# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 1 Background
The Weibull distribution is widely used in modeling lifetimes because it can effectively represent different failure patterns observed in real-life situations. Failure data can be classified into two types: complete data and incomplete (censored) data. Complete data refer to cases where the actual observed values are known for every observation in the dataset. On the other hand, censored data occur when the actual observed values are unknown for some or all of the observations. It is worth noting that there are various types of censoring; more information can be found in the work by [PERSON] [18]. Several studies have examined the application of the Weibull distribution in analyzing censored data. For example, [PERSON] et al. (2005) [19] and [PERSON] (2022) [20] extended the Weibull model and applied it to bladder cancer data, [PERSON] (2011) [21] and [PERSON] et al. (2016) [22] studied leukemia data, and [PERSON] et al. (2007) [23] focused on the head-and-neck-cancer trial. The Weibull distribution has an intriguing application in models involving a fraction of cure. This distribution is particularly well-suited for estimating the time it takes for cancer cells to produce detectable cancer. As a result, it has gained significant popularity in this field. Several studies, including those by [PERSON] et al. [24], [PERSON] and [PERSON] [25], [PERSON] et al. [26], [PERSON] et al. [27], and [PERSON] et al. [28], have extensively explored this topic. There are numerous extensions, generalizations, and modifications to the Weibull distribution. These developments have emerged to address the requirements of empirical datasets that exhibit characteristics beyond what can be effectively captured by a standard two-parameter Weibull model ([PERSON] and [PERSON] [17]). These extended models can be broadly classified into three groups: univariate, multivariate, and stochastic process models. Univariate models focus on enhancing the flexibility of the Weibull distribution for single variable analysis. Multivariate models extend the Weibull framework to handle multiple variables and their dependencies. Stochastic process models delve into time-dependent and dynamic variations of the Weibull distribution. Noteworthy references, such as [PERSON] et al. [29; 30], provide insights into these diverse extensions and shed light on the advancements made in modeling techniques beyond the traditional Weibull framework. [PERSON] et al. [31] derived the power-series extended Weibull class of distributions. [PERSON] et al. [32] and [PERSON] et al. [33] introduced a family of distributions encompassing well over forty variants. For more details about the generalizations and modifications of the Weibull distribution, see [PERSON] [30], [PERSON] and [PERSON] [34], [PERSON] and [PERSON] [35], and [PERSON] and [PERSON] [36], as well as the references therein. The analysis of the truncated Weibull distribution has been explored in many papers. For example, [PERSON] [37] proposed the left-truncated Weibull distribution. The right-truncated Weibull distribution has been analyzed by [PERSON] and [PERSON] [38]. The doubly-truncated Weibull distribution has been studied in work by [PERSON] and [PERSON] [39]. In summary, the Weibull distribution is a versatile probability distribution that finds widespread application in modeling the failure times of objects or systems. The distribution's characteristics and behaviors are determined by two key parameters: the shape parameter and the scale parameter. The shape parameter governs the shape of the distribution, allowing it to range from exponential to highly skewed distributions. On the other hand, the scale parameter influences the location and spread of the distribution. By manipulating these parameters, the Weibull distribution can effectively capture various failure patterns observed in real-world scenarios. Its flexibility and wide range of applications make it a valuable tool in reliability analysis and survival modeling. The main objectives of this manuscript are as follows:  
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