Weibull Distribution with Linear Shape Function

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Applied Sciences provides an advanced forum on all aspects of applied natural sciences, where special attention is paid to reproducibility, completeness of description of experimental and computational procedures, as well as practical value of the obtained results. The presented work, despite its mathematical novelty, is focused mainly on theoretical constructions and does not contain sufficient applied justification, experimental data and detailed interpretation of the model parameters in the context of practical applications. In this regard, the article does not correspond to the profile and goals of the journal Applied Sciences and, in my opinion, cannot be recommended for publication.

As for specific comments on the content of the work: the introduction provides a large number of sources, but the literary background seems mechanistic: the articles are mostly listed without analysis, and only three of them were published in the last three years. It is not shown which models are traditionally used in applied areas and why their capabilities are insufficient. The goals and objectives of the study are formulated vaguely, their relevance is not properly substantiated, and the practical value of the new model remains unclear. The authors focused on the LAW method but did not compare it with other modern approaches, as a result of which the effectiveness of the proposed solution is not confirmed. The model was tested only on two relatively small samples, which limits the possibility of making general conclusions about the universality of the proposed approach. Finally, the model parameters are described exclusively in mathematical form, without interpreting their applied meaning, which is especially important for a journal with an applied focus.

Author Response

in the file

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Review on the article “Weibull Distribution with Linear Shape Function” by Piotr Sulewski and Antoni Drapella

The authors presented the results of the study of the properties and possible applications of the new distribution, which is a modification of the classical Weibull distribution, in which the Weibull shape parameter is replaced by a shape function. A flexible version of the distribution with three and four parameters is presented, which the authors call the Weibull distribution with a linear shape function. Actually, the main defining idea of this article is the replacement of the Weibull shape parameter by a shape function. Estimates of the presented distribution are obtained based on theoretical and empirical reliability functions. The results of the work presented by the authors show the effectiveness of the approach. In particular, even a three-parameter distribution can compete in data modeling with distributions that have much more parameters.

The results of the work have wide practical application, which is noted in the article. An additional positive point of the article is a thorough review of the literature on various generalizations of the Weibull distribution.

The article makes a positive impression. The level of work is appropriate, the results are new and sufficiently substantiated.

A few questions and wishes to the authors regarding the improvement of the text of the article.

1. Unify the Abstract. Make the two parts logically connected into one readable text.
2. Definition 2 looks strange. Maybe avoid using the expression …has the form… in the definition, and write …is defined as…? In this definition there is a reference to the proof of Theorem 2. This is somehow confusing. Correct the logic of the thought.
3. If in Section 3 all the proofs are so trivial, then can the term Theorem be avoided?Maybe it would be better to say Proposition?
4. Question to the authors: are generalizations of the method possible for the case of a nonlinear function of the form? Of course, this is just a wish. But mentioning this could be implemented in the Conclusions, for example.

Author Response

in the file

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The article presents discussions using weibull distributions and despite be a mathematical article the authors shown the applications about the theory explained.

 

Positive Points

1. Showed the application of the theory presented ;
   2. deposited the code ;
   
   
   

suggestions:

1. to present metrics of convergence to the estimations procedures.
2. Use an example that the gaussian distribution can not be used. Usually engineer use gaussian ;
3. Explain the difference among the metrics AIC, BIC, HQIC,   KS
4. Compare the technique used to classical techniques 

Author Response

in the file

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript introduces a new family of lifetime models by replacing the constant Weibull shape parameter with a shape function and develops in detail the simplest linear version, available in three- and four-parameter forms. Closed-form expressions are derived for the cumulative failure function, density, hazard rate, and related quantities. An estimation method based on minimizing absolute differences between empirical and theoretical reliability functions is proposed alongside ML, LS, WLS, and LAW. The study includes Monte Carlo simulations and applications to two real datasets. The central question is whether a Weibull model with a time-varying shape parameter, particularly a linear shape function, can match the flexibility of more complex hybrid or bimodal models while requiring fewer parameters and avoiding the estimation challenges of mixture models.

The central innovation lies in transforming the static Weibull shape parameter into a deterministic function of time. This yields new analytical forms for key functions and introduces hybrid-like features such as a two-component density decomposition. The approach offers a balance between simple monolithic lifetime models and highly parameterized hybrid models, providing flexibility with fewer parameters. It can capture non-homogeneous populations and complex hazard rate patterns, including bimodality and bathtub shapes, without relying on explicit mixing proportions, which are difficult to estimate with small samples. The method is particularly relevant for reliability engineering, survival analysis, and predictive maintenance with IoT data. Together with a review of 165 generalizations of the Weibull distribution and the focused development of a practically implementable linear version, the work has the potential to contribute significantly to both theory and practice.

Compared with classical Weibull, Gamma, Gamma–Weibull, and explicit mixture or hybrid models, the proposed distribution can generate hazard rate shapes beyond the reach of monolithic models, such as bimodal or flat-bottom bathtub patterns, while using only three or four parameters rather than the five to eight typical of hybrid models. Unlike mixtures, it does not require an explicit mixing proportion, reducing data demands for identifiability and precision, as confirmed by Monte Carlo studies of parameter standard errors. The estimation method based on absolute reliability adds a methodological alternative to ML and squared-error fitting.

Strengths

The model is presented with closed-form expressions for CDF, PDF, HRF, HRAF, quantiles, and random number generation, which ensures reproducibility. Several estimation methods are compared, including ML, LS, WLS, LAW, and the proposed absolute-reliability method, making the study comprehensive. Monte Carlo simulations and real-data applications further demonstrate the practical relevance of the approach.

Weaknesses and required clarifications

The manuscript requires stronger treatment of identifiability and parameterization, including explicit conditions for identifiability and validity of the distribution. The statistical properties of the absolute-reliability estimator remain unstudied; theoretical or simulation-based evidence on consistency, asymptotic behavior, and efficiency is needed. More detail should be provided on optimization, including starting values, objective function behavior, and convergence.

The handling of censoring is a critical omission, as reliability and survival data often involve right-censoring. The estimators should be extended to this setting, or limitations must be clearly stated, with at least one censored example or simulation. Model selection and comparison should not rely solely on visual fit but should incorporate AIC, BIC, cross-validation, formal goodness-of-fit tests, and diagnostic plots.

Robustness requires further investigation through Monte Carlo designs with different sample sizes, shape regimes, and contaminated data. Bootstrap standard errors and confidence intervals should be reported in real-data examples. Comparisons with nonparametric hazard estimation would help validate model flexibility.

Reproducibility should be improved by making the R implementation publicly available (e.g., on GitHub or CRAN) with scripts, unit tests, and documentation. While the focus on a linear shape function is reasonable, alternative choices such as quadratic or log-linear functions should be briefly explored. Uncertainty quantification for hazard and survival functions should be provided through bootstrap or delta-method confidence bands.

The review of 165 generalizations is noteworthy, but the manuscript should clarify the selection process (search methods, inclusion criteria, timeframe) and present a concise summary table of the most relevant models and their rationale for inclusion. References should be updated to include recent work from the last five years on flexible lifetime models, hazard modulation, and nonparametric hazard estimation. The reference list should be complete, consistently formatted, and include DOIs or URLs.

 

Required revisions (high priority)

Provide a formal discussion of identifiability and validity conditions for the shape function and parameters.

Extend estimation methods to right-censored data (or explicitly state limitations) and include a censored-data example.

Study the statistical properties of the absolute-reliability estimator versus ML using theory or simulation.

 

Expand simulation studies to cover multiple sample sizes, alternative data-generating processes, and report bias, RMSE, coverage, and convergence.

Add diagnostic plots and formal goodness-of-fit tests for real datasets, with bootstrap confidence intervals.

Make all R code and replication scripts publicly available.

Recommended but lower priority

Provide theoretical conditions for when the HRF is bimodal or bathtub-shaped.

Compare parametric results with nonparametric hazard estimates.

Explore alternative shape functions and conduct a short sensitivity analysis.

If these revisions are addressed with theoretical clarifications, extended simulations, diagnostics, and reproducible code, the paper would be a strong candidate for publication in a reliability or applied statistics journal.

Author Response

in the file

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have substantially revised the article and demonstrated a high level of scientific research. Their responses to my comments are comprehensive. Although the study is primarily theoretical, it contains interesting ideas and is of significant academic interest to readers. The work can be accepted for publication in its current form.

Reviewer 4 Report

Comments and Suggestions for Authors

I appreciate the authors for accommodating my comments.