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The revision is sufficient to address the comments from the reviewers. The manuscript is clear and easy to follow. It can serve as a document for governments and researchers, who are interested in applying the delta-gamma distribution to rainfall data.
[# PeerJ Staff Note - this decision was reviewed and approved by Monika Mortimer, a PeerJ Section Editor covering this Section #]
The reviewers agree the work is solid and the results are reasonable. There are suggested changes from the three reviewers before the article is published. Please revise your manuscript accordingly.
In this manuscript, credible interval (CI) and highest posterior density (HPD) interval for the mean and the difference between the means of the delta-gamma distribution are constructed. For Bayesian method, the Jeffreys’ prior and uniform prior are taken into consideration. Also the CI and HPD intervals are derived based on fiducial quantities. The numerical computation is performed for comparing the efficacy of five proposed methods in terms of their coverage probabilities and expected lengths.
The structure of the manuscript is satisfactory. The paper is logically and technically in standard paper format. The computation part and simulation is reasonable.
The results and Tables are clear and acceptable. In general, after some minor revisions, the paper is suitable for acceptance.
I invite the authors to make minor revisions to improve the presentation, the comments are listed below:
1. Motivation of the paper needs to be illustrated.
2. In equation (8), the last term of the joint posterior density of θ will be exp(- x_n(1) /(2σ^2 )) instead of exp( x_n(1) /(2σ^2 )). Also the equations (10), (20) and (22) need to be corrected in similar manner.
3. Each equation should be ended either with “.” Or with “,”.
4. The resolution of all the plots (Figures 1 - 9) needs to be improved. I suggest to produce your graphics in PDF (do not convert it from png, jpeg, … , to pdf).
5. The line before equation (17), “Thus, the Fisher information …”, after the word ‘for’, "ϕ” is missing.
The authors presented well the methods and provided examples. They also uploaded the data files and the R code.
I have minor comments: In the abstract, discuss in one to two sentences at the start of the paragraph why this research is needed (or state the research question). The methods are acceptable, but I suggest checking the manuscript for typographical errors (especially regarding the English style).
The paper used data from weather stations. The authors uploaded the data files as supplementary information, which is good and highly encouraged. However, I recommend including in the paper a statement that the authors asked permission from the stations to publish and use the data for this research.
The methods are acceptable. Results can also be replicated as the authors uploaded the R files, and they provided examples.
1. In the introduction, rainfall is important enough. The review of rainfall is need more.
1. The independent of random sample or random variable is need to declare. It effects to your derivation of the likelihood function.
2. The description of HPD interval is need to explain more in both lines 89-92 and Algorithm 1.
1. In the results of simulation for the difference between the means of two delta-gamma distribution, the case of n>m is need to show or explain. Is it similar or difference from n<m.
2. Rainfall data from Example 1, were obtained by monthly with 50 observations from 1972-2021. Is it monthly?
3. Why is reason of data collection of Example 2 by January and February from 1976-2021.
4. The period of observations collections of 3 examples in the part of an empirical application is need to explain of reason in the manuscript.
1. The discussion should be separated from the results.
2. Add the limitation of your study in the discussion.
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