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

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.