# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 4 Inference  
In this section, we discuss the estimation procedure in a unified way for the WEI, WEI2, WEI3, WEI4, and WEI5 models from a classical approach. The asymptotic distribution of the estimators is also presented. Parameter EstimationLet \(Y_{1},\ldots,Y_{n}\) be \(n\) independent random variables, where each \(Y_{i}\), \(i=1,\ldots,n\), follows a WEI, WEI2, WEI3, WEI4, or WEI5 model. The log-likelihood function for \(\mathbf{\theta}^{\top}=(\mathbf{\beta}^{\top},\mathbf{\delta}^{\top})\) has the form  
\[\ell(\mathbf{\theta})=\sum_{i=1}^{n}\ell_{i}(\lambda_{i},\
u_{i};y_{i}),\]  
where  
\[\ell_{i}(\lambda_{i},\
u_{i};y_{i}) = \log\
u_{i}-\log\lambda_{i}+(\
u_{i}-1)[\log t_{i}-\log\lambda_{ i}]-\left[\frac{t_{i}}{\lambda_{i}}\right]^{\
u_{i}},\]  
with \(\lambda_{i}=k(\lambda_{i}^{*},\
u_{i})\), \(\lambda_{i}^{*}=g(\mathbf{x}_{i}^{\top}\mathbf{\beta})\) and \(\
u_{i}=h(\mathbf{z}_{i}^{\top}\mathbf{\delta})\). The function \(k(\cdot,\cdot)\) is defined in Table 2 for the different parameterizations of the WEI model. The maximum likelihood (ML) estimates of \(\mathbf{\beta}\) and \(\mathbf{\delta}\) are computed as the solution of the non-linear system \(\mathbf{U}(\mathbf{\theta})=\mathbf{0}_{q+r}\), where \(\mathbf{0}_{q+r}\) denotes a \((q+r)\times 1\) vector of zeros and \(\mathbf{U}(\mathbf{\theta})\) is given by  
\[\mathbf{U}(\mathbf{\theta})=\left[ \begin{array}{c}\partial\ell(\mathbf{\theta})/\partial \mathbf{\beta}\\ \partial\ell(\mathbf{\theta})/\partial\mathbf{\delta}\end{array} \right]=\left[ \begin{array} []{c}\mathbf{X}^{\top}\mathbf{A}_{\lambda}\mathbf{C}_{\lambda}\mathbf{D}_{ \lambda}\\ \mathbf{Z}^{\top}\mathbf{A}_{u}\mathbf{D}_{\
u}\end{array} \right],\]  
where \(\mathbf{A}_{u}=\text{diag}(a_{u}^{(1)},\ldots,a_{u}^{(n)})\), \(\mathbf{C}_{\lambda}=\text{diag}(c_{\lambda}^{(1)},\ldots,c_{\lambda}^{(n)})\), \(\mathbf{D}_{u}=\text{diag}(d_{u}^{(1)},\ldots,d_{u}^{(n)})\), for \(u\in\{\lambda,\
u\}\), \(a_{\lambda}^{(i)}=g^{\prime}(\mathbf{x}_{i}^{\top}\mathbf{\beta})\), \(a_{\
u}^{(i)}=h^{\prime}(\mathbf{z}_{i}^{\top}\mathbf{\delta})\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **WEI** & **WEI2** & **WEI3** & **WEI4** & **WEI5** \\ \hline \(k(\lambda,\
u)\) & \(\lambda\) & \(\lambda^{-1/\
u}\) & \(\lambda[\
u\{\Gamma(1+1/\
u)\}^{\
u}]^{-1}\) & \(-\lambda\log(1-q)\) & \(\lambda\left[1-\frac{1}{\
u}\right]^{1/\
u}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Function \(k(\lambda,\
u)\) for the different parameterizations of the WEI model. \[c_{\lambda}^{(i)} =\left\{ \begin{array}{ll}1&\text{, for the WEI model}\\ -\frac{1}{\
u}\lambda^{-1/\
u-1}&\text{, for the WEI2 model}\\ \left[\
u\{\Gamma(1+1/\
u)\}\right]^{-1}&\text{, for the WEI3 model}\\ -\log(1-q)&\text{, for the WEI4 model}\\ \left[1-1\
u\right]^{1/\
u}&\text{, for the WEI5 model}\end{array} \right.\] \[d_{\lambda}^{(i)} =-\frac{v_{i}}{\lambda_{i}}+\frac{v_{i}}{\lambda_{i}}\left[\frac {t_{i}}{\lambda_{i}}\right]^{v_{i}}_{i},\] \[d_{\
u}^{(i)} =\frac{1}{v_{i}}+\log t_{i}-\log\lambda_{i}-\left[\frac{t_{i}}{ \lambda_{i}}\right]^{v_{i}}[\log t_{i}-\log\lambda_{i}].\]  
The Fisher information matrix for models can be written as  
\[\mathbf{I}(\boldsymbol{\theta})=\mathbb{E}\bigg{[} \begin{array}{cc}- \partial^{2}\ell(\boldsymbol{\theta})/\partial\boldsymbol{\beta}\boldsymbol {\beta}^{\top}&-\partial^{2}\ell(\boldsymbol{\theta})/\partial\boldsymbol{ \beta}\partial\delta^{\top}\\ \cdot&-\partial^{2}\ell(\boldsymbol{\theta})/\partial\boldsymbol{\delta} \partial\delta^{\top}\end{array} \bigg{]}=\widetilde{\mathbf{X}}^{\top} \mathbf{W}(\boldsymbol{\theta})\widetilde{\mathbf{X}},\]  
where  
\[\widetilde{\mathbf{X}} =\left( \begin{array}{cc}\mathbf{X}&\boldsymbol{0}_{p\times 1}\\ \boldsymbol{0}_{q\times 1}&\mathbf{Z}\end{array} \right)\qquad\text{and}\] \[\mathbf{W}(\boldsymbol{\theta}) =\left[