Estimating Common Mean in Heteroscedastic Variances Model

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article studies the Bayesian estimator with non-informative prior distribution, and studies the approximate expression of its variance and the limit state, which has certain theoretical significance.

The abstract should further explain the background and motivation of the study.

The motivation and necessity (especially the necessity) of the research problem should be fully explained in the introduction, and corresponding arguments should be provided.

The structure of the article is very rough, with only three chapters. It should be further refined, otherwise it will be difficult to meet the standards of an academic paper.

The article lacks the necessary content of an academic paper, such as conclusions and acknowledgments.

The number of references is too small.

There are very few references in the past five years, only 2/20. There are also very few references in the past ten years, only 3/20. 60% (12/20) of the references are before 2000. The current references can only make readers think that the research question of this article is not a hot and meaningful topic in the relevant field. It is even more impossible for readers to clearly feel the necessity of the research question of this article, or even question the necessity of the research question.

In line 76, how do you understand the formula $x_j\sim \sigma^{-1}_j p((x_j-\mu)/\sigma_j)$? Is $x_j$ a sample (that is, is $x_j$ a random variable)? If so, why does $x_j$ follow a distribution with parameter “$x_j$”?

Lines 83-84 state that omega is a function of x_1,\cdots,x_n, but line 79 gives an expression for omega, which is in conflict with the expression given in line 79.

In formula 1, if there exists an indicator j_0 such that x_{j_0}=x_i, what should we consider at this time? The same is true for formula 2.

The formulas in lines 113-114 should be placed in a formula environment.

Is there a requirement for sigma_i to be non-zero? If not, will sigma_i be equal to 0? If so, how should we discuss or consider this?

The research question in this article is too theoretical and old-fashioned, and is more suitable for a pure mathematics journal.

 

 

Author Response

 

 

 

To the referees of
Estimating Common Mean in a Heteroscedastic Variances Model
by Andrew L. Rukhin submitted to Mathematics

Many thanks for your comments which I tried to address in the revision.

The abstract and  the introduction are expanded to emphasize importance
of this work for repeatability challenge. The modern application to the
citizen science projects is stressed.

Two subsections and the conclusion section are added.

Some modern references are added.

Confusing notation (former line 76) is  modified.

Clarification about the estimated weights is added.

Distinct nature of x's is stressed now.

The formula for the parities (line 113-14?) is displayed now.

All sigma's are assumed to be positive.

Have tried  to emphasize the modern nature of this work whch acknowledges
heterogeneous uncertainties and their influence on statistical inference.
The stated Aims&Scope of "Mathematics" includes theoretical papers in
" all areas of pure and applied mathematics'' in  particular in
mathematical statistics. I hope that this theoretical work will stimulate
the use of its results in applications.
~                                                                           
~                                

 

 

Reviewer 2 Report

Comments and Suggestions for Authors

Review Report on Estimating Common Mean in a Heteroscedastic Variance Model

The paper presents a Bayesian approach to estimating the common mean in a heteroscedastic variance model. The authors derive Bayes estimators using a non-informative prior and explore their limiting behavior. The study employs orthogonal polynomials to obtain recursive formulas and discusses applications in meta-analysis and measurement uncertainty.

Major Methodological Issues:

1. Misinterpretation of Heteroscedastic Variance Models

• The paper assumes that variances are unrestricted but later introduces Bayesian estimators with reference priors.
• If variances are truly unrestricted, estimating the common mean using Bayesian methods may lead to ill-defined posterior distributions.

Recommendation:

• Justify why Bayesian estimation is appropriate under extreme heteroscedasticity.
• Verify whether the posterior distribution is proper.

2. Problems with the Non-Informative Prior Choice

• The chosen prior: π(μ,σ1,...,σn)=[∏jσj−a]σ−b may result in an improper posterior distribution if a or b are not properly chosen.

Recommendation:

• Provide conditions under which the posterior is normalizable.
• Explore alternative priors to ensure proper Bayesian inference.

3. Unjustified Use of Orthogonal Polynomials

• The paper uses orthogonal polynomials to derive recursive formulas but does not justify why they are necessary.
• Their role in Bayesian estimation remains unclear.

Recommendation:

• Explain whether these polynomials improve computational efficiency or numerical stability.
• If unnecessary, consider removing them from the derivations.

4. Incorrect Interpretation of Pitman Estimator

• The paper claims that when s² = 0, the Bayes estimator reduces to a Pitman estimator.
• The Pitman estimator applies to location families, but it is unclear if it applies to this setting.

Recommendation:

• Define minimax properties clearly.
• If the term “Pitman estimator” is used loosely, revise its description.

5. Misrepresentation of Maximum Likelihood Estimation (MLE)

• The manuscript claims that the likelihood function "reaches infinity at each data point," making MLE undefined.
• However, MLE exists but may be inefficient in heteroscedastic settings.

Recommendation:

• Compare Bayesian vs. MLE performance.
• If MLE is inefficient, provide empirical evidence supporting Bayesian superiority.

6. Missing Empirical Validation

• No real-world data or simulations are included.
• The proposed estimator’s practical performance is unknown.

Recommendation:

• Include Monte Carlo simulations with different variance structures.
• Apply the method to a real dataset (e.g., meta-analysis or measurement uncertainty data).

7. Sensitivity to Outliers

• The estimator relies on a weighted sum, making it sensitive to extreme values.
• There is no discussion of robustness.

Recommendation:

• Analyze breakdown points to test estimator stability.
• Compare with robust estimators such as Huber or Tukey M-estimators.

 

Editorial and Formatting Issues:

1. Issues in the Title

• "Estimating Common Mean in a1 Heteroscedastic Variances Model2" has formatting errors.
• Suggested correction: "Estimating the Common Mean in a Heteroscedastic Variance Model".

2. Abstract Issues

• Typos and grammatical errors:
• "The setting with a possible group of homeogeneous observations..." → "The setting considers a possible group of homogeneous observations..."
• "Under normality condition..." → "Under the normality condition..."

3. Inconsistent Notation and Formatting

• Subscripts and summation symbols are inconsistently formatted.
• Ensure proper use of σ²ᵢ instead of σ2 i throughout.

4. Citation Errors

• "Spiegelhalter, 2025" refers to a future publication, which may be incorrect.
• "Box and Tiao, 1992" should be "Box and Tiao, 1973".
• "Rukhin, 2017" is cited, but "Rukhin, 2023" is referenced elsewhere inconsistently.

Recommendation:

• Verify and correct all citations.
• Ensure that reference years match the in-text citations.

Overall Rating: Major Revisions Required

Comments on the Quality of English Language

The quality of English language in the manuscript requires major revision due to frequent grammatical errors, awkward sentence constructions, and inconsistent terminology. Several sentences are overly complex or unclear, making it difficult to follow the logical flow of ideas. For example, the phrase "The setting with a possible group of homeogeneous observations which have the same unknown variance is considered." can be rewritten as "We consider a setting where a group of homogeneous observations shares the same unknown variance." Additionally, improper word choices, inconsistent notation (e.g., σ2i\sigma2_i instead of σi2\sigma^2_i), and lack of transition words disrupt readability. The abstract and conclusion also suffer from unclear phrasing and insufficient summarization of key findings. Long paragraphs should be broken down for better readability, and technical terms should be used consistently. To improve the manuscript, a thorough proofreading by a native English speaker or professional editor is highly recommended. Additionally, using writing tools like Grammarly or Hemingway Editor can help refine clarity and readability.

Author Response

Please see  my response above

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The current version of the article has met the publication requirements and can be published after the article is adjusted to the format of academic papers.

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

The article still needs revision.

 Major Issue

Typographical and Grammatical Errors

Numerous typographical errors, formatting inconsistencies, and grammatical mistakes are present throughout the manuscript. Examples include:

• "homeogeneous" should be "homogeneous"
• "varianceis" should be "variance is"
• "neccessary" should be "necessary"
• "repeatabiity" should be "repeatability"
• "sigma−2" notation is inconsistent in different sections.
• Repeated words such as "the the" (Line 6), "in in" (Line 79).
•  
• Clarity and Readability Issues

• Some explanations are overly complex and lack intuitive descriptions. The paper would benefit from a clearer exposition of key ideas before diving into technical details.
• Inconsistent spacing, indentation, and text alignment make it difficult to follow the mathematical derivations.
• Some sections contain misplaced colons and unnecessary line breaks, possibly due to incorrect PDF conversion.
• Mathematical and Notational Errors

• Inconsistent use of subscripts and superscripts.
• Incorrectly formatted probability expressions.
• Incomplete or unclear mathematical explanations in derivations of Bayes estimators.
• The notation for variance calculations needs standardization.

1. Logical and Conceptual Issues

• Certain arguments lack sufficient explanation and transition smoothly between concepts.
• Some sections repeat content unnecessarily.
• Citations and references should be formatted consistently.
• The role of the proposed estimators in practical applications could be elaborated further.

 

Comments on the Quality of English Language

revision needed

Author Response

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