Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The strategy for testing circular uniformity here is to adopt the NNTS family as an alternative model. This idea is well-motivated: to create an alternative model that includes the null model as a special case but can also be much more flexible. The challenges in implementing the strategy include (i) the choice of a test statistic to reflect deviation from the null based on data when the null is false, (ii) estimation of the quantiles of the null distribution of a proposed test statistic, (iii) choice of the model complexity for the alternative model. 

I think the authors made efforts to address all three challenges, but can improve in (ii) & (iii). Indeed, estimating the null distribution of a test statistic that involves nonregularity maximum likelihood estimation (MLE) can be a hard problem. But there are some carefully designed bootstrap methods out there that still allow one to estimate the null distribution (and thus estimate the p-values). I acknowledge that it may be unfair to expect the authors try out these unconventional bootstrap methods, but it can be helpful to include some discussions/review/comments on these relevant bootstrap-based strategies and argue benefits of your current adopted method of estimating the critical values based on certain regression models. I actually saw more works on the bootstrap-based methods than the regression-based methods of estimating these critical values when one faces irregularity MLEs, and hence cannot help wondering about the validity of the latter. 

The authors did make recommendations on the issue of (iii) by suggesting choosing a larger M (between two candidates). But a larger M leads to more parameters to estimate and thus can potentially lower the efficiency and the power of a test. Hence I believe this can be addressed more satisfactorily. Can one use a model selection criterion to choose M? If so, how would this extra step of model selection impact the operating characteristics of a test statistic? 

Lastly, in the spirit of flexible modeling, I would recommend the authors include in the simulation a setting where one does not generate data from NNTS, but from some circular distribution (not uniform) that is outside of the family of NNTS. One would expect your proposed tests (and maybe competing methods too) can work well here too when a data set does not come from the null or the alternative. 

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

appliedmath-2877054:  Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions." By Duránet al.

 This manuscript presents a very interesting statistical investigation on family of circular distributions based on nonnegative trigonometric sums. The practical application of the proposed tests to real data on the time of occurrence of earthquakes and the flying orientation of home pigeons is presented. The power of the NNTS circular uniformity test based on the generalized likelihood ratio (NNTS2) presents the largest power over the NNTS test based on the standardized maximum likelihood estimator, NNTS1, Pycke test, and modified Hermans-Rasson test in this simulation studies.

The manuscript is original. It appears well. I believe that the paper can interest many readers of "Applied Math" So, The manuscript contains some interesting information which (in my opinion) should be published in Applied Math. However, I have some minor suggestions:

 1-      the authors should read the manuscript carefully, and should correct all the typographical and grammatical errors through out the manuscript.

2-      The authors should discuss the point that their theory is valid for some applications in earthquakes.

3-      Figure 4 needs some discussions?

4-      The authors should briefly refer to some new references in this works.

5-      Authors should explain the following sentence((  The interpolated critical values for the generalized likelihood ratio NNTS2 test for any sample size were obtained by using regression models that showed an excellent fit)) ?.

 6-      The discussion section in the manuscript is primarily a presentation of the obtained results with little interpretive discussion.

 

 

Comments on the Quality of English Language

No comments

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper can be accepted but only after minor revision. 

1. Mathematics subject classification is missing

2. abstract needs to be improve

3. Aims of the paper should clearly be written

4. English-wise mistakes should be remove

5. Conclusion should be modify

6. some up to date references should be add

Comments on the Quality of English Language

English is fine

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Please find attached.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Minor editing recommended.

Author Response

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Author Response File: Author Response.pdf

Reviewer 5 Report

Comments and Suggestions for Authors Paper "Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions" submitted to AppliedMath (appliedmath-2877054)

Summary: This paper establishes the closed-munder summation property of a family of circular distributions based on nonnegative trigonometric sums (NNTS). NNTS circular distributions are a viable option for usage as alternative models to test for circular uniformity in order to identify distinct violations from the circular uniformity null hypothesis because of their flexibility in modeling multimodality and skewness. The circular uniform distribution belongs to the NNTS family; nevertheless, when the parameters are estimated using the maximum likelihood approach, it corresponds to a point on the boundary of the parameter space in the NNTS parameter space, suggesting that the regularity constraints are not satisfied. By taking into account the generalized likelihood ratio and the standardized maximum likelihood estimator, two NNTS tests for circular homogeneity were created. The critical values of the proposed NNTS circular uniformity tests were determined by simulation, taking into account the nonregularity criterion. Regression model fitting was then used to interpolate the values for any sample size. By creating NNTS models that are near to the circular uniformity null hypothesis, the validity of the suggested NNTS circular uniformity tests was assessed.   Evaluation: The paper is well written and very interesting; the theory is clear, the simulation and applications are convincing. This is a great paper. I just formulated just one comment before publication; Improve the structure of the paper, some parts are a bit drafty in the reading, mainly the theoretical part with the equations. Perhaps put some under the form of Proposition. This is a great paper and I recommend it for publication in AppliedMath. Comments on the Quality of English Language

Professional quality

Author Response

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Author Response File: Author Response.pdf

Reviewer 6 Report

Comments and Suggestions for Authors

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed all my comments very carefully. 

Reviewer 4 Report

Comments and Suggestions for Authors

I don't have further comments.

Comments on the Quality of English Language

English is fine overall.