Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper introduces and presents an application of the Laplace Logistic Unit (LLU) distribution  in real data modeling, in the domain of dynamic and regression analysis. It presents a comparison of the LLU distribution with some existing, well-known and frequently used unit distributions. For this purpose, 3 datasets (series) were presented which were transformed using the mentioned logistic function and thus obtained a corresponding regression model with an output variable that can be seen as the realization of independent RVs with LLU distribution.

I would advice that the authors might complete the article by an application with Entropy measures, Gini index and Lorenz Curve.

Author Response

RESPONSE TO REVIEWER 1

Reviewer comment: This paper introduces and presents an application of the Laplace Logistic Unit (LLU) distribution  in real data modeling, in the domain of dynamic and regression analysis. It presents a comparison of the LLU distribution with some existing, well-known and frequently used unit distributions. For this purpose, 3 datasets (series) were presented which were transformed using the mentioned logistic function and thus obtained a corresponding regression model with an output variable that can be seen as the realization of independent RVs with LLU distribution.

Authors response: The authors would like to express their great gratitude to Reviewer 1 for positive evaluation of our paper, as well as for useful comments that helped us significantly improve the manuscript. In this version, we have thoroughly revised the manuscript (the parts marked in red), and below are the responses to all suggestions and comments.

Issue 1. I would advice that the authors might complete the article by an application with Entropy measures, Gini index and Lorenz Curve.

• Authors response: Thank you very much for this comment. All of the Reviewer's suggestions have been implemented in this revised version. A new subsection 2.3 has been added, in which the incomplete moments of the LLU distribution are first presented. Based on them, the Lorenz curve, Gini index and entropy measures of this distribution are then described. (Please, see pages 9-12.)

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attached document.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please proofread for some minor grammatical mistakes.

Author Response

RESPONSE TO REVIEWER 2

Reviewer comment: In this paper, the authors proposed a new two-parameter unit stochastic distribution, the Laplace-logistic unit (LLU) distribution. The authors studied the basic properties, such as the moment, cumulative distribution function, hazard function and quantile function, of the proposed distribution and provided an estimation method for the parameters. Overall, this paper is well written and below are my comments and concerns on the paper.

Authors response: The authors would like to express their great gratitude to Reviewer 2 for positive evaluation of our paper, as well as for useful comments that helped us significantly improve the manuscript. In this version, we have thoroughly revised the manuscript (the parts marked in red), and below are the responses to all suggestions and comments.

Issue 1. In the simulation study, the authors generated samples from the LLU distribution. What is the algorithm used to generate the samples?

• Authors response: Thank you for this comment. Generating the sample is simply done by using the R-package "distr", as mentioned in the manuscript itself (please see Lines 177-178 on page 14). Let us point out that this package allows the definition of an arbitrary stochastic distribution, so the LLU distribution is also implemented in this way. Moreover, using this package, a random sample (x) can be directly generated by applying the R-function "x <- r(distr_name)(n)", where n is the sample size. The authors are ready to provide the corresponding code in "R", if the Reviewer considers it necessary.

Issue 2. I was confused on the comparisons of the proposed distribution with other two distribution. The authors mentioned in line 202-204 on page 15 that one of the reasons of the choice of estimation methods of these two distributions is the comparison not only with their distributions, but also with regard to different estimation procedures. Since different estimation procedures were used, is it possible that the superior performance of LLU was due to the estimation methods instead of the nature of the distribution? It seems to me that estimation of parameters in LLU is based on order-statistic, which is more of a nonparametric estimation method. Will the results change if similar estimation methods were used for the other two distributions? More justifications are needed here.

• Authors response: Thank you very much for this helpful comment. In this version, we explained the choice of MM estimators for the Beta distribution, i.e. the fact that these estimators are explicitly given and have some asymptotic properties. Regarding the Kumaraswamy distribution, instead of the MLE estimators, we now used percentile-based (say PC) estimators. The main reason is that they are more efficient than the MLE estimators, as shown in Ref. [27]. Also, PC estimators are more similar to quantile (say Q) estimators of the LLU distribution used here. It is worth mentioning that our motivation for introducing Q-estimators is not "non-parametric". As shown in Ref. [18], the MLE estimator of the scale parameter of the Laplacian distribution is the sample median, and our idea is based on this fact. (Please see Lines 151-152 on page 12.)

Issue 3. When testing the normality, only the Anderson-Darling test was used? I would recommend to include one more normality test such as the Shapiro-Wilks test or the Kolmogorov-Smirnov test.

• Authors response: Thank you for this comment. In this revised version, the results of the Shapiro-Wilks normality test are also included. (Please see Lines 187-193 on page 16, Lines 195-200 on page 17, and the newly added parts in Tables 1-3 on page 15.)

Issue 4. Beta distribution is commonly used as a prior distribution for the parameter p in a binomial distribution in Bayesian inference. Any comments on the usage of the proposed LLU in Bayesian inference?

• Authors response: Thank you very much for this comment and interesting idea. In this revised version, a new Remark 5 has been added describing Bayesian inference with the LLU distribution as prior versus the binomial distribution. (Please see the red parts on pages 8 and 9.)

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

 

The authors presented what they called   LLU (Laplace-Logistic Unit) distribution and developed classic derivations of properties of the new distribution, the mathematics is correct. The study included considerations on consistency issues. The importance of the proposal was illustrated.

They claim thatch the proposal behaves better than other existent distributions for data fitting .

I recommend to give a look to the role it may have in Data Science and AI, where the nature of data needs of logistic distribution based methods in Machine Learning for example.

To my consideration the  list of references is excessive as the basics of the ideas referred are present in a smaller set of entries.

 

 

 

  . Recommendations to the editor:

The changes are minimal.

 

Recommendations to the author

 

Conclusions

To publish the paper after these changes are made

 

Comments for author File: Comments.pdf

Author Response

RESPONSE TO REVIEWER 3

Reviewer comment: The authors presented what they called   LLU (Laplace-Logistic Unit) distribution and developed classic derivations of properties of the new distribution, the mathematics is correct. The study included considerations on consistency issues. The importance of the proposal was illustrated.

Authors response: The authors would like to express their great gratitude to Reviewer 3 for positive evaluation of our paper, as well as for useful comments that helped us significantly improve the manuscript. In this version, we have thoroughly revised the manuscript (the parts marked in red), and below are the responses to all suggestions and comments.

Issue 1. I recommend to give a look to the role it may have in Data Science and AI, where the nature of data needs of logistic distribution based methods in Machine Learning for example.

• Authors response: Thank you very much for this comment. In our opinion, the manuscript provides a significant and diverse number of applications of the LLU distribution. Nevertheless, we are very grateful to Reviewer 3 for this suggestion, and especially for the above-mentioned idea about the application of logistics functions. In this revised version, we have made some remarks in the conclusion, and we will certainly keep this in mind in the future research. (Please, see the red parts on page 19.)

Issue 2. To my consideration the  list of references is excessive as the basics of the ideas referred are present in a smaller set of entries.

• Authors response: Thank you for this comment. In this revised version, the number of reference entries has been significantly reduced. (Please, see page 20.)

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all my review comments and I do have no further concerns.