Tight bounds for the median of a gamma distribution

Richard F. Lyon

doi:10.1371/journal.pone.0288601

Peer Review History

Original SubmissionDecember 20, 2022
22 May 2023 Decision Letter - Pablo Martin Rodriguez, Editor PONE-D-22-34782 Tight bounds for the median of a gamma distribution PLOS ONE Dear Dr. Lyon, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. You will see that the two reviewers are advising that you revise your manuscript. Note that it is a minor revision so please consider making the suggested changes. Please submit your revised manuscript by Jun 22 2023 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. 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Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes ****** 5. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: In this paper, the author investigates various bounds (including tight bounds) for the median the scaled Gamma distribution. The results and proofs are quite technical and the author is nonetheless precise in their treatment. The weaknesses I see are that the review of the literature is quite lacking (there are many works on very close topics such as asymptotic bounds for the gamma, Poisson and negative binomial distributions that are not mentioned, even very recent ones. Including them could help motivate the results of the paper a bit more and help the reader put the results in perspective. Also, the figures could benefit from a legend or a clearer labelling of the different curves. Other than that, the conclusions and the proofs look accurate. I found the paper to be well-written, easy to follow, and of interest to researchers working on asymptotic bounds for the gamma, Poisson and negative binomial distributions. My recommendation is to accept the paper if the author can address the weaknesses mentioned above. Reviewer #2: I appreciated reading this paper which presents clearly new bounds for the median of a standard gamma distribution. The paper is well written and clear. The problem addressed by the article is interesting and the results seem to be relevant to probability theory and could be potentially useful. However, the paper needs minor revisions before it can be accepted for publication. Thus, should the author answer adequately to my comments and suggestions below, then I would recommend the paper for publication in Plos One. ****** 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No Reviewer #2: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. Attachments Attachment Submitted filename: Report - PONE- - D - 22 - 34782.pdf https://doi.org/10.1371/journal.pone.0288601.r001
Revision 1
25 Jun 2023 Author Response Rebuttal Letter / Response to Reviewers by Richard F. Lyon PONE-D-22-34782 24 June 2023 I appreciate the positive reviews from both reviewers. I mostly made changes in line with their suggestions, and added a few more bits along the way to help clarify (such as the new section "Toward a proof" that might help others complete the work, and an example of a third-order rational function interpolation with low relative error). Reviewer #1: …The weaknesses I see are that the review of the literature is quite lacking (there are many works on very close topics such as asymptotic bounds for the gamma, Poisson and negative binomial distributions that are not mentioned, even very recent ones. Including them could help motivate the results of the paper a bit more and help the reader put the results in perspective. I have added three references to very recent related works. But I don’t understand them well enough to say much about how related they are, so I just said, "In addition to the works mentioned above, there have been several more recent works on asymptotic properties and bounds for medians and other quantiles of gamma distributions and of the closely related Poisson and negative binomial (or Polya or Pascal) distributions, with a variety of interesting approaches \\cite{priore2022approximate, ouimet2023refined, pinelis2021monotonicity}. … Also, the figures could benefit from a legend or a clearer labelling of the different curves. I found that Figure 2 had an extra curve (from a corollary) that should not have been there, and that confused the interpretation of some of the labels on the curves, so I simply removed that (it was a thin solid red curve in the original figure). Now Figure 2 seems more clear. For Figures 1 and 3 the labels on the curves seem clear and unambiguous. That leaves Figure 4, which I agree was a bit confusing and crowded, with the (conjectured) interpolated bounds being described in the caption but not in the figure itself. So I managed to squeeze in a legend, and removed the 53% and 54% quantile curves that were in that area, leaving the presentation more symmetric between 48% and 52% and leaving more room for the legend. Reviewer #2 …the paper needs minor revisions … my comments and suggestions below … Most of the suggestions were of the form “might be better to add a full stop after the equation” or “might be better to add a comma after the equation”. I followed all of those, but I think I also added another comma or two. In addition, many were “might be better to add the symbol ⇒ before the equation and a full stop after the equation”, which I also agreed with and did. In each case I followed the typical LaTeX styling advice of separating the end punctuation from the math by a thin-space. Where I already had full stops after a few equations, I moved them away by a thin-space. Most of these changes are hard to spot in the diff, as the full stop or comma are too short for underlining to show up, and the blue color is easy to miss. But they’re there. I enumerate responses to the numbered suggestions that were more than these. These 3 I group and treat out of order: 1. p.1-9: γ should be defined. 5. p.2-10: Again, γ should be defined. 12. p.4-13: Again, γ should be defined. In the abstract I added “(with $\\gamma$ being the Euler--Mascheroni constant)”. In the text, on p.2, I added “(with $\\gamma \\approx 0.5772157$ being the Euler--Mascheroni constant)” which includes its approximate value. Thank you for noticing this was needed. I think it does not need to be said again on p.4. 2. p.1-17: might be “...probability density function (PDF)...” instead of “...PDF...”. I took the liberty to add a possessive, this way: “The gamma distribution's probability density function (PDF) is …” 3. p.1-19: might be x>0 instead of x≥0. Note that if k=1 and x=0, we have p1(0) = 0/0, which is an indeterminate form. Yes, x>0. Done. 4. p.1-23: might be “... cumulative distribution function (CDF)...” instead of “...CDF... ”. Done 9. p.4-2: the author uses p(k,x) to denote the PDF, but in p.1 the notation used is pk(x). I switched to using the subscript k consistently for both the PDF p_k(x) and the CDF P_k(x). 17. p.5-18: might be helpful to clarify that ν(k) > L4(k) for all k > 0. I added “for all k > 0” and also added some clarifying words to the start of the paragraph, “Since the denominator and the exponent are positive…”. And I removed the redundant comment that followed: “The derivation holds for all k > 0.” 31. p.7-27: might be “when k < 1” (as in the theorem statement) instead of for all positive k. I fixed it the other way. The theorem statement now says for all positive k, as in the table of theorems to prove. I re-checked that the proof is valid over that full domain. 50. p.11-22: avoid acronym like RHS and LHS without say its respective meaning. Reworded as “Dividing by the positive right-hand side gives the equivalent condition…”. Other changes I made, which you may notice in the diffs: I changed “The positivity constraint is met” to “The constraint is met” in one place where the constraint inequality is not a comparison to 0. Where I had “rather a” I changed to “rather than a”. I changed “which is completes the proof” to “this completes the proof” in the proof of Lemma 4. In the Fig. 4 caption I used a code font (\\texttt) for the Matlab function name gammaincinv. I think that’s all. I tried to restrain my tendency to keep tweaking. Attachments Attachment Submitted filename: Rebuttal Letter.pdf https://doi.org/10.1371/journal.pone.0288601.r002
2 Jul 2023 Decision Letter - Pablo Martin Rodriguez, Editor Tight bounds for the median of a gamma distribution PONE-D-22-34782R1 Dear Dr. Lyon, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Pablo Martin Rodriguez Academic Editor PLOS ONE Additional Editor Comments (optional): Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #1: All comments have been addressed Reviewer #2: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #2: Yes ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: Yes Reviewer #2: N/A ****** 4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes Reviewer #2: Yes ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: After thorough consideration and addressing all mentioned points, I find this version acceptable. The paper can be accepted. Reviewer #2: The former report pointed out some minor revisions. These has been corrected in the revised version of the authors. More generally, all of the modifications made by the authors (included those that were not requested) are satisfactory. Thus, I recommend the paper for publication in Plos One. ****** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No Reviewer #2: No ******** https://doi.org/10.1371/journal.pone.0288601.r003
Formally Accepted
29 Aug 2023 Acceptance Letter - Pablo Martin Rodriguez, Editor PONE-D-22-34782R1 Tight bounds for the median of a gamma distribution Dear Dr. Lyon: I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department. If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org. If we can help with anything else, please email us at plosone@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Professor Pablo Martin Rodriguez Academic Editor PLOS ONE https://doi.org/10.1371/journal.pone.0288601.r004

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