IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2023i1p56-d1306508.html
   My bibliography  Save this article

An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations

Author

Listed:
  • Yolanda M. Gómez

    (Departamento de Estadística, Facultad de Ciencias, Universidad del Bío-Bío, Concepción 4081112, Chile
    These authors contributed equally to this work.)

  • Diego I. Gallardo

    (Departamento de Estadística, Facultad de Ciencias, Universidad del Bío-Bío, Concepción 4081112, Chile
    These authors contributed equally to this work.)

  • Carolina Marchant

    (Faculty of Basic Sciences, Universidad Católica del Maule, Talca 3480112, Chile
    These authors contributed equally to this work.)

  • Luis Sánchez

    (Institute of Statistics, Universidad Austral de Chile, Valdivia 5110566, Chile
    These authors contributed equally to this work.)

  • Marcelo Bourguignon

    (Statistics Department, Federal University of Rio Grande do Norte, Natal 59078-970, RN, Brazil
    These authors contributed equally to this work.)

Abstract

The Weibull distribution is a versatile probability distribution widely applied in modeling the failure times of objects or systems. Its behavior is shaped by two essential parameters: the shape parameter and the scale parameter. By manipulating these parameters, the Weibull distribution adeptly captures diverse failure patterns observed in real-world scenarios. This flexibility and broad applicability make it an indispensable tool in reliability analysis and survival modeling. This manuscript explores five parameterizations of the Weibull distribution, each based on different moments, like mean, quantile, and mode. It meticulously characterizes each parameterization, introducing a novel one based on the model’s mode, along with its hazard and survival functions, shedding light on their unique properties. Additionally, it delves into the interpretation of regression coefficients when incorporating regression structures into these parameterizations. It is analytically established that all five parameterizations define the same log-likelihood function, underlining their equivalence. Through Monte Carlo simulation studies, the performances of these parameterizations are evaluated in terms of parameter estimations and residuals. The models are further applied to real-world data, illustrating their effectiveness in analyzing material fatigue life and survival data. In summary, this manuscript provides a comprehensive exploration of the Weibull distribution and its various parameterizations. It offers valuable insights into their applications and implications in modeling failure times, with potential contributions to diverse fields requiring reliability and survival analysis.

Suggested Citation

  • Yolanda M. Gómez & Diego I. Gallardo & Carolina Marchant & Luis Sánchez & Marcelo Bourguignon, 2023. "An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations," Mathematics, MDPI, vol. 12(1), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:56-:d:1306508
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/1/56/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/1/56/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Knut Røed & Tao Zhang, 2002. "A note on the Weibull distribution and time aggregation bias," Applied Economics Letters, Taylor & Francis Journals, vol. 9(7), pages 469-472.
    2. Almalki, Saad J. & Nadarajah, Saralees, 2014. "Modifications of the Weibull distribution: A review," Reliability Engineering and System Safety, Elsevier, vol. 124(C), pages 32-55.
    3. Silva, Rodrigo B. & Bourguignon, Marcelo & Dias, Cícero R.B. & Cordeiro, Gauss M., 2013. "The compound class of extended Weibull power series distributions," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 352-367.
    4. Jia, Xiang & Wang, Dong & Jiang, Ping & Guo, Bo, 2016. "Inference on the reliability of Weibull distribution with multiply Type-I censored data," Reliability Engineering and System Safety, Elsevier, vol. 150(C), pages 171-181.
    5. M. E. Ghitany & E. K. Al-Hussaini & R. A. Al-Jarallah, 2005. "Marshall-Olkin extended weibull distribution and its application to censored data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(10), pages 1025-1034.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xiang Jia & Saralees Nadarajah & Bo Guo, 2020. "Inference on q-Weibull parameters," Statistical Papers, Springer, vol. 61(2), pages 575-593, April.
    2. Almalki, Saad J. & Nadarajah, Saralees, 2014. "Modifications of the Weibull distribution: A review," Reliability Engineering and System Safety, Elsevier, vol. 124(C), pages 32-55.
    3. Jia, Xiang & Guo, Bo, 2022. "Reliability analysis for complex system with multi-source data integration and multi-level data transmission," Reliability Engineering and System Safety, Elsevier, vol. 217(C).
    4. He, Bo & Cui, Weimin & Du, Xiaofeng, 2016. "An additive modified Weibull distribution," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 28-37.
    5. Jia, Xiang & Cheng, Zhijun & Guo, Bo, 2022. "Reliability analysis for system by transmitting, pooling and integrating multi-source data," Reliability Engineering and System Safety, Elsevier, vol. 224(C).
    6. Boikanyo Makubate & Fastel Chipepa & Broderick Oluyede & Peter O. Peter, 2021. "The Marshall-Olkin Half Logistic-G Family of Distributions With Applications," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(2), pages 120-120, March.
    7. Hadeel S Klakattawi, 2022. "Survival analysis of cancer patients using a new extended Weibull distribution," PLOS ONE, Public Library of Science, vol. 17(2), pages 1-20, February.
    8. Starling, James K. & Mastrangelo, Christina & Choe, Youngjun, 2021. "Improving Weibull distribution estimation for generalized Type I censored data using modified SMOTE," Reliability Engineering and System Safety, Elsevier, vol. 211(C).
    9. Szymkowiak, Magdalena & Iwińska, Maria, 2016. "Characterizations of Discrete Weibull related distributions," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 41-48.
    10. Saralees Nadarajah & Božidar Popović & Miroslav Ristić, 2013. "Compounding: an R package for computing continuous distributions obtained by compounding a continuous and a discrete distribution," Computational Statistics, Springer, vol. 28(3), pages 977-992, June.
    11. Christian N. Brinch, 2011. "Non‐parametric identification of the mixed proportional hazards model with interval‐censored durations," Econometrics Journal, Royal Economic Society, vol. 14(2), pages 343-350, July.
    12. Rasool Roozegar & Saralees Nadarajah & Eisa Mahmoudi, 2022. "The Power Series Exponential Power Series Distributions with Applications to Failure Data Sets," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 44-78, May.
    13. Manudeep Bhuller & Christian N. Brinch & Sebastian Königs, 2017. "Time Aggregation and State Dependence in Welfare Receipt," Economic Journal, Royal Economic Society, vol. 127(604), pages 1833-1873, September.
    14. Hadeel Klakattawi & Dawlah Alsulami & Mervat Abd Elaal & Sanku Dey & Lamya Baharith, 2022. "A new generalized family of distributions based on combining Marshal-Olkin transformation with T-X family," PLOS ONE, Public Library of Science, vol. 17(2), pages 1-29, February.
    15. Sandeep Kumar Maurya & Saralees Nadarajah, 2021. "Poisson Generated Family of Distributions: A Review," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 484-540, November.
    16. Zubair Ahmad, 2020. "The Zubair-G Family of Distributions: Properties and Applications," Annals of Data Science, Springer, vol. 7(2), pages 195-208, June.
    17. E. M. Almetwally & H. M. Almongy & M. K. Rastogi & M. Ibrahim, 2020. "Maximum Product Spacing Estimation of Weibull Distribution Under Adaptive Type-II Progressive Censoring Schemes," Annals of Data Science, Springer, vol. 7(2), pages 257-279, June.
    18. Devendra Kumar & Neetu Jain & Mazen Nassar & Osama Eraki Abo-Kasem, 2021. "Parameter Estimation for the Exponentiated Kumaraswamy-Power Function Distribution Based on Order Statistics with Application," Annals of Data Science, Springer, vol. 8(4), pages 785-811, December.
    19. Bentaha, Mohand-Lounes & Voisin, Alexandre & Marangé, Pascale, 2020. "A decision tool for disassembly process planning under end-of-life product quality," International Journal of Production Economics, Elsevier, vol. 219(C), pages 386-401.
    20. Jose K. K. & Sivadas Remya, 2015. "Negative Binomial Marshall–Olkin Rayleigh Distribution and Its Applications," Stochastics and Quality Control, De Gruyter, vol. 30(2), pages 89-98, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:56-:d:1306508. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.