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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
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    References listed on IDEAS

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    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. 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.
    5. 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.
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