Improving the Robustness of the Theil-Sen Estimator Using a Simple Heuristic-Based Modification

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article is well-written and the literature review adequate. The motivation is also clear and the results from the related numerical analysis show the effectiveness on the proposed approach to have a robust Theil-Sen estimator in the case of simple linear regression problem.

The proposed method can be useful in practice but I would suggest the author to consider adding a numerical example on the use of this method. This will help those interested in using the proposed RTS estimator to better understand how to do it.

Finally, I have some (very) minor comment that the author might want to consider and the revise these parts in the manuscript:

1.  Isn't s a parameter of the procedure? Doesn't it necessary for the user to provide it? I didn't see something in the Exp-A and Exp-B, please clarify this point.

2. I am a bit confused with the OLSc estimator. It is the OLS estimator calculated only for the correct data points in data set. In practice, it is not possible to whether a data point in the dataset is a correct one or not (and thus, it is an outlier), right? So, probably its use is only for comparative purposes for the presented experiments. This is also holds for the number of outliers in the dataset. Users have to investigate first which of these values are outliers, right? And probably, the correct (or not) decisions will affect the estimator. Please comment and clarify, if necessary.

3. Page 11 of 15: "... and the probably that the breakdown point value for the RTS estimator is equal to the breakdown point value of the RM estimator". Did you prove this? It is not clear if this is concluded by your results it is property that the RTS estimator might have it (but more research is needed on this). Please clarify.

4. Line 155, RTS instead of rTS 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper proposes a novel modification to the Theil-Sen estimator aimed at enhancing its robustness. The authors introduce a heuristic-based approach that mitigates the influence of outliers and extreme values, thereby improving the accuracy and reliability of the estimator. The methodology is well-articulated, and the empirical results demonstrate significant improvements in robustness without compromising computational efficiency.

 

I recommend the acceptance of this paper with minor revisions. The proposed heuristic-based modification to the Theil-Sen estimator is a valuable contribution to the field, offering enhanced robustness and maintaining simplicity and computational efficiency. Addressing the minor corrections and suggestions outlined above will further strengthen the paper.

 

I.              The Theil-Sen estimator is a robust, non-parametric method for estimating the slope of a trend line, particularly useful when dealing with datasets containing outliers. While it is computationally intensive for large datasets, its robustness makes it a valuable tool in many practical applications. Other related estimators offer varying balances between robustness and computational efficiency, providing options for different types of data and outlier distributions. I think you need to motivate the Theil-Sen estimator by showing its importance in real life phenomena.

II.           The compromise between OLS and “Least Absolute Deviations (LAD) Estimator”, minimizing a combination of squared residuals for small errors and absolute residuals for large errors. It will be great if you can add some results related to the LAD Estimator” and/or Rousseeuw's Least Median of Squares (LMS) Estimator and provide some comparative notations.

III.        It is noted that the paper still needs to some real data applications. I am wondering if you can provide an application to identify trends in climatic data, such as temperature and precipitation over time.

IV.       Please add some potential future points for future works.

V.           The paper did not provide any appropriate details about the software used, the packages applied, the algorithms, etc. I believe that the author can avoid this inability to help others resume and develop their work.

VI.       The English is good, however the paper still need to be revised.

Comments on the Quality of English Language

The English is good, however the paper still need to be revised.

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf