# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 2 Weibull Parameterizations
For a set of \(p\) covariates observed for each individual and the intercept term, say \(\mathbf{x}_{i}^{\top}=(1,x_{1i},x_{2i},\ldots,x_{pi})\), and for the four parameterizations, it is typical to include a regression structure in \(\lambda\), as follows  
\[S(t;\lambda,\
u)=\exp\left(-\left[\frac{t}{\lambda}\right]^{\
u}\right),\quad t,\
u,\lambda>0,\]0  
where \(\boldsymbol{\beta}=(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{p})\), and \(g(\cdot)\) is a twice differentiable function, such as \(g:\mathbb{R}\to\mathbb{R}^{+}\). The most common choice is \(g(u)=\exp(u)\), which also facilitates the interpretation of the regression coefficients. In each model, such interpretation depends on its specific parameterization (WEI, WEI2, WEI3, WEI4, and WEI5). However, it will be demonstrated that when a regression structure is incorporated solely into \(\lambda\), as in Equation (1), without affecting the shape parameter \(\
u\), then all five models share the same probability density function (PDF) and, consequently, the same log-likelihood function. Furthermore, it will also be demonstrated that introducing a dual regression structure for both \(\lambda\) and \(\
u\) in the WEI, WEI2, WEI3, and WEI4 models results in different PDFs among the models. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Mean** & **Mode** & \(\boldsymbol{q}\)**-Quantile** & **Variance** \\ \hline WEI & \(\lambda\Gamma\Bigl{(}1+\frac{1}{\
u}\Bigr{)}\) & \(\left\{\begin{array}{cc}\lambda\Bigl{[}\frac{(\
u-1)}{\
u}\Bigr{]}^{1/\
u} &,\text{if }\
u>1\\ 0&,\text{if }\
u\leq 1\end{array}\right.\) & \(\lambda[-\log(1-q)]^{1/\
u}\) & \(\lambda^{2}\omega(\
u)\) \\ WEI2 & \(\lambda^{-1/\
u}\Gamma\Bigl{(}1+\frac{1}{\
u}\Bigr{)}\) & \(\left\{\begin{array}{cc}\Bigl{[}\frac{(\
u-1)}{\lambda\
u}\Bigr{]}^{1/\
u} &,\text{if }\
u>1\\ 0&,\text{if }\
u\leq 1\end{array}\right.\) & \(\left[\frac{-\log(1-q)}{\lambda}\right]^{1/\
u}\) & \(\lambda^{-2/\
u}\omega(\
u)\) \\ WEI3 & \(\lambda\) & \(\left\{\begin{array}{cc}\frac{\lambda}{\Gamma\bigl{(}1+\frac{1}{\
u} \bigr{)}}\Bigl{[}\frac{(\
u-1)}{\
u}\Bigr{]}^{1/\
u}&,\text{if }\
u>1\\ 0&,\text{if }\
u\leq 1\end{array}\right.\) & \(\frac{\lambda}{\Gamma\bigl{(}1+\frac{1}{\
u}\bigr{)}}[-\log(1-q)]^{1/\
u}\) & \(\lambda^{2}\frac{\omega(\
u)}{\Gamma^{2}\bigl{(}1+\frac{1}{\
u}\bigr{)}}\) \\ WEI4 & \(\frac{\lambda\Gamma\bigl{(}1+\frac{1}{\
u}\bigr{)}}{\bigl{[}-\log(1-q)\bigr{]} ^{1/\
u}}\) & \(\left\{\begin{array}{cc}\lambda\Bigl{[}\frac{(\
u-1)}{\bigl{[}-\log(1-q) \bigr{]}}\Bigr{]}^{1/\
u}&,\text{if }\
u>1\\ 0&,\text{if }\
u\leq 1\end{array}\right.\) & \(\lambda\) & \(\lambda^{2}\Bigl{[}\log^{2}(1-q)\Bigr{]}^{-2/\
u}\omega(\
u)\) \\ WEI5 & \(\lambda\Bigl{[}1-\frac{1}{\
u}\Bigr{]}^{-1/\
u}\Gamma\Bigl{(}1+\frac{1}{\
u} \Bigr{)}\) & \(\lambda\) & \(\lambda\Bigl{[}\frac{-\
u\log(1-q)}{(\
u-1)}\Bigr{]}^{1/\
u}\) & \(\lambda^{2}\Bigl{[}1-\frac{1}{\
u}\Bigr{]}^{-1/\
u}\omega(\
u)\) \\ \hline \hline \end{tabular} NOTE: \(\omega(\
u)=\Gamma\bigl{(}1+\frac{2}{\
u}\bigr{)}-\Gamma^{2}\bigl{(}1+ \frac{1}{\
u}\bigr{)}\).  
\end{table} 
Table 1: Different measures for the five parameterization of the Weibull distribution. In all parameterizations, the parameter space for \((\lambda,\
u)\) is \(\mathbb{R}_{+}^{2}\), except for the WEI5 model, where the parameter space is \(\mathbb{R}_{+}\times(1,\infty)\).