Confidence Intervals for Mean and Difference between Means of Delta-Lognormal Distributions Based on Left-Censored Data
Round 1
Reviewer 1 Report
The authors studied methods of developing confidence intervals for the mean and difference between means of delta-lognormal (zero-inflated lognormal) distributions based on left-censored data. However, I have a few concerns.
1) The methodology part should be better organized so that the readers can understand better. e.g., which methods are proposed by the authors themselves? If there is new methods proposed by authors, they should be presented in separate sections.
2) The frequentist properties doesn't seem to be quite sufficient to be published yet. For an appropriate confidence interval, its coverage probability should be close to its nominal level. As one can see, none of these methods produce "good" coverage. This part should be significantly improved.
Language is not a big problem.
Author Response
To Reviewer 1,
Thank you for your report, we revised the manuscript in accordance with your advice.
- The methodology part should be better organized so that the readers can understand better. e.g., which methods are proposed by the authors themselves? If there are new methods proposed by authors, they should be presented in separate sections.
Response: We proposed the new methods.
Lines 142-143 (Page 4 of the revised manuscript), we added “Three novel approaches were proposed to construct confidence intervals for the mean of DLN distribution based on left-censored data.”.
Lines 273-274 (Page 8 of the revised manuscript), we added “Four novel approaches were presented to estimate confidence intervals for the difference between means of DLN distributions based on left-censored data.”.
- The frequentist properties doesn’t seem to be quite sufficient to be published yet. For an appropriate confidence interval, its coverage probability should be close to its nominal level. As one can see, none of these methods produce “good” coverage. This part should be significantly improved.
Response: Coverage probability and average length were used to compare the performance of the proposed confidence intervals at a nominal confidence level of 0.95. The best confidence interval was defined as the one with a coverage probability greater than or equal to 0.95 and the shortest average length.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
This study considered the challenges of constructing confidence intervals for the mean and difference between means of delta-lognormal distributions containing left-censored data and applies them to compare the two daily rainfall average areas in Thailand. I have the following comments and concerns:
1. Avoid using abbreviations in the abstract.
2. It is not clear whether the mean, for example, belongs to the log-normal distribution or normal distribution. The authors said that the random variable X follows the log-normal distribution, then used Y=log(X) and obtain the properties of Y. My question now is what is the distribution of Y?
3. More detail should be added regarding the Bayesian approach. What are the used priors? How to get the posterior distribution?
4. More discussion should be added regarding the left censoring with examples.
5. In the Bayesian approach, the authors used confidence intervals based on the Bayesian approach. This is incorrect, use credible interval instead of confidence interval.
6 . It is known that the mean of the log-normal distribution is positive. How the authors selected negative values for mu in the simulation, see Table 1.
7. The same thing in real data analysis. The estimate of mu is negative. How?
Kindly, see my comments.
Author Response
Dear Reviewer,
Thank you very much for your useful report. We revised the manuscript in accordance with your advice.
- Avoid using abbreviations in the abstract.
Response: Abstract (Page 1 of the revised manuscript), we revised it.
- It is not clear whether the mean, for example, belongs to the log-normal distribution or normal distribution. The authors said that the random variable X follows the log-normal distribution, then used Y=log(X) and obtain the properties of Y. My question now is what is the distribution of Y?
Response: Lines 127-128 (Page 3 of the revised manuscript), we added “Since has the normal distribution.”.
- More detail should be added regarding the Bayesian approach. What are the used priors? How to get the posterior distribution?
Response: Lines 184-186 (Page 5 of the revised manuscript), we added “The prior distribution is based on the experimenter’s belief. In this paper, the Jeffreys Independence prior is used, which is defined as .”.
- More discussion should be added regarding the left censoring with examples.
Response: Lines 576-578 (Page 22 of the revised manuscript), we added “such as particulate matter 2.5 data and rainfall data. Owen and DeRouen [21] estimated the mean for LN distribution containing zeroes and left-censored values.”.
- In the Bayesian approach, the authors used confidence intervals based on the Bayesian approach. This is incorrect, use credible interval instead of confidence interval.
Response: We changed from “confidence interval” to “credible interval” in the revised manuscript.
- It is known that the mean of the log-normal distribution is positive. How the authors selected negative values for mu in the simulation, see Table 1.
Response: Lines 450-453 (Page 14 of the revised manuscript), we added “Note that the estimate of is the maximum likelihood estimator for the censored log-normal distribution. The estimate of is the range between -(infinity) and + (infinity). If is greater than zero, then is positive value.”.
- The same thing in real data analysis. The estimate of mu is negative. How?
Response: The estimate of mu is not the mean of the log-normal distribution, but the estimate of mu is the maximum likelihood estimator of censored log-normal distribution. It is the range between -(infinity) and + (infinity).
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
1. Sorry for the confusion in the first round of review. I agree with the authors' response on comment 2. A "good" coverage probabilities should be greater than or equal to the nominal level. But, even though most of the coverage probabilities produced by the methods proposed are greater than nominal level. The results are still considered as to be too conservative.
Can be improved
Author Response
Response to the Reviewer 1
(i) Does the introduction provide sufficient background and include all relevant references?
(ii) Are all the cited references relevant to the research?
(iii) Is the research design appropriate?
(iv) Are the methods adequately described?
Response: Discussion, Lines 577-580 (Page 22 of the revised manuscript), we improved “Owen and DeRouen [21] proposed the point estimators for the mean for LN distribution containing zeroes and left-censored values. This study extends to confidence intervals for the mean and the difference between means of DLN distributions that include left-censored data.”.
(i) Sorry for the confusion in the first round of review. I agree with the authors’ response on comment 2. A “good” coverage probabilities should be greater than or equal to the nominal level. But, even though most of the coverage probabilities produced by the methods proposed are greater than nominal level. The results are still considered as to be too conservative.
Response: The Bayesian credible interval (in this case) is a conservative confidence interval because its coverage probability is close to 1.00. However, as the sample size increases, the coverage probabilities of the Bayesian credible interval tend to the nominal confidence interval of 0.95.
Reviewer 2 Report
I recommend the publication of the paper.
I think the english language is well.
Author Response
Response to the Reviewer 2
- (i) Does the introduction provide sufficient background and include all relevant references?
(ii) Are all the cited references relevant to the research?
(iii) Is the research design appropriate?
Response: Discussion, Lines 577-580 (Page 22 of the revised manuscript), we improved “Owen and DeRouen [21] proposed the point estimators for the mean for LN distribution containing zeroes and left-censored values. This study extends to confidence intervals for the mean and the difference between means of DLN distributions that include left-censored data.”.
- I recommend the publication of the paper.
Response: Thank you very much.
Round 3
Reviewer 1 Report
I have no more comments
Ok
