Estimation of Gumbel Distribution Based on Ordered Maximum Ranked Set Sampling with Unequal Samples

Round 1

Reviewer 1 Report

The submitted manuscript investigates the estimation of parameters of the Gumbel distribution with ordered maximum ranked set sampling. I think that the manuscript is clearly written, but some aspects could be improved.
Detailed comments:
1.    There are obvious errors in the use of the English language. The manuscript should undergo careful proofreading.
2.    It would be preferable if the authors would have included line numbers to ease the review process (see the MDPI LaTeX template). Moreover, there are not even page numbers. This is quite annoying in the review.
3.    Shouldn’t it read “Gumbel” instead of “Gumble” throughout the manuscript?
4.    Equations: Please write “exp” instead of “e” and “log” instead of “ln” because this is more usual in statistical literature.
5.    P. 4: Write “differentiable”.
6.    Sect. 4: Motivate the sample sizes in the simulation study. Do they appear in applied research?
7.    Sect. 4: Give a reference to the R software.
8.    Table 4: The information in the table should be included in the text, and the table can be safely removed.
9.    Do not use vertical lines in tables.

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Author Response

Thank you for careful reading and giving useful comments. The responses for each and every comments are gven below:

1. Thank you for your valuable suggestion. We have gone through the manuscript and  tried to reduce the error in the revised manuscript.
2. Thank you for this suggestion.  According to your suggestion, we have modified the latex file into the given format of this journal. 
3. Thank you for this sgestion. In the revised manuscript, it has been canged from Gumble to Gumbel. 
4. Thank you for this useful suggestion. In the revised manuscript  we have modified “exp” instead of “e” and “log” instead of “ln", according to your suggestion.
5. Thank you for this comment. We have modified  “differentiable” according to your suggestion in the revised manuscript. 
6. Thank you for your valuable comment. In todays world we need to reduce the time and cost in the experiment to get a decission from that. For this reason we have done the simulation study taking small sampe sizes. One may do the same for large sample size also. 
7.  Thank you for this valuable suggestion. R package reference has been incorporated in the revised manuscript.
8.  Thank you for this comment. We have modified the tables in the revised manuscript. 
9. Thank you for this comment. We have modified the tables in the revised manuscript according to your suggestion. 

Reviewer 2 Report

Please, read the attached file review.pdf

Comments for author File: Comments.pdf

The English language needs little editing. I explain this in my report.

Author Response

Thank you for your careful reading and valuable comments. The point-by-point responses to your comments are given below. 

1. According to your suggestion, some distributional properties have been incorporated in the revised manuscript. (See page 2)
2. Thank you for this valuable comment. In this article we fixed our focus on the parameter estimation under ordered minimum ranked set sampling with unequal sample sizes. That is why we did not try to show such properties. 
3. Thank you for this careful reading and suggestion. According to your suggestion, the typing errors have been rectified in the revised manuscript. 
4. Thank you for this valueable comments. The performance of the estimates of the parameters has been compared based on absolute bias, relative efficiency, and mean squared error. These statistical properties has been elaborated in the revised manuscript. (See page 8)
5. Thank you for this comment. To obtain the estimates, we have used Optim package to obtain the parameter estimates.  It ahs been updated in the revised manuscript. 
6. Thank you for this comment. According to your suggestion, we have mentioned all the pacakge name s in the revised manuscript. 
7. Thank you for careful reading and valuable comments. We have updated all the typo errors in the revised manuscript. 

Reviewer 3 Report

The work is solid and interesting. The paper investigates the estimation of unknown parameters of the Gumbel distribution using the method of moments (MOM) across various sample selection techniques: Simple Random Sample (SRS), Ranked Set Sampling (RSS), Maximum Ranked Set Sampling (MRSS), and Ordered Maximum Ranked Set Sampling (OMRSS) due to small sample sizes. It emphasizes the efficiency improvements in estimator accuracy through these methods compared to traditional SRS, especially for small sample sizes. A Monte Carlo simulation study demonstrates the effectiveness of these techniques, and the analysis of two datasets validates the adaptability of the Gumbel distribution estimation based on these sampling techniques.

The study aims to enhance the accuracy and efficiency of estimating the Gumbel distribution parameters by comparing different sampling methods, particularly focusing on small sample sizes.The paper employs the method of moments for parameter estimation, incorporating four sampling techniques to address the challenges posed by small sample sizes. A Monte Carlo simulation study underpins the comparative analysis, supplemented by the application of these methods to real datasets.

Here findings of the work:

Sample Selection Impact: The study illustrates that OMRSS and MRSS significantly improve the accuracy of parameter estimates over traditional SRS and RSS, especially in scenarios of small sample sizes.
 Simulation Study: The Monte Carlo simulation demonstrates that OMRSS-based estimators yield the most accurate estimates with minimal biases, followed closely by MRSS, across varying sample sizes and parameter values.
 Real Data Application: Application to real datasets corroborates the simulation study's findings, showcasing the practical utility of OMRSS and MRSS in improving estimation accuracy.

Strengths:

    Innovative Sampling Techniques: The introduction of OMRSS with unequal samples as a novel approach for parameter estimation in small sample contexts is a significant contribution.
    Comprehensive Analysis: The combination of theoretical exposition, simulation studies, and real data application provides a robust validation of the proposed methods.
    Clear Presentation: The methodology and results are presented clearly, with detailed explanations of the mathematical underpinnings and statistical procedures.

Areas for Improvement:

    Methodological Rigor: While the simulation study is comprehensive, further analysis on the sensitivity of the estimators to assumptions about the underlying distribution could strengthen the findings.
    Comparison with Other Methods: Expanding the comparative analysis to include other estimation techniques, such as Maximum Likelihood Estimation (MLE) or Bayesian methods, could provide a broader context for the efficacy of the proposed methods.
    Practical Implications: Discussing the implications of these findings in specific applied contexts (e.g., hydrology, meteorology) where the Gumbel distribution is commonly used could enhance the paper's relevance to practitioners.

This study makes a valuable contribution to the field of statistical estimation for the Gumbel distribution, especially in the context of small sample sizes. The innovative use of OMRSS and MRSS offers a promising avenue for research and application in statistical sampling and estimation. Further exploration and validation in broader contexts would be beneficial for solidifying the practical utility and theoretical foundations of these methods.

The English can be improved considering a revision of the text.

Author Response

Thank you for your careful reading and valuable suggestions. We will try to give point-by-point response to your comments below. 

1. Thank you for this valuable comment. The aim of sensitivity analysis is to assess whether results obtained by applying a given estimation method are sufficiently reliable or not. In this article, we have assesed the perofrmance based on absolute bias, mean squared error and relative efficiency.
2. Thank you for this valuable comment. In most of the article in statistical literature one may find MLE and Bayes. In this article we are focusing on method of moment estimator based on different ranked set sampling. 
3. Thank you for your valuable comment. According to your suggestion, we have added the implications of hydrology and meterelogy in the revised manuscript.  

Reviewer 4 Report

I enjoyed reviewing this paper and I do not have any specific comments.

Author Response

Thank you for your careful reading. 

Round 2

Reviewer 1 Report

no further comments

Author Response

Thank you for your kind comment. 

Reviewer 2 Report

The revision made improves the paper. However, additional revision is necessary. Written in that way, the paper is not motivated at all. I suggest once again  the authors to discuss their model in relation to other  exponential-style distributions -- for example different Kies families, their Hausdorff saturations, and applicability to real-world data.

Also, there is a problem with formulas referring as well as the literature sources referring. Thus, the manuscript is difficult to read and check.

Author Response

Thank you for your careful reading and valuable suggestions. According to your suggestion, some new refernces have been added regarding the use of Gumbel distribution in statistical literature. As this article is focusing on the estimation based on different ranked set sampling, we did not add the Hausdroff properties and all. We are taking that suggestion positively to compare the Gumbel distribution with some other exponantial family distribution along with hausdroff properties for future direction of our research.  

Round 3

Reviewer 2 Report

The revision made is satisfactory.