Comparative Analysis of the Existence and Uniqueness Conditions of Parameter Estimation in Paired Comparison Models
Round 1
Reviewer 1 Report
In this paper, the authors address an interesting problem in statistics about the existence and uniqueness of MLEs in some certain circumstances. I find this work interesting and enjoy reading it. Moreover, the results that the authors obtained are believable. So, it is definitely worth publishing. However, a few aspects of the paper can be improved via a revision. See my comments in the enclosed PDF file for specific suggestions.
Comments for author File: 
Comments.pdf
Author Response
Thank you for your work.
Our responses to your comments are included in the attached file.
Author Response File: Author Response.pdf
Reviewer 2 Report
The paper is interesting, well written and consistent with the journal. The main defect of the work is that a section dedicated to real data application is completely missing. In my opinion it is necessary to make the paper suitable for publication.
The first part of the introduction, "Comparisons in pairs...can garantee more definite results." is not clear and it should be explained better.
I don't agree with the sentence on page 1, line 24-26, "Nevertheless, due to the lack of stochastic background, the usual statistical tools, like confidence intervals, testing hypothesis are out of the possibilities". In fact, some papers propose a composite indicator in a stochastic framework, based on sample data and defined according to an inferential approach that consists of a multiple test (with pairwise comparisons). Please, correct the sentence and update the literature review.
The construction of some sentences should be improved by a native English speaker.
Author Response
Thank you for your work.
Our responses to your comments are included in the attached file.
Author Response File: Author Response.pdf
Reviewer 3 Report
The paper deals with an interesting topic, which refers to paired comparison stochastic modeling. A literature review is presented in some detail, while some generalizations are also provided. The main contribution of the paper seems to be the generalization of Ford’s necessary and sufficient condition for the existence and the uniqueness of the corresponding estimator in the case of two options. Moreover, the authors provide some advances for the case of three options. The paper could be reconsidered for publication in the journal after addressing the following issues.
Line 68: Replace “is” by “are”. There are quite a few typos throughout the lines of the manuscript that should be corrected .
Section 2: Provide a short illustrative example of the investigated model. The implementation of formulae (3)-(6) within the example is also welcome.
Section 3: Figures 2-5 seems to be quite interesting. However, the argumentation upon them is really poor. The authors are highly recommended to provide explanatory comments referring to the usefulness of these figures.
Section 4: The authors used Monte Carlo simulation for their investigation. The authors are kindly requested to provide more details about the simulation scheme. For instance, they should refer to the software, the number of iterations and the algorithm that has been followed.
The paper deals with an interesting topic, which refers to paired comparison stochastic modeling. A literature review is presented in some detail, while some generalizations are also provided. The main contribution of the paper seems to be the generalization of Ford’s necessary and sufficient condition for the existence and the uniqueness of the corresponding estimator in the case of two options. Moreover, the authors provide some advances for the case of three options. The paper could be reconsidered for publication in the journal after addressing the following issues.
Line 68: Replace “is” by “are”. There are quite a few typos throughout the lines of the manuscript that should be corrected .
Section 2: Provide a short illustrative example of the investigated model. The implementation of formulae (3)-(6) within the example is also welcome.
Section 3: Figures 2-5 seems to be quite interesting. However, the argumentation upon them is really poor. The authors are highly recommended to provide explanatory comments referring to the usefulness of these figures.
Section 4: The authors used Monte Carlo simulation for their investigation. The authors are kindly requested to provide more details about the simulation scheme. For instance, they should refer to the software, the number of iterations and the algorithm that has been followed.
Author Response
Thank you for your work.
Our responses to your comments are included in the attached file.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
All my comments have been taken into account by the authors and the current version of the manuscript is suitable for publication after a few minor changes due to the following small typos:
- page 7, line 241: replace "The first example is a very simple example With its help it is easy..." with "The first example is a very simple example. With its help it is easy..." (add full stop after "...example")
- page 8, line 259, 260, 262, 265: after equation (7), (8), (9) and (10), full stop should be added to mark the end of the sentence.
- page 9, line 287-288: replace "We can see that although the official points are equal in the case of France and Australia, the strength of France is larger than the strength of Australia." with "We can see that, although the official points are equal in the case of France and Australia, the strength of France is larger than the strength of Australia." (add comma after "...that")
Reviewer 3 Report
All issues raised by the reviewer have been appropriately addressed in the revised manuscript. I recommend the acceptance of the paper in present form.
