# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 2 Weibull Parameterizations  
As discussed in Section 1, the Weibull distribution is one of the most used models in reliability analysis because it is a parsimonious model with a simple expression for the density, survival, and hazard functions, and with some interesting properties. For instance, it can assume a decreasing, increasing, or constant hazard rate, depending only on the shape parameter \(v\) (\(<\)1, \(>\)1, or \(=\)1, respectively). However, the Weibull distribution has many parameterizations in the literature, depending on the study. The accelerated failure time (AFT) model, which is employed as the base parameterization, is used for the Weibull model; it has hazard and survival functions given by  
\[h(t;\lambda,\
u)=\frac{\
u}{\lambda}\left[\frac{t}{\lambda}\right]^{\
u-1}, \quad t,\
u,\lambda>0.\]  
and  
\[S(t;\lambda,\
u)=\exp\left(-\left[\frac{t}{\lambda}\right]^{\
u}\right),\quad t,\
u,\lambda>0,\]  
respectively, with \(\lambda\) and \(\
u\) parameters of scale and shape, respectively. This parameterization is referred to as WEI\((\lambda,\
u)\). An alternative parameterization of the Weibull model is associated with the proportional hazards model, in which the hazard and survival functions are given by  
\[h(t;\lambda,\
u)=\
u\lambda t^{\
u-1},\quad t,\
u,\lambda>0\]  
and  
\[S(t;\lambda,\
u)=\exp(-\lambda t^{\
u}),\quad t,\
u,\lambda>0,\]  
respectively. This parameterization is referred to as WEI\((\lambda,\
u)\), where \(\lambda\) and \(\
u\) act as the scale and shape parameters, respectively. A third alternative parameterization of the Weibull model is related to its mean. [PERSON] et al. [PERSON] et al. (2018) propose a reparameterization that expresses the Weibull distribution in terms of the process mean, enabling straightforward monitoring of the Weibull mean. In this parameterization, the hazard and survival functions are given by  
\[h(t;\lambda,\
u)=\frac{\
u}{\gamma}\left[\frac{t}{\gamma}\right]^{\
u-1}, \quad t,\
u,\lambda>0\]  
where \(\gamma=\lambda/\Gamma(1+1/\
u)\) and  
\[S(t;\lambda,\
u)=\exp\left(-\left[\frac{t}{\gamma}\right]^{\
u}\right),\quad t,\
u,\lambda>0,\]  
respectively. This parameterization is referred to as WEI\((\lambda,\
u)\). In this case, \(\mathbb{E}(T)=\lambda\). Recently, a fourth parameterization for the Weibull distribution was introduced by [PERSON] et al. [PERSON] et al. (2018). In this parameterization, the hazard and survival functions are given by  
\[h(t;\lambda,\
u)=-\log(1-q)\frac{\
u}{\lambda}\left[\frac{t}{\lambda}\right]^{ \
u-1},\quad t,\
u,\lambda>0.\]  
where \(q\in(0,1)\) and  
\[S(t;\lambda,\
u)=\exp\left(\log(1-q)\left[\frac{t}{\lambda}\right]^{\
u} \right),\quad t,\
u,\lambda>0,\]  
respectively. This parameterization is referred to as WEI\((\lambda,\
u)\). In this case, \(\lambda\) represents the \(q\)-quantile of the distribution. Alternatively, a fifth parameterization of the Weibull is proposed, based on its mode. The main motivation for re-parameterizing the model in terms of this measure is associated with the robustness of the mode (above the mean, for example) and the fact that it is a measure that has been used more frequently in the literature in recent years. For instance,see [PERSON] and [PERSON] [42], [PERSON] [43], and [PERSON] et al. [44]. In this parameterization, the hazard and survival functions are given by  
\[h(t;\lambda,\
u)=\frac{[\
u-1]}{\lambda}\bigg{[}\frac{t}{\lambda}\bigg{]}^{\
u- 1},\quad t,\
u,\lambda>0.\]  
and  
\[S(t;\lambda,\
u)=\exp\biggl{(}-\biggl{[}1-\frac{1}{\
u}\biggr{]}\bigg{[}\frac{t }{\lambda}\bigg{]}^{\
u}\biggr{)},\quad t,\lambda>0,\
u>1.\]  
This parameterization is referred to as WEIS\((\lambda,\
u)\). In this case, \(\lambda\) represents the mode of the distribution. Table 1 presents the mean, mode, \(q\)-quantile, and variance for the five parameterizations of the Weibull distribution explored in this study.