Sharper Concentration Inequalities for Median-of-Mean Processes

Round 1

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

The article is devoted to the topic, which is important both from a theoretical point of view and in view of possible applications of the results in big data processing processes.

The work is well written. The results are justified.

The comments concern the quality of the presentation of the results.

1. It is not clear why the authors start the numbering with Theorem 2, and then Theorem 9, etc. This confuses the reader. The same applies to the numbering of definitions, lemmas.... I suggest that the authors clearly number the statements, namely Theorem 1, Theorem 2, ... Definition 1, Definition 2 ..., Remark 1, Remark 2,...

2. It is not clear why the proofs are presented beyond the main text of the article, in the Appendix A. I think they should be included after the relevant statements in the main text of the article.

3. What is the point of Chapter 4, the size of which is very small. And the proof of Theorem 11 (which will have a different numbering) will make it larger and this will improve the presentation of the authors' results.

4. Authors must check the correctness of the list of references. 5. In the Сonclusions, the authors should more clearly note the novelty of the obtained results in comparison with previous works. And in general, the Conclusions need to be expanded.

Acceptable

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Reviewer 3 Report

This study proposes an interesting variance-dependent MoM estimation method using the tail probability of Binomial distribution. Compared to the McDiarmid and classical Hoeffding methods, this method performs better under mild conditions.

In reviewing this study, I find its methodology robust and its findings compelling. However, I have a small suggestion that may enhance the understanding of a specific point. In lines 128-129 of section 4, the statement reads: "Sometimes we can't find each outlier directly, but we can get a rough idea of the total number of outliers." This concept might be further illuminated by the inclusion of a tangible real-world example. Could the authors possibly elucidate a scenario in which practitioners, though unable to pinpoint each individual outlier, can nevertheless discern a general sense of the total number of outliers? Such an addition would likely provide greater clarity and enrich the reader's comprehension of the subject.

I found a grammatical error in the sentence in lines 47-48. You might want to revise it to "To get a clear picture of robust estimation via a non-asymptotic viewpoint, variance-dependent MoM methods based on Binomial tail probability are mainly studied, including uncontaminated and contaminated cases. The paper proceeds as follows."

Author Response

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Reviewer 4 Report

The paper addressed problems around concentration inequalities, which should attract the audience. In my impression, the authors have a good ability to tackle technical problems. My primary concern is the correctness of the proof of Theorem 2.

(A1). Chebyshev's inequality bounds <= sigma^2\epsilon^2, but the authors said <= sigma^2\(epsilon^2 B^2), which is sharper than Chebyshev's inequality. 

Page 9, the first line. Why B \in N can be equated with [2+delta]sigma^2/epsilon^2?

In addition, the descriptions are a bit rough. The points I found are as follows.

Page 3 the last line. 200 should be 800. 

Page 5 line 13 from the bottom. Is |\psi(x)| \in B_L correct?

Page 5 line 11 from the bottom. Please define P.

Page 7 line 10. Please define |\hat{X}|_{b,G}. Is P \bf{P}?

Author Response

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Round 2

Reviewer 1 Report

New version of the paper contains new serious mathematical mistakes. It seems that mathematical base of authors have significant gaps.  

Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

The list of literature and data is not issued in accordance with the rules of the MDPS. But he changes made have improved the overall impression of the article. I offer to accept the paper.

Sufficient

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Reviewer 4 Report

Thank you for clarifying the issue which I commented.

Author Response

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