Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions
Round 1
Reviewer 1 Report
The manuscript focuses on Birnbaum-Saunders (BS) distribution. The manuscript is well-written and organized. Overall, a study on this distribution seems interesting since it can be an alternative to commonly used distributions for assessing the damage caused by component fails. The numerical experiments using real data from Thailand are also interesting. Here are some of my comments:- The results are shown in long tables, sometimes difficult to read and follow. I would suggest showing some of the results using plots.
- What would be the extension of the proposed methods in correlated data? Some comments would be helpful.
- There have been some advances in the Gaussian process and Bayesian optimization to solve a similar model in recent years. The authors are encouraged to cite some of these Bayesian approaches in their introduction to provide some machine learning context for general readers:
- "Additive Manufacturing Melt Pool Prediction and Classification via Multifidelity Gaussian Process Surrogates." Integrating Materials and Manufacturing Innovation (2022): 1-19.
- "Particle filters for partially-observed Boolean dynamical systems." Automatica 87 (2018): 238-250.
- "Mechanical behavior predictions of additively manufactured microstructures using functional Gaussian process surrogates." npj Computational Materials 7.1 (2021): 1-11.
- Some description of the complexity ion form of O() would improve the overall paper.
Author Response
Response to the Reviewer
Paper Title: Simultaneous Confidence Intervals for all Pairwise Differences Between the Coefficients of Variations of Multiple Birnbaum-Saunders Distributions
Dear Reviewers,
We are grateful for your comments to our manuscript. We revised the manuscript in accordance with your advice. Here below is one-by-one response to your comments.
Lists of correction
Reviewer 1:
- The results are shown in long tables, sometimes difficult to read and follow. I would suggest showing some of the results using plots.
Response: We summarized the results in Tables 2, 3 and 4 by using line graph as in Fig. 1, 2 and 3. In addition, we added some description in Simulation study settings and results as follows. “In addition, Figures 1, 2 and 3 summarize the coverage probabilities and the average lengths of the methods in Tables 2, 3 and 4, respectively.”
**Additional Change**
- In Table 1, we changed sample sizes in scenarios 43-48 from (30,502,1002) to (505) and in scenarios 49-50 from (505) to (30,502,1002)
- Hence, there are some changes in Table 3. (See Latex file for track modification.)
- What would be the extension of the proposed methods in correlated data? Some comments would be helpful.
Response: We added some comments about correlated data in Conclusion section as follows. “In practice, several random variables are not independent of each other. Therefore, correlated data that follows the BS distributions is of interest for future research.”
- There have been some advances in the Gaussian process and Bayesian optimization to solve a similar model in recent years. The authors are encouraged to cite some of these Bayesian approaches in their introduction to provide some machine learning context for general readers:
"Additive Manufacturing Melt Pool Prediction and Classification via Multifidelity Gaussian Process Surrogates." Integrating Materials and Manufacturing Innovation (2022): 1-19.
"Particle filters for partially-observed Boolean dynamical systems." Automatica 87 (2018): 238-250.
"Mechanical behavior predictions of additively manufactured microstructures using functional Gaussian process surrogates." npj Computational Materials 7.1 (2021): 1-11.
Response: Since it is difficult to cite some of these three citations in Introduction section, we cite two of these three citations in “The BayCrI approach” section (after we describe the concept of Bayesian) as follows.
“Recently, the application of Bayesian method can be seen in [32, 33].”
We added two References as follows.
[32] Saunders, R.; Rawlings, A.; Birnbaum, A.; Iliopoulos, A.; Michopoulos, J.; Lagoudas, D.; Elwany, A. Additive Manufacturing Melt Pool Prediction and Classification via Multifidelity Gaussian Process Surrogates. Integr. Mater. Manuf. Innov. 2022, 1–19.
[33] Saunders, R.; Butler, C.; Michopoulos, J.; Lagoudas, D.; Elwany, A.; Bagchi, A. Mechanical behavior predictions of additively manufactured microstructures using functional Gaussian process surrogates. npj. Comput. Mater. 2021, 7, 81.
- Some description of the complexity ion form of O() would improve the overall paper.
Response: We have already improved it in the following sections: The PB approach, The GCI approach and The MOVER based on GCI approach.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors use different approaches to study the SCI of the CV based on the BS distribution. I list my comments as follow,
1. The content of this article is very rich, there are presented the parametric bootstrap (PB) approach, generalized confidence interval (GCI) approach, method of variance estimates recovery (MOVER) approach, and the highest posterior density (HPD) approach. However, all of them come from previous researches by others. It is difficult for readers to find innovative content and contributions in this paper. The paper seems like a simple combination of many methods, what is the main innovation of this paper? I think the author's creativity needs to be clarified.
2. For your simulation and modelling analysis was used software, you only mentioned in R software, and I hope you can add the corresponding codes in the Appendix.
3. You have too many Scenarios and results and without any clear explanations, please make the reader known why you choose these cases in Table 1, and which situations you would like to analysis? In addition, you compare different values of k (k=3,5,10), but did not address any useful findings about it.
4. In your simulation study, as the power increased, the AL also changed larger. I suggest authors to conduct extensive simulation exercise to compare the AL for fixed power of each approach or vice versa.
Author Response
Response to the Reviewer
Paper Title: Simultaneous Confidence Intervals for all Pairwise Differences Between the Coefficients of Variations of Multiple Birnbaum-Saunders Distributions
Dear Reviewers,
We are grateful for your comments to our manuscript. We revised the manuscript in accordance with your advice. Here below is one-by-one response to your comments.
Lists of correction
Reviewer 2:
- The content of this article is very rich, there are presented the parametric bootstrap (PB) approach, generalized confidence interval (GCI) approach, method of variance estimates recovery (MOVER) approach, and the highest posterior density (HPD) approach. However, all of them come from previous researches by others. It is difficult for readers to find innovative content and contributions in this paper. The paper seems like a simple combination of many methods, what is the main innovation of this paper? I think the author's creativity needs to be clarified.
Response: In the previous research for example Wang [1] constructed the generalized confidence interval (GCI) for α, the mean, quantiles, and a reliability function of the BS distribution. Wang et al. [2] investigated Bayesian credible intervals for α and β using the inverse-gamma prior. Hence, they focus on estimating confidence interval for the parameters of BS distribution ( and ) but in this study we are interested in constructing SCIs for comparing all pairwise differences between the CVs of several BS distributions. In addition, PB, MOVER and the HPD interval were commonly applied to construct confidence interval or SCIs interval for the parameter of interest for several distribution [3,4,5]. Furthermore, there have not been any publications about the statistical comparison of all pairwise differences between the CVs of several BS distributions, we applied the concept of PB, GCI, MOVER based on ACI and GCI, BayCrI and the HPD interval to construct SCIs for comparing all pairwise differences between the CVs of several BS distributions. Therefore, in introduction section, we changed from
“Hence, the purpose of this study is to propose SCIs for all pairwise differences between the CVs of several BS distributions using the percentile bootstrap (PB), GCI, MOVER, BayCrI, and HPD interval approaches.” to
“Hence, the purpose of this study is to apply the concepts of the percentile bootstrap (PB), GCI, MOVER based on ACI and GCI, BayCrI, and HPD interval approaches to construct SCIs for all pairwise differences between the CVs of several BS distributions.”
[1] Wang, B.X. Generalized interval estimation for the Birnbaum-Saunders distribution. Comput. Stat. Data Anal. 2012, 56, 4320–4326.
[2] Wang, M.; Sun, X.; Park, C. Bayesian analysis of Birnbaum-Saunders distribution via the generalized ratio-of-uniforms method. Comput. Stat. 2016, 31, 207–225.
[3] Hasan, M. S.; Krishnamoorthy, K. Improved confidence intervals for the ratio of coefficients of variation of two lognormal distributions. Journal of Statistical Theory and Applications. 2017, 16, 345- 353.
[4] Maneerat, P.; Niwitpong, S.A.; Niwitpong, S. A Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand. PeerJ 2020, 8, e8502.
[5] Puggard, W.; Niwitpong, S.A.; Niwitpong, S. Bayesian estimation for the coefficients of variation of Birnbaum–Saunders distributions. Symmetry 2021, 13, 2130.
- For your simulation and modelling analysis was used software, you only mentioned in R software, and I hope you can add the corresponding codes in the Appendix.
Response: The R code that support the finding of this study is lengthy and contains numerous variables. Therefore, we summarized the process of our R code for each method by using an algorithm which is commonly used by several researcher [1], [2], [3].
[1] Niu, C.; Guo, X.; Zhu, L. Comparison of several Birnbaum–Saunders distributions. J. Stat. Comput. Simul. 2014, 84, 2721–2733.
[2] Farias, R.B.A.; Lemonte, A.J. Bayesian inference for the Birnbaum–Saunders nonlinear regression model. Stat. Methods Appl. 2011, 20, 423-438
[3] Wang, M.; Sun, X.; Park, C. Bayesian analysis of Birnbaum-Saunders distribution via the generalized ratio-of-uniforms method. Comput. Stat. 2016, 31, 207–225.
- You have too many Scenarios and results and without any clear explanations, please make the reader known why you choose these cases in Table 1, and which situations you would like to analysis? In addition, you compare different values of k (k=3,5,10), but did not address any useful findings about it.
Response: We set many scenarios under k=3, 5 and 10 sample cases with different parameter values and different sample sizes to compare the performance of the proposed methods. Therefore, we added more details in Simulation study settings and results section as “We compare the effectiveness of the proposed methods in various circumstances. Therefore, we consider k=3, 5 and 10 sample cases.”
In addition, we analyzed every scenario to determine which method is the most effective. Furthermore, in this study, the results for k=3, 5 and 10 were similar. Hence, in Simulation study settings and results section, we added more description about these results as follows. “As the simulation results of these three scenarios were similar, we can draw the following conclusions.”
- In your simulation study, as the power increased, the AL also changed larger. I suggest authors to conduct extensive simulation exercise to compare the AL for fixed power of each approach or vice versa.
Response: In this study, is the shape parameter and is the scale parameter. When decreases, the density of BS distribution become more symmetric around , which is the median of the distribution. In addition, the variance increases when the value of increases. Hence, in simulation study, we used different value of parameter and different sample sizes to evaluate the performance of the proposed methods. For , the value for were kept fixed at 1.00 since there are scale parameter without loss of generality. Furthermore, we estimate SCIs at the nominal confidence level 1-=0.95.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Thank you.
