Confidence Intervals for Common Coefficient of Variation of Several Birnbaum–Saunders Distributions

Round 1

Reviewer 1 Report

The authors  study confidence intervals based on the generalized confidence interval (GCI), method of variance estimates recovery (MOVER), large-sample, Bayesian credible interval  (BayCrI), and highest posterior density interval (HPDI) methods  to estimate the common CV of several BS distributions. Their performances were studied numerically through Monte Carlo simulation in terms of their coverage probabilities and average lengths. They shiow that HPDI produced the shortest average length for all cases, thus resulting the most appropriate for constructing the confidence interval for the common  CV of several BS distributions. 

 The paper is correctly written and presents a correct methodological approach. It is well presented, organized and structured.

The topic is important or interesting enough to both scholars and reliability engineers.  There are enough and appropriate references, and   the paper is reasonably self-contained.

Decision: The manuscript is appropriate for the journal in terms of both scope and quality.  The reviewer deems it is acceptable as it is 

Author Response

Thank you for your kind reply. We are grateful for your comments to our manuscript.

Reviewer 2 Report

Common remark. Traditionally, CV is used for symmetrical distributions, without havy tails, where sd gives a good idea of ​​the spread of the parameters. Using CV for a highly skewed distribution looks less reasonable.

Author Response

Dear Reviewer,

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:

1. Traditionally, CV is used for symmetrical distributions, without heavy tails, where sd gives a good idea of ​​the spread of the parameters. Using CV for a highly skewed distribution looks less reasonable

            Response: Many statistical procedures concerning the CV are based on the normal distribution. However, several phenomena do not agree with the normality assumption due to asymmetry or to the presence of heavy- and light tails in the distribution of the data. Therefore, it is useful and necessary to have inferential procedures that allow us to estimate parameters, or any function of the parameters, CV in particular, for a non-normal population. There are several researchers have investigated statistical inference for the CV of asymmetrical distributions or highly skewed distributions. For example, Pang et al. [1] proposed confidence interval for CV of Weibull, log- normal, and gamma distributions and performed simulation studies for the results. Soliman et al. [2] proposed the estimation of the coefficient of variation for non-normal model using progressive first-failure-censoring data. El-Din et al. [3] examined the estimation of the coefficient of variation for Lindley distribution based on progressive first failure censored data. For more details, see [4-6].

[1] Pang, W.K.; Leung, P.K.; Huang, W.K.; Liu, W. On interval estimation of the coefficient of variation for the three-parameter Weibull, log-normal and gamma distribution: A simulation-based approach. European J. Oper. Res. 2005, 164, 367-377.

[2] Soliman, A.A.; Ellah, A.H.A.; Abou-Elheggag, N.A.; Modhesh, A.A. Estimation of the coefficient of variation for non-normal model using progressive first-failure-censoring data. Journal of Applied Statistics. 2012, 39, 2741-2758.

[3] El-Din, M.M.M.; Amein, M.M.; El-Raheem, A.M.A.; El-Attar, H.E.; Hafez, E.H. Estimation of the coefficient of variation for Lindley distribution based on progressive first failure censored data. Journal of Statistics Applications & Probability. 2019, 8, 83-90.

[4] Curto, J.D.; Pinto, J.C. The coefficient of variation asymptotic distribution in the case of non-iid random variables.  Journal of Applied Statistics. 2009, 36, 21-32.

[5] Yue, Z.; Baleanu, D. Inference about the ratio of the coefficients of variation of two independent symmetric or asymmetric populations. Symmetry. 2019, 11, 824.

[6] Hsieh, H.K. Inferences on the coefficient of variation of an inverse gaussian distribution. Communications in Statistics - Theory and Methods. 1990, 19, 1589-1605.

Author Response File: Author Response.pdf

Reviewer 3 Report

The manuscript titled "Confidence Intervals for Common Coefficient of Variation of Several Birnbaum–Saunders Distributions" is well written, better presented, correct scientifically and contains novel ideas. Although this manuscript has many positives, but yet it has some serious issues which needs to be resolved before making any decision. Hence I recommend the following Major Revisions:

1. Abstract of the manuscript is not convincing and it should be rewritten, so that it clearly presents the work proposed in the manuscript.

2. Language quality of the manuscript needs improvement.

3. Literature review and historical background in the introduction section should be improved.

4. Presentation of the manuscript needs comprehensive improvement.

5. The manuscript falls in self citation issues e.g. the author Sa-Aat Niwitpong has 7 citations and Suparat Niwitpong has 6 citations. Make sure no author should have more than 3 to 4 citations.

Author Response

Dear Reviewer,

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 3:

1. Abstract of the manuscript is not convincing and it should be rewritten, so that it clearly presents the work proposed in the manuscript.

            Response: In Abstract, we changed from “In statistics, the coefficient of variation (CV) is commonly used to analyze variation within a dataset while the common CV of several independent populations can be used to analyze variation among them.” to “Furthermore, the coefficient of variation (CV) is commonly used to analyze variation within a dataset. In some situations, the independent samples are collected from different instruments or laboratories. Consequently, it is of importance to make inference for the common CV.”

2. Language quality of the manuscript needs improvement.

Response: Our manuscript has been proved English language, style and spelling by professional. 

3. Literature review and historical background in the introduction section should be improved.

            Response: We delete some citation in introduction as follows:

• “Sangnawakij and Niwitpong [18] introduced three confidence intervals for the common CV of several gamma distributions by employing normal approximation and MOVER based on the score and Wald methods.”
• “The concepts of the adjusted GCI and computational approaches were used by Thangjai et al. [22] to estimate confidence intervals for the common CV of several normal populations.”
• “They also applied these four methods to construct confidence intervals for the variance and difference between the variances of BS distributions [14].”

We added two new citations in introduction as follows:

• “Verrill and Johnson [23] proposed a likelihood ratio-based confidence interval for a common CV of several normal distributions.”
• “Liu and Xu [11] introduced a new confidence interval for the common CV of several normal distributions based on the concept of the confidence distribution interval.”

In addition, in page 4, line 1, we changed from “According to Thangjai et al. [22] and Thangjai and Niwitpong [23],…” to “According to Thangjai and Niwitpong [20] and Yosboonruang et al. [32],…”

4. Presentation of the manuscript needs comprehensive improvement.

Response:  In abstract, we changed from “large-sample” to “large-sample (LS)”

• In introduction, we changed from “large-sample” to “LS”
• In section 2.3, we changed from “The large-sample approach” to “The LS approach” and “large-sample” to “LS”
• In Simulation study and results, Application of the confidence interval methods with real data and Conclusions, we changed from “large-sample” to “LS”
• In Table 2, 3, 4, and 8, we changed from “LSCI” to “LS” , “BayCI” to BayCrI” and “HPD” to “HPDI”

5. The manuscript falls in self citation issues e.g. the author Sa-Aat Niwitpong has 7 citations and Suparat Niwitpong has 6 citations. Make sure no author should have more than 3 to 4 citations.

Response: In References, we deleted three references as follows.

• “ Sangnawakij, P.; Niwitpong, S.A. Interval estimation for the common coefficient of variation of gamma distributions. Thailand Statistician. 2020, 18, 340–353.”
• “ Thangjai, W.; Niwitpong, S.A.; Niwitpong, S. Adjusted generalized confidence intervals for the common coefficient of variation of several normal populations. Communications in Statistics-Simulation and Computation. 2020, 49, 194-206.”
• “ Puggard, W.; Niwitpong, S.A.; Niwitpong, S. Confidence intervals for the variance and difference of variances of Birnbaum-Saunders distributions. Journal of Statistical Computation and Simulation. 2022, 92, 2829–2845.”

We added two new references as follows.

• “ Liu X.; Xu, X. A note on combined inference on the common coefficient of variation using confidence distributions. Electronic Journal of Statistics. 2015, 9, 219–233.”
• “ Verrill, S.; Johnson, R. A. Confidence bounds and hypothesis tests for normal distribution coefficients of variation. Commun. Statist. Theor. Meth. 2007, 36, 2187–2206.”

Additional changes

            In Table 1:

• Scenarios 25-30, we changed from “(100₂)” to “(100₃)
• Scenarios 37-42, we changed from “(30₂, 50₃)” to “(30₂, 50₂, 100)
• Scenarios 43-48, we changed from “(30₂, 50₃)” to “(30, 50₂, 100₂)
• Scenarios 49-54, we changed from “(30₂, 50₃)” to “(50₅)
• Scenarios 55-60, we changed from “(30₂, 50₃)” to “(50₂, 100₃)
• Scenarios 67-72, we changed from “(30₅, 50₅)” to “(30₅, 50₃, 100₂)
• Scenarios 73-78, we changed from “(30₅, 50₅)” to “(30₃, 50₄, 100₃)
• Scenarios 79-84, we changed from “(30₅, 50₅)” to “(50₆, 100₄)

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Satisfied with the revised version and hence recommend acceptance in current form.

Author Response

Thank you for your comments. A revised version is improved by a native English teacher.  

Author Response File: Author Response.pdf