-descending coefficients that aid estimation accuracy in the case of anomalies under systematic sampling.

The remaining article explores the development and application of re-descending M-estimators for robust average approximation (estimation) in systematic sampling design with multiple sections. Section 2 provides an overview of re-descending M-estimators, which is one of the potential solutions under outlier-contaminated conditions. The adapted estimators are also provided in the same section. In Section 3, the theoretical framework behind the proposed estimator is explained based on a systematic sampling design. The statistical features of the estimators under study are also provided in the same section. In Section 4, the real-world applications and the simulation studies are separated into two main subsections. In the real-world application, the study evaluates the effectiveness of the developed estimators through well-known datasets, including the mtcars and Trees datasets, where comparisons are made against adapted estimators based on MSE and PRE. The results are also compared using a simulation study for varying statistical scenarios. Section 4.2 interprets the findings. Finally, Section 6 summarizes the important findings of the study, emphasizes the advantage of re-descending M-estimators in systematic sampling, and points out future research directions.

2. M-Estimators Based Average Estimation

OLS is a fundamental tool in regression analysis in many applications. However, outliers are present in the real world, and the assumption of a normal error distribution for the method is unrealistic. Therefore, OLS is not used if there are anomalies in the data (Dunder et al. [14]), even in the case of large datasets. In order to mitigate this drawback, re-descending regression methods have been developed as an extension of OLS in order to strengthen its robustness. In large samples with skewed logistic distribution, OLS is still vulnerable to distortion by outliers; robust M-estimators are more effective.

Huber built on research studying the effect of outliers on linear regression models using the M-estimator. With the central values reaching close to one and the extreme values near zero, this estimator assigns weights close to one to the central values and reduces the influence of extreme values. While not explicitly stated, the M-estimator provides a reasonable method of inference within the context of likelihood maximizing when the error normality assumption about the model may not be valid. With regard to robust M-estimators, instead of squared error terms used in the OLS, M-estimators use the following symmetric loss function:

$$\underset{\widehat{\mathsf{\Theta }}}{min}\sum _{i=1}^N\psi \left({ϵ}_i\right).$$

(1)

These estimators are constructed by adjusting the loss function $\psi (.)$ to ensure a re-descending property (see Refs. [15,16,17,18,19,20,21]). These estimators demonstrate a great ability to mitigate the influence of outliers, thus improving the reliability of statistical inference.

The arithmetic average is a key measure employed in statistical analysis. When a strong positive linear association exists between the auxiliary and primary study variables in survey sampling, ratio estimation provides an effective approach for estimating the population mean. Ratio estimation was first introduced by Cochran during the post-1950 period and has since been a major methodological breakthrough in research, except for in agricultural studies. For a comprehensive discussion on ratio-type estimators, readers may refer to Refs. [22,23,24].

Kadilar and Cingi [25] were the first to use ratio-type estimation incorporating OLS regression coefficients. However, traditional OLS-based estimators are incredibly sensitive to outliers, and are prone to delivering very bad estimates in their presence. To address this issue, Kadilar et al. [26], Zaman et al. [27], and Koc and Koc [28] introduced robust regression techniques that enhance the stability and accuracy of ratio estimators in simple random sampling. Based on this work, we suggest an adapted class of intuitive mean estimators based on re-descending coefficients under systematic sampling.

2.1. $T_{{\mathit{tt}}_{\mathsf{\Delta }1}}-T_{{\mathit{tt}}_{\mathsf{\Delta }4}}$ Estimators with [19] Coefficients

To handle datasets consisting of outliers, Ref. [19] developed an estimator that penalizes large residuals using a weighting function. This estimator is adjustable with parameters c and a, which determine its robustness, efficiency, and flexibility in accounting for robust regression models. The corresponding objective function (OF), influence function (IF), and weight function (WF) are expressed as follows:

OF:

$$\theta \left({\lambda }_l\right)=k^2{\left(1+{\left({\displaystyle \frac{{\lambda }_l}{c}}\right)}^2\right)}^{-a},\left|{\lambda }_l\right| \ge 0$$

IF:

$$\psi \left({\lambda }_l\right)=\left\{\begin{array}{cc}{\lambda }_l{\left(1+{\left({\displaystyle \frac{{\lambda }_l}{c}}\right)}^2\right)}^{-a-1},\hfill & |{\lambda }_l| < c,\hfill \\ 0,\hfill & |{\lambda }_l| \ge c\hfill \end{array}\right.$$

WF:

$$v\left({\lambda }_l\right)=\left\{\begin{array}{cc}{\left(1+{\left({\displaystyle \frac{{\lambda }_l}{c}}\right)}^2\right)}^{-a-1},\hfill & |{\lambda }_l| < c,\hfill \\ 0,\hfill & |{\lambda }_l| \ge c\hfill \end{array}\right.$$

The adapted family of mean estimators under systematic sampling utilizing the re-descending coefficient ${\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}1\right)}$ proposed by Ref. [19] is defined as follows:

$$T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}=[{\overline{y}}_s+\widehat{\mathsf{\Gamma }}\left(\mathrm{t}1\right)\left({\overline{X}}_{d\left(s\right)}\right)]\left({\displaystyle \frac{D_{\mathrm{j}}{\overline{x}}_s+H_{\mathrm{j}}}{D_{\mathrm{j}}\overline{X}+H_{\mathrm{j}}}}\right),j=1,\dots ,4$$

(2)

Note that ${\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}=\overline{X}-{\overline{x}}_{\mathrm{s}}$. The MSE of $T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}$ is derived using a Taylor series expansion:

$$\mathrm{MSE}\left(T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}\right)=\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2+{(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}1\right)})}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2-2(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}1\right)}){\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}\right),$$

(3)

where:

$${\vartheta }_{\mathrm{x}}=1+(n-1)r_{\mathrm{x}},{\vartheta }_{\mathrm{y}}=1+(n-1)r_{\mathrm{y}},t^{*}=\sqrt{{\displaystyle \frac{{\vartheta }_{\mathrm{y}}}{{\vartheta }_{\mathrm{x}}}}}.$$

(4)

2.2. $T_{{\mathit{tt}}_{\mathsf{\Delta }5}}-T_{{\mathit{tt}}_{\mathsf{\Delta }8}}$ Estimators with [29] Coefficients

In the work of Khan et al. [29], a novel estimator was introduced using a hyperbolic tangent function. The estimator is particularly effective in handling asymmetric heavy-tailed distributions. Khan et al. [29] used the hyperbolic cosine function as a weight function in order to minimize sensitivity. The OF, IF, and WF are formulated as follows:

OF:

$$\rho \left(e_l\right)=c{tan}^{-1}\left(tanh\left({\displaystyle \frac{e_l}{2c}}\right)\right),\left|e_l\right|\ge 0.$$

IF:

$$\psi \left(e_l\right)={\displaystyle \frac{e_l}{cosh\left(\frac{e_l}{2c}\right)}},\left|e_l\right|\ge 0.$$

WF:

$$w\left(e_l\right)={\displaystyle \frac{1}{cosh\left(\frac{e_l}{2c}\right)}},\left|e_l\right|\ge 0.$$

The modified class of mean estimators under systematic sampling utilizing the re-descending M-estimator coefficient ${\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}2\right)}$ from Khan et al. [29] is given by

$$T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}=[{\overline{y}}_s+\widehat{\mathsf{\Gamma }}\left(\mathrm{t}2\right)\left({\overline{X}}_{d\left(s\right)}\right)]\left({\displaystyle \frac{D_{\mathrm{j}}{\overline{x}}_s+H_{\mathrm{j}}}{D_{\mathrm{j}}\overline{X}+H_{\mathrm{j}}}}\right),j=5,\dots ,8$$

(5)

The MSE of the estimators is given by

$$\mathrm{MSE}\left(T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}\right)=\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2+{(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}2\right)})}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2-2(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}2\right)}){\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}\right)$$

(6)

2.3. $T_{{\mathit{tt}}_{\mathsf{\Delta }9}}-T_{{\mathit{tt}}_{\mathsf{\Delta }12}}$ Estimators with [20] Coefficients

Ref. [20] introduced a re-descending-type estimator that employs polynomial-driven weight functions. When residuals exceed a defined threshold h, the weight function smoothly reduces to zero. This method ensures a high breakdown point, making it resistant to leverage points. OF, IF, and WF are defined as follows:

OF:

$$\rho \left(e_l\right)=\left\{\begin{array}{cc}{\displaystyle \frac{e_l^6}{h^4}}+{\displaystyle \frac{e_l^{10}}{2h^8}}-{\displaystyle \frac{2e_l^6}{h^4}}+{\displaystyle \frac{e_l^2}{2}}-{\displaystyle \frac{2e_l^6(3e_l^4-5h^4)}{15h^8}},\hfill & |e_l| \le h,\hfill \\ {\displaystyle \frac{4h^2}{15}},\hfill & |e_l| > h.\hfill \end{array}\right.$$

IF:

$$\psi \left(e_l\right)=\left\{\begin{array}{cc}{e}_l{\left(1-{\left({\displaystyle \frac{e_l}{h}}\right)}^2\right)}^2{\left(1+{\left({\displaystyle \frac{e_l}{h}}\right)}^2\right)}^2,\hfill & |e_l| < h,\hfill \\ 0,\hfill & |e_l| \ge h.\hfill \end{array}\right.$$

WF:

$$w\left(e_l\right)=\left\{\begin{array}{cc}{\left(1-{\left({\displaystyle \frac{e_l}{h}}\right)}^2\right)}^2{\left(1+{\left({\displaystyle \frac{e_l}{h}}\right)}^2\right)}^2,\hfill & |e_l| < h,\hfill \\ 0,\hfill & |e_l| \ge h.\hfill \end{array}\right.$$

The modified class of mean estimators using the re-descending M-estimator coefficient ${\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}3\right)}$ proposed by Anekwe and Onyeagu [20] is given by

$$T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}=[{\overline{y}}_s+\widehat{\mathsf{\Gamma }}\left(\mathrm{t}3\right)\left({\overline{X}}_{d\left(s\right)}\right)]\left({\displaystyle \frac{D_{\mathrm{j}}{\overline{x}}_s+H_{\mathrm{j}}}{D_{\mathrm{j}}\overline{X}+H_{\mathrm{j}}}}\right),j=9,\dots ,12$$

(7)

The MSE of these estimators is given by

$$\mathrm{MSE}\left(T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}\right)=\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2+{(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}3\right)})}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2-2(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}3\right)}){\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}\right)$$

(8)

2.4. $T_{{\mathit{tt}}_{\mathsf{\Delta }13}}-T_{{\mathit{tt}}_{\mathsf{\Delta }16}}$ Estimators with [30] Coefficients

Ref. [30] defined a parameterized robustness method for re-descending estimator. The weight function of the estimator is purposely designed to vary the parameters of its robustness and efficiency, m and b, so that some desired balance can be achieved. The OF, IF, and WF are defined as follows:

OF:

$$\rho \left(e_l\right)=\left\{\begin{array}{cc}{\displaystyle \frac{m^2}{2b}}\left[1-{\left(1+{\left({\displaystyle \frac{e_l}{m}}\right)}^2\right)}^{-b}\right],\hfill & |e_l| \ge 0\hfill \end{array}\right.$$

IF:

$$\psi \left(e_l\right)=\left\{\begin{array}{cc}{e}_l{\left[1+{\left({\displaystyle \frac{e_l}{m}}\right)}^2\right]}^{-b-1},\hfill & |e_l| \ge 0\hfill \end{array}\right.$$

WF:

$$w\left(e_l\right)=\left\{\begin{array}{cc}{\left[1+{\left({\displaystyle \frac{e_l}{m}}\right)}^2\right]}^{-b-1},\hfill & |e_l| \ge 0\hfill \end{array}\right.$$

The modified class of mean estimators utilizing the re-descending M-estimator coefficient ${\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}4\right)}$ proposed by Raza et al. [30] is expressed as

$$T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}=[{\overline{y}}_s+\widehat{\mathsf{\Gamma }}\left(\mathrm{t}4\right)\left({\overline{X}}_{d\left(s\right)}\right)]\left({\displaystyle \frac{D_{\mathrm{j}}{\overline{x}}_s+H_{\mathrm{j}}}{D_{\mathrm{j}}\overline{X}+H_{\mathrm{j}}}}\right),j=13,\dots ,16$$

(9)

The MSE of these estimators is formulated as

$$\mathrm{MSE}\left(T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}\right)=\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2+{(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}4\right)})}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2-2(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}4\right)}){\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}\right)$$

(10)

2.5. $T_{{\mathit{tt}}_{\mathsf{\Delta }17}}-T_{{\mathit{tt}}_{\mathsf{\Delta }20}}$ Estimators with [31] Re-Descending

Raza et al. [31] suggested a re-descending-type estimator of a higher-order polynomial, which minimizes the effect of anomalies while keeping the accuracy in data regions away from the tails. The estimator’s OF, IF, and WF are defined as follows:

OF:

$$\rho \left(e_l\right)=\left\{\begin{array}{cc}{\displaystyle \frac{e_l^2}{810b^6}}\left[e_l^8-30b^2{e}_l^4+405b^4\right],\hfill & |e_l| \le b,\hfill \\ {\displaystyle \frac{18b^2}{405}},\hfill & |e_l| > b.\hfill \end{array}\right.$$

IF:

$$\psi \left(e_l\right)=\left\{\begin{array}{cc}{e}_l{\left[1-{\left({\displaystyle \frac{e_l}{3b}}\right)}^4\right]}^2,\hfill & |e_l| \le b,\hfill \\ 0,\hfill & |e_l| > b.\hfill \end{array}\right.$$

WF:

$$w\left(e_l\right)=\left\{\begin{array}{cc}{\left[1-{\left({\displaystyle \frac{e_l}{3b}}\right)}^4\right]}^2,\hfill & |e_l| \le b,\hfill \\ 0,\hfill & |e_l| > b.\hfill \end{array}\right.$$

The modified class of mean estimators using the re-descending M-estimator coefficient ${\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}5\right)}$ proposed by Raza et al. [31] is expressed as

$$T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}=[{\overline{y}}_s+\widehat{\mathsf{\Gamma }}\left(\mathrm{t}5\right)\left({\overline{X}}_{d\left(s\right)}\right)]\left({\displaystyle \frac{D_{\mathrm{j}}{\overline{x}}_s+H_{\mathrm{j}}}{D_{\mathrm{j}}\overline{X}+H_{\mathrm{j}}}}\right),j=17,\dots ,20$$

(11)

The MSE of these estimators is given by

$$\mathrm{MSE}\left(T_{{\mathrm{tt}}_{\mathsf{\Delta }j}}\right)=\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2+{(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}5\right)})}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2-2(K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}5\right)}){\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}\right)$$

(12)

In the estimators $T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$,

$$K_{{\mathrm{tt}}_{\mathsf{\Delta }j}}={\displaystyle \frac{D_{\mathrm{j}}\overline{Y}}{D_{\mathrm{j}}\overline{X}+H_{\mathrm{j}}}},\mathrm{for}j=1,\dots ,20$$

where $D_{\mathrm{j}}$ and $H_{\mathrm{j}}$ take values:

$$(D_{\mathrm{j}},H_{\mathrm{j}})=\left\{\begin{array}{cc}(0,0),\hfill & j=1,5,9,13,17\hfill \\ (1,H_c),\hfill & j=2,6,10,14,18\hfill \\ (1,H_b),\hfill & j=3,7,11,15,19\hfill \\ (H_c,H_b),\hfill & j=4,8,12,16,20\hfill \end{array}\right.$$

(13)

In Equation (13), $H_{\mathrm{c}}$ represents the coefficient of variation, and $H_{\mathrm{b}}$ denotes the kurtosis coefficient of X.

The tuning constants $a,c,h,b$, and m control the aggressiveness of the respective re-descending influence functions and thus directly control the robust-efficiency trade-off. These constants are not treated as free design parameters in this research but are instead chosen based on the original formulations and recommendations in the relevant robust regression references [19,20,29,30,31]. Smaller values of $a,c$, or h tend to mean that more extreme residual values have been rejected/down-weighted (some robustness at the cost of efficiency), whereas larger values imply fewer of them have been thrown out (less robustness at the cost of a large amount of contaminated data). On the same note, m and b can control the smoothness and tail decay of the weight function, and thus the extreme observations can be either well diminished or hard discarded. The approach that is used in practice is to choose these parameters based on robust scale approximations of the residuals and to select them such that the central values have almost equal weights, while extreme values are heavily suppressed.

The importance of robust estimation methods increases in real-life situations when the data is noisy and contaminated [32,33]. It is noteworthy, as highlighted in the Introduction, that even though a number of strong estimation methods have been used to approximate the average parameter in the presence and absence of anomalies, including Huber M-estimators [34], quantile regression [28], and jackknife-based methods [4], these approaches have already been established. While these methods are utilized in simple or stratified random sampling, to the best of our knowledge, no current study has applied re-descending M-estimators to the estimation of means with a systematic sampling design in environments characterized by outlier influence. This use of re-descending M-estimators is the gist and the novelty of our work: the creation of 20 adapted estimators specifically designed to improve the robustness in the systematic sampling case.

3. Proposed Family: ${\mathit{T}}_{{\mathbf{Q}}_{\mathbf{1}}}-{\mathit{T}}_{{\mathbf{Q}}_{\mathbf{5}}}$

The data points that deviate significantly from other data points in the dataset are called outliers. Their presence can have a huge effect on estimating the mean as a way of measuring the center point of the distribution. In most cases, traditional mean estimators such as the population mean ($\overline{Y}$) are approximated through OLS regression coefficients. However, these regression coefficients are greatly influenced by the existence of anomalies, making the approximates (estimates) of the population mean ($\overline{y}$) unreliable. To counteract this problem, a robust alternative is provided by M-estimators for data that are non-normal and contaminated by extreme values. Also, re-descending coefficients are less sensitive to outliers (Raza et al. [30,31]). OLS and re-descending regression, as different techniques for modeling the link between a response variable and a predictor, differ in their optimization criterion. We extend the notion of the adapted class $T_{\mathrm{tt}}$ to a novel class of robust re-descending-coefficient-driven estimators. The developed family of estimators within the systematic random sampling design is

$$T_{{\mathrm{Q}}_i}=\left({\overline{y}}_{\mathrm{s}}+{\widehat{\mathsf{\Gamma }}}_{\left({\mathrm{t}}_i\right)}\left({\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}\right)\right)\mathrm{for}i=1,2,...,5,$$

(14)

where ${\widehat{\mathsf{\Gamma }}}_{\left({\mathrm{t}}_i\right)}$ represents the five re-descending coefficients $({\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}1\right)},{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}2\right)},{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}3\right)},{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}4\right)},{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}5\right)})$ used in this context. The notations used in $T_{{\mathrm{Q}}_i}$ retain their standard interpretation as discussed in previous sections. The MSE corresponding to the developed estimator family is given by

$$\mathrm{MSE}\left(T_{{\mathrm{Q}}_i}\right)=\left(V\left({\overline{y}}_{\mathrm{s}}\right)-2{\mathsf{\Gamma }}_{\left({\mathrm{t}}_i\right)}\mathrm{cov}({\overline{x}}_{\mathrm{s}},{\overline{y}}_{\mathrm{s}})+{\mathsf{\Gamma }}_{\left({\mathrm{t}}_i\right)}^{2}V\left({\overline{x}}_{\mathrm{s}}\right)\right).$$

(15)

Inserting the values of $V\left({\overline{y}}_{\mathrm{s}}\right)$, $V\left({\overline{x}}_{\mathrm{s}}\right)$, and $\mathrm{cov}({\overline{x}}_{\mathrm{s}},{\overline{y}}_{\mathrm{s}})$ from Ref. [11] into Equation (15) results in:

$$\mathrm{MSE}\left(T_{{\mathrm{Q}}_i}\right)=\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2-2{\mathsf{\Gamma }}_{\left({\mathrm{t}}_i\right)}{\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}+{\mathsf{\Gamma }}_{\left({\mathrm{t}}_i\right)}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2\right).$$

(16)

It is essential to highlight that our estimation process incorporates five distinct re-descending coefficients. To enhance robustness, these coefficients are determined iteratively in order to dampen the extreme values. The weight functions, $w\left(e_l\right)$, applied by each estimator, are specifically structured to reduce the effect of large residuals. Initially, the OLS technique is employed for the initial estimate of ${\widehat{\mathsf{\Gamma }}}_{\left({\mathrm{t}}_i\right)}$. Subsequently, an iterative weighted least squares algorithm is applied, where residual weights are updated in each iteration t according to

$${\widehat{\mathsf{\Gamma }}}^{(t+1)}={\displaystyle \frac{\sum w\left(e_l^{\left(t\right)}\right)X_i{Y}_i}{\sum w\left(e_l^{\left(t\right)}\right)X_i^2}}.$$

Repeat the iteration until the convergence criterion $|{\widehat{\mathsf{\Gamma }}}^{(t+1)}-{\widehat{\mathsf{\Gamma }}}^{\left(t\right)}| < ϵ$ is satisfied, with $ϵ$ being defined as a predefined tolerance threshold.

For clarity, the five key members of the developed family and their MSE are presented in a compact form as follows:

$$T_{{\mathrm{Q}}_i}=\left\{\begin{array}{c}\left({\overline{y}}_{\mathrm{s}}+{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}1\right)}\left({\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}\right)\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(1\right),\\ \left({\overline{y}}_{\mathrm{s}}+{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}2\right)}\left({\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}\right)\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(2\right),\\ \left({\overline{y}}_{\mathrm{s}}+{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}3\right)}\left({\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}\right)\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(3\right),\\ \left({\overline{y}}_{\mathrm{s}}+{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}4\right)}\left({\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}\right)\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(4\right),\\ \left({\overline{y}}_{\mathrm{s}}+{\widehat{\mathsf{\Gamma }}}_{\left(\mathrm{t}5\right)}\left({\overline{X}}_{\mathrm{d}\left(\mathrm{s}\right)}\right)\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(5\right).\end{array}\right.$$

(17)

$$\mathrm{MSE}\left(T_{{\mathrm{Q}}_i}\right)=\left\{\begin{array}{c}\left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2-2{\mathsf{\Gamma }}_{\left(\mathrm{t}1\right)}{\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}1\right)}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(1\right),\\ \left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2-2{\mathsf{\Gamma }}_{\left(\mathrm{t}2\right)}{\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}2\right)}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(2\right),\\ \left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2-2{\mathsf{\Gamma }}_{\left(\mathrm{t}3\right)}{\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}3\right)}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(3\right),\\ \left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2-2{\mathsf{\Gamma }}_{\left(\mathrm{t}4\right)}{\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}4\right)}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(4\right),\\ \left({\displaystyle \frac{1-f}{n}}\right)\left({\vartheta }_{\mathrm{y}}{S}_{\mathrm{y}}^2-2{\mathsf{\Gamma }}_{\left(\mathrm{t}5\right)}{\vartheta }_{\mathrm{x}}{t}^{*}{S}_{\mathrm{xy}}+{\mathsf{\Gamma }}_{\left(\mathrm{t}5\right)}^2{\vartheta }_{\mathrm{x}}{S}_{\mathrm{x}}^2\right),\hspace{1em}\mathrm{for} \left(i\right)=\left(5\right).\end{array}\right.$$

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We should also note that when finding the expressions of MSE, the first-order Taylor series approximation was applied. This method is standard in the analysis of robust estimation schemes, as it is simple and can give a sensible analytical estimate of the behavior of estimators in conditions of mild regularity. First-order expansion is a representation of the principal linear characteristic of the estimator at the actual parameter, which is usually sufficient with large samples or in moderate deviations. Although higher-order terms may provide more accurate approximations, especially where there are extreme outliers or heavy tails, they have less effect on bias and variance in practice in systematic sampling problems. In addition, there is no guarantee that adding terms of higher order adds the same amount of complexity to an algebric structure or that this doing so provides an equivalent improvement in accuracy. Therefore, the first-order approximation is a trade-off between features of easy analysis and empirical validity, which is borne out by the fact that the theoretical MSEs have the same behavior as the ones in our simulation studies, as shown in Section 4.

The efficiency conditions of the developed family can be determined by comparing the MSE of $T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$ with $T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$:

$$\mathrm{Efficiency}\mathrm{conditions}=\left\{\begin{array}{c}{\mathsf{\Gamma }}_{\left(\mathrm{t}1\right)}>{\displaystyle \frac{S_{XY}}{S_X^2}}-{\displaystyle \frac{1}{2}}{K}_{{\mathrm{tt}}_{\mathsf{\Delta }j}},\hspace{1em}\mathrm{for} j=1,2,3,4,\hfill \\ {\mathsf{\Gamma }}_{\left(\mathrm{t}2\right)}>{\displaystyle \frac{S_{XY}}{S_X^2}}-{\displaystyle \frac{1}{2}}{K}_{{\mathrm{tt}}_{\mathsf{\Delta }j}},\hspace{1em}\mathrm{for} j=5,6,7,8,\hfill \\ {\mathsf{\Gamma }}_{\left(\mathrm{t}3\right)}>{\displaystyle \frac{S_{XY}}{S_X^2}}-{\displaystyle \frac{1}{2}}{K}_{{\mathrm{tt}}_{\mathsf{\Delta }j}},\hspace{1em}\mathrm{for} j=9,10,11,12,\hfill \\ {\mathsf{\Gamma }}_{\left(\mathrm{t}4\right)}>{\displaystyle \frac{S_{XY}}{S_X^2}}-{\displaystyle \frac{1}{2}}{K}_{{\mathrm{tt}}_{\mathsf{\Delta }j}},\hspace{1em}\mathrm{for} j=13,14,15,16,\hfill \\ {\mathsf{\Gamma }}_{\left(\mathrm{t}5\right)}>{\displaystyle \frac{S_{XY}}{S_X^2}}-{\displaystyle \frac{1}{2}}{K}_{{\mathrm{tt}}_{\mathsf{\Delta }j}},\hspace{1em}\mathrm{for} j=17,18,19,20.\hfill \end{array}\right.$$

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4. Numerical Performance Evaluation

We evaluated our proposed and adaptive estimators in this section using real and synthetic data.

4.1. Applications to Real-World Data

4.1.1. Cars Dataset

The mtcars dataset available in R is a widely recognized real-world dataset originally published in the 1974 Motor Trend US magazine. It was derived from the 1973–74 Road Test dataset by Henderson and Velleman [35]. In this study, we considered miles per gallon (mpg) as the research variable and horsepower (hp) as the supplementary variable, with an intra-class correlation of $r_X\cong r_Y\cong r_W=-0.3192$. This dataset is a good candidate for systematic sampling, as it is structured and can be ordered systematically.

4.1.2. Trees Dataset

We also examined the estimators using the trees dataset, which originates from Murthy [36]. In this case, the supplementary variable is strip length, while the research variable is the timber volume for 176 forest strips. The intra-class correlation was calculated as $r_X\cong r_Y\cong r_W=-0.1510$.

Table 1. Cars and Trees data.

	Cars Data	Trees Data
N	32	176
n	4	4
$\rho$	−0.7761	0.9520
$S_y$	6.026948	606.6869
$S_x$	68.56287	19.58067
$S_{xy}$	−320.7321	11309.95
${\mathsf{\Gamma }}_{\left(\mathrm{t}1\right)}$	0.1059073	31.19974
${\mathsf{\Gamma }}_{\left(\mathrm{t}2\right)}$	0.0965251	31.98072
${\mathsf{\Gamma }}_{\left(\mathrm{t}3\right)}$	0.1072613	31.19974
${\mathsf{\Gamma }}_{\left(\mathrm{t}4\right)}$	0.1052937	31.56664
${\mathsf{\Gamma }}_{\left(\mathrm{t}5\right)}$	0.09973342	31.19974

presents the key characteristics that are observed in the datasets. The visualizations in Figure 1 and Figure 2 show the data distributions, highlighting asymmetry along with outliers emerging from these datasets. Also, it is worth noting that the Trees data in Figure 2 shows leverage-type outliers. The computed MSE and PRE outcomes for these real datasets are provided in

Table 2. MSEs for Cars data.

Adapted
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$
	4.353679	4.341886	4.278283	4.196810
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$
	4.102995	4.091558	4.029883	3.950901
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$
	4.390490	4.378645	4.314765	4.232931
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$
	4.337048	4.325278	4.261802	4.180491
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$
	4.187855	4.176295	4.113962	4.034128
Proposed
$T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$	$T_{{\mathrm{Q}}_1}$	$T_{{\mathrm{Q}}_2}$	$T_{{\mathrm{Q}}_3}$	$T_{{\mathrm{Q}}_4}$	$T_{{\mathrm{Q}}_5}$
	1.45605	1.31742	1.47669	1.446748	1.363962

,

Table 3. PREs for Cars data.

Estimators	${\mathit{T}}_{{\mathbf{Q}}_1}$	${\mathit{T}}_{{\mathbf{Q}}_2}$	${\mathit{T}}_{{\mathbf{Q}}_3}$	${\mathit{T}}_{{\mathbf{Q}}_4}$	${\mathit{T}}_{{\mathbf{Q}}_5}$
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	299.0062	330.4700	294.8269	300.9287	319.1936
$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	298.1962	329.5749	294.0283	300.1135	318.3290
$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	293.8281	324.7471	289.7212	295.7173	313.6659
$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	288.2326	318.5627	284.2039	290.0858	307.6926
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	281.7895	311.4416	277.8509	283.6012	300.8145
$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	281.0040	310.5735	277.0763	282.8107	299.9760
$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	276.7683	305.8920	272.8998	278.5477	295.4543
$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	271.3438	299.8968	267.5512	273.0884	289.6636
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	301.5343	333.2642	297.3197	303.4730	321.8924
$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	300.7209	332.3651	296.5176	302.6543	321.0240
$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	296.3336	327.5162	292.1917	298.2389	316.3406
$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	290.7134	321.3046	286.6500	292.5825	310.3409
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	297.8640	329.2077	293.7007	299.7791	317.9743
$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	297.0557	328.3143	292.9036	298.9656	317.1114
$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	292.6962	323.4960	288.6051	294.5781	312.4576
$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	287.1118	317.3241	283.0988	288.9578	306.4962
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	287.6175	317.8830	283.5975	289.4668	307.0360
$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	286.8237	317.0056	282.8147	288.6678	306.1886
$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	282.5427	312.2741	278.5935	284.3593	301.6185
$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	277.0597	306.2142	273.1872	278.8411	295.7654

,

Table 4. MSEs for Trees data.

Adapted
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$
	85,952.86	60,073.03	6252.913	8895.110
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$
	89,173.06	62,737.57	6738.394	9658.905
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$
	85,952.86	60,073.02	6252.912	8895.109
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$
	87,457.90	61,317.03	6473.204	9246.151
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$
	85,952.86	60,073.03	6252.913	8895.110
Proposed
$T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$	$T_{{\mathrm{Q}}_1}$	$T_{{\mathrm{Q}}_2}$	$T_{{\mathrm{Q}}_3}$	$T_{{\mathrm{Q}}_4}$	$T_{{\mathrm{Q}}_5}$
	4750.646	4918.023	4750.646	4821.494	4750.646

and

Table 5. PREs for Trees data.

Estimators	${\mathit{T}}_{{\mathbf{Q}}_1}$	${\mathit{T}}_{{\mathbf{Q}}_2}$	${\mathit{T}}_{{\mathbf{Q}}_3}$	${\mathit{T}}_{{\mathbf{Q}}_4}$	${\mathit{T}}_{{\mathbf{Q}}_5}$
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	1809.2878	1747.7115	1809.2878	1782.7017	1809.2878
$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	1264.5233	1221.4872	1264.5234	1245.9421	1264.5233
$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	131.6224	127.1428	131.6224	129.6883	131.6224
$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	187.2400	180.8676	187.2400	184.4887	187.2400
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	1877.0723	1813.1891	1877.0724	1849.4902	1877.0723
$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	1320.6114	1275.6664	1320.6114	1301.2060	1320.6114
$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	141.8416	137.0143	141.8416	139.7574	141.8416
$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	203.3177	196.3981	203.3177	200.3301	203.3177
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	1809.2877	1747.7114	1809.2878	1782.7016	1809.2877
$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	1264.5233	1221.4872	1264.5233	1245.9421	1264.5233
$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	131.6224	127.1428	131.6224	129.6883	131.6224
$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	187.2400	180.8676	187.2400	184.4886	187.2400
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	1840.9686	1778.3141	1840.9686	1813.9170	1840.9686
$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	1290.7092	1246.7819	1290.7093	1271.7432	1290.7092
$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	136.2594	131.6221	136.2595	134.2572	136.2594
$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	194.6293	188.0054	194.6293	191.7694	194.6293
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	1809.2878	1747.7115	1809.2878	1782.7017	1809.2878
$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	1264.5233	1221.4872	1264.5233	1245.9421	1264.5233
$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	131.6224	127.1428	131.6224	129.6883	131.6224
$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	187.2400	180.8676	187.2400	184.4887	187.2400

.

4.2. Simulation Study

Within the framework of systematic sampling, a comprehensive simulation experiment was conducted to evaluate the behavior of the developed re-descending estimators. This study consisted of an analytical investigation of a variety of artificially constructed populations $(1,2,3)$, some of which specifically included outliers. In these populations, X was generated from $\mathrm{Uniform}(1,100)$, $\mathrm{Gumbel}(50,10)$, and $\mathrm{Weibull}(2,50)$, respectively. Y was defined using a linear model:

$$Y=2X+ϵ,$$

where $ϵ\sim N(0,10)$. To introduce anomalies into the dataset, random deviations were added to selected observations of Y using $N(100,20)$, as anomalies significantly deviate from the majority of the data. The datasets generated by this method contained standard data points and extreme outliers that reflect the case in the real world, where outliers are caused by measurement error or some rare event.

The simulated dataset consisted of 100 observations, including both regular and outlier points. Figure 3, Figure 4 and Figure 5 display the distribution of the independent variable Y using violin plots generated in R software. The violin density distributions in the plots are shown in gray and red, representing regular data points and data points with outliers, respectively. Each violin at different Y values shows the width, which represents the data density. A wider section indicates a higher concentration, while a narrower section represents a lower concentration in the data. The black dots represent individual observations, clearly distinguishable from the rest in the red violin. The legend on the right differentiates between regular observations and outliers.

It is important to note that we used a convergence tolerance of $ϵ={10}^{-6}$ across all re-descending coefficient calculations to achieve consistency and fairness among all estimators. The simulation-based MSE and PRE outcomes for these experimental populations are reported in

Table 6. MSEs from the first simulated data.

Adapted
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$
	102.76081	101.38606	96.89423	91.13955
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$
	95.28268	93.96780	89.67497	84.18337
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$
	102.81353	101.43836	96.94516	91.18866
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$
	102.72157	101.34712	96.85631	91.10298
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$
	100.91878	99.55853	95.11491	89.42396
Proposed
$T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$	$T_{{\mathrm{Q}}_1}$	$T_{{\mathrm{Q}}_2}$	$T_{{\mathrm{Q}}_3}$	$T_{{\mathrm{Q}}_4}$	$T_{{\mathrm{Q}}_5}$
	15.0517	14.7064	15.05524	15.04908	14.93762

,

Table 7. PREs for the first simulated data.

Estimators	${\mathit{T}}_{{\mathbf{Q}}_1}$	${\mathit{T}}_{{\mathbf{Q}}_2}$	${\mathit{T}}_{{\mathbf{Q}}_3}$	${\mathit{T}}_{{\mathbf{Q}}_4}$	${\mathit{T}}_{{\mathbf{Q}}_5}$
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	682.7188	698.7487	682.5587	682.8377	687.9330
$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	673.5853	689.4008	673.4273	673.7026	678.7298
$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	643.7426	658.8574	643.5916	643.8547	648.6592
$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	605.5098	619.7269	605.3678	605.6153	610.1344
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	633.0358	647.8992	632.8873	633.1460	637.8706
$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	624.3001	638.9583	624.1536	624.4088	629.0681
$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	595.7796	609.7682	595.6398	595.8833	600.3298
$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	559.2946	572.4266	559.1634	559.3920	563.5662
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	683.0690	699.1072	682.9088	683.1880	688.2859
$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	673.9328	689.7564	673.7747	674.0501	679.0799
$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	644.0810	659.2037	643.9299	644.1931	649.0001
$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	605.8361	620.0609	605.6940	605.9416	610.4632
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	682.4581	698.4819	682.2980	682.5769	687.6703
$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	673.3266	689.1360	673.1686	673.4438	678.4691
$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	643.4907	658.5996	643.3397	643.6027	648.4053
$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	605.2669	619.4783	605.1249	605.3723	609.8896
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	670.4808	686.2234	670.3235	670.5975	675.6015
$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	661.4436	676.9740	661.2885	661.5588	666.4953
$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	631.9212	646.7585	631.7730	632.0313	636.7475
$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	594.1119	608.0614	593.9725	594.2153	598.6494

,

Table 8. MSEs from second simulated data.

Adapted
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$
	36.13900	35.99061	32.45623	25.19268
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$
	34.51767	34.37499	30.98315	24.06637
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$
	36.12775	35.97940	32.44598	25.18480
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$
	36.02500	35.87701	32.35247	25.11286
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$
	35.99728	35.84938	32.32724	25.09347
Proposed
$T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$	$T_{{\mathrm{Q}}_1}$	$T_{{\mathrm{Q}}_2}$	$T_{{\mathrm{Q}}_3}$	$T_{{\mathrm{Q}}_4}$	$T_{{\mathrm{Q}}_5}$
	14.84067	14.71397	14.83959	14.82988	14.82729

,

Table 9. PREs for the second simulated data.

Estimators	${\mathit{T}}_{{\mathbf{Q}}_1}$	${\mathit{T}}_{{\mathbf{Q}}_2}$	${\mathit{T}}_{{\mathbf{Q}}_3}$	${\mathit{T}}_{{\mathbf{Q}}_4}$	${\mathit{T}}_{{\mathbf{Q}}_5}$
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	243.5133	245.6101	243.5310	243.6905	243.7329
$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	242.5134	244.6016	242.5310	242.6899	242.7322
$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	218.6979	220.5810	218.7138	218.8570	218.8951
$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	169.7543	171.2160	169.7667	169.8779	169.9075
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	232.5883	234.5910	232.6052	232.7576	232.7981
$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	231.6270	233.6214	231.6438	231.7955	231.8359
$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	208.7719	210.5696	208.7871	208.9239	208.9602
$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	162.1650	163.5613	162.1768	162.2830	162.3113
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	243.4374	245.5336	243.4551	243.6146	243.6570
$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	242.4378	244.5254	242.4554	242.6143	242.6565
$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	218.6288	220.5114	218.6447	218.7879	218.8261
$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	169.7012	171.1624	169.7135	169.8247	169.8543
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	242.7451	244.8353	242.7628	242.9218	242.9641
$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	241.7479	243.8295	241.7655	241.9238	241.9660
$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	217.9987	219.8758	218.0145	218.1573	218.1953
$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	169.2165	170.6736	169.2288	169.3397	169.3692
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	242.5583	244.6469	242.5759	242.7348	242.7771
$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	241.5618	243.6417	241.5793	241.7375	241.7796
$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	217.8287	219.7043	217.8445	217.9872	218.0252
$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	169.0858	170.5417	169.0981	169.2089	169.2383

,

Table 10. MSEs from third simulated data.

Adapted
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$
	65.00710	63.70964	57.46972	52.12881
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$
	62.11380	60.85445	54.80528	49.63902
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$
	65.33604	64.03432	57.77307	52.41262
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$
	63.46263	62.18537	56.04658	50.79826
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}-T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$
	65.34649	64.04464	57.78271	52.42165
Proposed
$T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$	$T_{{\mathrm{Q}}_1}$	$T_{{\mathrm{Q}}_2}$	$T_{{\mathrm{Q}}_3}$	$T_{{\mathrm{Q}}_4}$	$T_{{\mathrm{Q}}_5}$
	14.70678	14.70564	14.71223	14.6955	14.71242

and

Table 11. PREs for the third simulated data.

Estimators	${\mathit{T}}_{{\mathbf{Q}}_1}$	${\mathit{T}}_{{\mathbf{Q}}_2}$	${\mathit{T}}_{{\mathbf{Q}}_3}$	${\mathit{T}}_{{\mathbf{Q}}_4}$	${\mathit{T}}_{{\mathbf{Q}}_5}$
$T_{{\mathrm{tt}}_{\mathsf{\Delta }1}}$	442.0214	442.0557	441.8575	442.3605	441.8517
$T_{{\mathrm{tt}}_{\mathsf{\Delta }2}}$	433.1992	433.2328	433.0386	433.5316	433.0330
$T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$	390.7703	390.8007	390.6255	391.0702	390.6204
$T_{{\mathrm{tt}}_{\mathsf{\Delta }4}}$	354.4544	354.4819	354.3229	354.7263	354.3184
$T_{{\mathrm{tt}}_{\mathsf{\Delta }5}}$	422.3481	422.3809	422.1915	422.6722	422.1861
$T_{{\mathrm{tt}}_{\mathsf{\Delta }6}}$	413.7851	413.8172	413.6317	414.1026	413.6263
$T_{{\mathrm{tt}}_{\mathsf{\Delta }7}}$	372.6532	372.6822	372.5151	372.9392	372.5102
$T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$	337.5248	337.5510	337.3997	337.7838	337.3953
$T_{{\mathrm{tt}}_{\mathsf{\Delta }9}}$	444.2580	444.2925	444.0933	444.5989	444.0875
$T_{{\mathrm{tt}}_{\mathsf{\Delta }10}}$	435.4069	435.4407	435.2454	435.7409	435.2398
$T_{{\mathrm{tt}}_{\mathsf{\Delta }11}}$	392.8330	392.8634	392.6873	393.1344	392.6822
$T_{{\mathrm{tt}}_{\mathsf{\Delta }12}}$	356.3841	356.4118	356.2520	356.6576	356.2474
$T_{{\mathrm{tt}}_{\mathsf{\Delta }13}}$	431.5196	431.5531	431.3596	431.8507	431.3541
$T_{{\mathrm{tt}}_{\mathsf{\Delta }14}}$	422.8348	422.8676	422.6780	423.1593	422.6726
$T_{{\mathrm{tt}}_{\mathsf{\Delta }15}}$	381.0936	381.1231	380.9523	381.3860	380.9473
$T_{{\mathrm{tt}}_{\mathsf{\Delta }16}}$	345.4071	345.4339	345.2791	345.6721	345.2746
$T_{{\mathrm{tt}}_{\mathsf{\Delta }17}}$	444.3291	444.3636	444.1644	444.6700	444.1586
$T_{{\mathrm{tt}}_{\mathsf{\Delta }18}}$	435.4770	435.5108	435.3156	435.8112	435.3099
$T_{{\mathrm{tt}}_{\mathsf{\Delta }19}}$	392.8985	392.9290	392.7529	393.2000	392.7478
$T_{{\mathrm{tt}}_{\mathsf{\Delta }20}}$	356.4455	356.4732	356.3133	356.7190	356.3087

. For more details about the calculation of table results, refer to [37].

4.3. Results Discussion

4.3.1. Cars Data

Table 2 results highlight the superior effectiveness of ( $T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$ ) estimators relative to the adapted ones. The MSE values reveal that the developed estimators exhibit significantly reduced error margins relative to the best-performing adapted ones, which is $T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$ with an MSE of 3.950901. The MSE of the best-performing estimator, $T_{{\mathrm{Q}}_2}$, is 1.31742, which is about three times smaller. This reduction in error indicates that the developed estimators are better in dealing with the variability in the Cars dataset, especially for the outliers. Additionally, in Table 3, the PRE values further support this presumption. $T_{{\mathrm{Q}}_2}$ yields a PRE of 333.2642, greatly outperforming the traditional adapted estimators. The consistency of the proposed estimators’ PRE values demonstrates their robustness and adaptability to systematic sampling scenarios across all proposed estimators.

4.3.2. Trees Data

Table 4 results demonstrate the superior efficiency of ($T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$) estimators in comparison with the adapted estimators for the Trees dataset. The MSE values demonstrate a significant reduction in estimation error compared to the best-performing adapted estimator, $T_{{\mathrm{tt}}_{\mathsf{\Delta }3}}$, which has an MSE of 6252.913. In contrast, $T_{{\mathrm{Q}}_1}$ performs best, with an MSE of 4750.646. This reduction indicates the robustness of the developed estimators for systematic sampling within extreme data scenarios. Additionally, the PRE values in Table 5 also support this finding, as the proposed estimators exhibit much higher levels of efficiency due to outlier severity, with $T_{{\mathrm{Q}}_1}$ having the highest PRE of 1809.2878.

4.3.3. First Simulated Data

Table 6 presents the results that show a considerable gain in estimation accuracy and efficiency using the proposed re-descending M-estimators over the adapted estimators. The developed MSE values are substantially reduced, and the best-performing proposed estimator, $T_{{\mathrm{Q}}_2}$, has an MSE of 14.7064, which is lower than the best adapted estimator, $T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$, with an MSE of 84.18337. This shows that the developed estimators are more robust and accurate in the case of simulated data contaminated with outliers. In Table 7, the PRE values further confirm that the proposed estimators work well, with the $T_{{\mathrm{Q}}_2}$ estimator obtaining the highest PRE of 699.1072, which exceeds the ones obtained for the adapted estimators by a significant margin.

4.3.4. Second Simulated Data

Table 8 shows that the developed re-descending M-estimators perform more accurately than the adapted ones in the second simulated dataset. An appreciable reduction in the value of the MSE is evidenced by the lower values of the MSE, with the best adapted estimator, $T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$, having an MSE value of 24.06637. However, $T_{{\mathrm{Q}}_2}$ performs the best, with an MSE of 14.71397. A substantial decrease in error compared to a systematic sample with a skewed or heavy-tailed distribution confirms that the proposed estimators are robust for systematic sampling. Additionally, as shown in Table 9, $T_{{\mathrm{Q}}_2}$ has the highest PRE of 245.6101, which is significantly higher than the corresponding adapted estimators.

4.3.5. Third Simulated Data

Table 10 shows the strong performance of ($T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$) estimators compared to adapted estimators in the third simulated dataset. The estimation error is significantly decreased, as the best -performing proposed estimator, $T_{{\mathrm{Q}}_4}$, has an MSE of only 14.6955, which has achieved lower MSEs compared to the best adapted estimator, which is $T_{{\mathrm{tt}}_{\mathsf{\Delta }8}}$, with an MSE of 49.63902. The existence of this substantial reduction in error shows the robustness of the proposed estimators, particularly when handling systematic sampling, where the distributions are skewed or have heavy tails. Additionally, these PRE values from Table 11 coincide with the findings, as the proposed estimators continue to have consistently higher efficiency, with the $T_{{\mathrm{Q}}_4}$ showing the highest PRE of 444.6700.

Based on the correlation of all the real and simulated populations. It is recommended to use proposed estimators for moderate and high correlation in the presence of outliers.

Even though some of the suggested estimators may seem algebraically identical, they vary in the selection of tuning parameters, which dictate the shape and vigor of the re-descending weight functions. The parameters are important to the trade-off between robustness and efficiency, in which a large parameter will tend to smooth outliers, and a small parameter will tend to smooth attenuation. To estimate the sensitivity of our estimators to these parameter choices, we utilized performance measures like MSE and PRE, which are sensitive to the choice of parameters. However, the relative performance of our estimators did not change much over a wide range of values. These results indicate that the estimators are relatively consistent to moderate variations of the tuning constants.

This significant efficiency gain confirms that the developed estimators perform much more efficiently than traditional adapted ones in terms of accuracy and robustness. Thus, they seem to be appropriate for use in the real world when data contain outliers or exhibit asymmetric distributions.

5. Limitations of the Study

One of the significant limitations of the proposed estimators is their particular design in terms of datasets that contain outliers; they are incredibly effective when used in such cases, but their benefits can decrease when using clean and normally distributed data. Based on this, they are mostly advised to be used in situations where outliers are probable or known to exist.

6. Conclusions

Overall, this study evaluated comprehensive, robust mean estimators, such as re-descending M-estimators, in systematic sampling. The disadvantage of traditional mean estimators is that they remain very susceptible to outliers, resulting in a large bias in the estimates and low efficiency of the estimators. To overcome this problem, we developed estimators ($T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$) that limit the impact of extreme values on the estimated parameters while maintaining statistical efficiency under systematic sampling with bias. For the analysis, real datasets (mtcars and trees) and simulated datasets were used, and empirically, the performance of these estimators was consistently more favorable in terms of MSE compared with conventional adapted estimators. PRE values were also calculated to confirm the accuracy of the developed ($T_{{\mathrm{Q}}_1}-T_{{\mathrm{Q}}_5}$) estimators across different asymmetric datasets. These estimators offer increased accuracy and are the recommended choice as a reliable alternative to standard robust estimation techniques. For future research and in the study of high-dimensional data, these estimators can be extended to rare-event and cluster sampling [38].