Confidence Intervals for the Coefficient of Variation in Delta Inverse Gaussian Distributions

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper addresses the construction of confidence intervals (CIs) for the coefficient of variation (CV) in the context of Delta-Inverse Gaussian (Delta-IG) distributions, which are useful for data exhibiting zero-inflation and positive skewness. The authors evaluate seven CI construction methods, including Generalized Confidence Interval (GCI), Adjusted Generalized Confidence Interval (AGCI), Bootstrap Percentile Confidence Interval (BPCI), Fiducial Confidence Interval (FCI), Fiducial Highest Posterior Density Confidence Interval (F-HPDCI), Bayesian Credible Interval (BCI), and Bayesian Highest Posterior Density Confidence Interval (B-HPDCI), through Monte Carlo simulations and empirical applications.

Strengths:

1. The paper presents a novel focus on Delta-IG distributions, which are relevant for zero-inflated and asymmetric data. The extension of CI estimation to this context fills a notable gap in the literature.

2. The inclusion of seven CI estimation methods, combined with a thorough comparison of coverage probabilities (CPs) and average widths (AWs), provides a well-rounded assessment. The use of Monte Carlo simulations strengthens the robustness of the results.

3. The application to accident count data in India provides a real-world validation of the proposed methodologies and illustrates the practical significance of the research.

Recommendations for Improvement:

1. Strengthen the theoretical foundation of the methods by providing deeper explanations of why certain methods (e.g., AGCI, F-HPDCI) outperform others, particularly under different sample sizes and parameter settings. A comparative theoretical analysis could enhance the depth of the paper.

2. Include additional graphical representations of the simulation outcomes (beyond just the two figures) to help readers more easily discern the performance differences among the methods. Perhaps a comparison of methods in terms of computational efficiency might also be beneficial.

3. Explain more clearly the choice of hyperparameters in the Bayesian approaches (BCI, B-HPDCI). Discuss how different prior distributions might influence the results and provide guidance on selecting priors in practice.

Recommendation: Minor revisions

Comments on the Quality of English Language

Can be improved.

Author Response

Dear Reviewer 1,

Please see attached.

Sincerely,

Sa-Aat Niwitpong

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The study addresses the challenge of estimating confidence intervals (CIs) for the coefficient of variation (CV) in accident count data, which often has a Delta Inverse Gaussian (Delta-IG) distribution according to the authors. To address this, the authors compare seven CI construction methods: Generalized CI (GCI), Adjusted Generalized CI (AGCI), Bootstrap Percentile CI (BPCI), Fiducial CI (FCI), Fiducial Highest Posterior Density CI (F-HPDCI), Bayesian Credible Interval (BCI), and Bayesian Highest Posterior Density Confidence Interval (B-HPDCI). Through Monte Carlo simulations, they assess these methods based on coverage probability (CP) and average width (AW),  which are crucial for balancing precision and accuracy. The results indicate that AGCI and F-HPDCI methods generally outperform the others CIs for the CV, demonstrating superior performance across both CP and AW metrics. Finally, these methodologies are applied to a real collision counts data.

Despite the significance and potential impact of these findings, several critical issues need to be addressed before the manuscript is suitable for publication. In its current form, it requires further refinement. Attached is a detailed list of the issues that need attention.

Comments for author File: Comments.pdf

Author Response

Dear Reviewer 2,

Please see attached.

Sincerely,

Sa-Aat Niwitpong

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript develops several Confidence Interval (CI) estimators for the coefficient of variation (CV) statistic of the Delta Inverse Gaussian (Delta-IG) distribution. The Delta-IG is an important distribution for non-negative random variables, characterized by its asymmetry and right-skewed shape (i.e., with longer tail on the right side of the distribution), which is useful for representing extreme values in one direction, such as waiting times, stochastic processes, and accident counts; while the Delta-IG distribution can account for any excess zeros in the distribution. The CI estimators studied includes the Generalized Confidence Interval (GCI), Adjusted Generalized Confidence Interval (AGCI), Bootstrap Percentile  Confidence Interval (BPCI), Fiducial Confidence Interval (FCI), Fiducial Highest Posterior Density Confidence Interval (F-HPDCI), Bayesian Credible Interval (BCI), and Bayesian Highest Posterior Density Confidence Interval (B-HPDCI). The manuscript reports the results of an extensive Monte Carlo simulation study, comparing the accuracies of these different CI estimators (relative to the true data-generating parameters) in terms of average widths (AW) and coverage probability (CP). Interestingly, this study revealed that F-HPDCI and AGCI estimators tended to be the most accurate, especially given that fiducial confidence intervals have seen a recent resurgence in research interest. Also the manuscript reports the results of these CI estimators, through the analysis of a real dataset on accident counts in India. According to my careful reading of the manuscript, this is a well-motivated, sufficiently-novel, technically correct, and obviously well-written manuscript that is suitable for publication in this fine journal, Symmetry.

Section 2.3, describes the "Bootstrap Percentile Confidence Interval" procedure, including through Algorithm 2.  The name of this bootstrap procedure suggests that it is a nonparametric bootstrap procedure, based on (repeatedly) resampling n observations from the original dataset of n observations, also referred to as the nonparametric "n out of n bootstrap" (e.g., from Bickel, 2008, Stat Sinica).  A closer look at Algorithm 2 shows that it is instead, a parametric bootstrap procedure, based on (repeatedly) drawing n i.i.d. samples (with replacement) from the assumed Delta-IG distribution model estimated from the data through maximum likelihood. (So perhaps this procedure should be renamed to the "Parametric Bootstrap Percentile Confidence Interval"). One potential drawback of this parametric bootstrap approach, is that it assumes that the Delta-IG distribution model is the true data-generating (population) distribution of the data, which implies that the corresponding  "(Parametric) Bootstrap Percentile Confidence Interval" can be inaccurate when the model is wrong (i.e., mis-specified) for the data.

As a suggestion, the manuscript should also compare results between the nonparametric "n out of n bootstrap" CI,
the nonparametric "k out of n bootstrap" CI,
and the "k out of n subsampling" CI (which samples data observations without replacement), 
with optimal choice of k if necessary (though using k = sqrt(n) is a useful default choice that works asymptotically).
For more details, see for example:

Bickel, P. J., & Sakov, A. (2008). On the choice of m in the m out of n bootstrap and confidence bounds for extrema. Statistica Sinica, 967-985.
Politis, D. N., Romano, J. P., & Wolf, M. (2001). On the asymptotic theory of subsampling. Statistica Sinica, 1105-1124.

and the references therein.

One remarkable aspect about the "k out of n subsampling" CI subsampling method is that it can provide consistent and accurate CI estimates with very few assumptions and for statistics with difficult or misbehaving (e.g., non-smooth or discontinuous) sampling distributions. This explains the popularity of this CI estimation method. Similarly for the  "k out of n bootstrap" CI.

Please consider providing a weblink (via Github?) to the code used to generate all the results reported in this manuscript. This way, interested readers can attempt to reproduce and verify all the CI estimation equations and results of this manuscript. 

Details and Suggestions
Line 15:  if the accident counts data...then the Delta Inverse Gaussian...
Line 16:  distribution is more suitable.
Line 18:  ...objective is to establish...
Line 19:  ...methods, namely: the Generalized...
Line 31:  Non-negative observations with excess of zeros...
Line 59:  ...and proposed a range of approaches.
Line 70:  ...to the three-parameter...
Line 79:  ...studies examining...
Line 100:  In expressing the Delta-IG(...) distribution, use a dash '-' instead of a larger math minus sign symbol.
etc.

Author Response

Dear Reviewer 3,

Please see attached.

Sincerely,

Sa-Aat Niwitpong

Author Response File: Author Response.pdf