# An In-Depth Review of the Weibull Model with a Focus on Various Parameterizations
## 4 Inference
\begin{array}{cc}\mathbf{A}_{\mathbf{i}}\mathbf{C}_{ \lambda}\mathbf{V}_{\lambda\lambda}\mathbf{C}_{\lambda}\mathbf{A}_{\mathbf{ \lambda}}&\mathbf{A}_{\mathbf{\lambda}}\mathbf{V}_{\lambda\
u}\mathbf{C}_{ \lambda}\mathbf{A}_{\mathbf{\
u}}\\ \mathbf{A}_{\mathbf{\
u}}\mathbf{V}_{\lambda\
u}\mathbf{C}_{\lambda}\mathbf{A}_ {\mathbf{\lambda}}&\mathbf{A}_{\mathbf{\
u}}\mathbf{V}_{\
u\
u}\mathbf{A}_{ \
u}\end{array} \right],\]  
with \(\mathbf{V}_{uv}=\text{diag}(V_{uv}^{(1)},\ldots,V_{uv}^{(n)})\), for \(u,v\in\{\lambda,\
u\}\), where  
\[V_{\lambda\lambda}^{(i)} =-\mathbb{E}\bigg{(}\frac{\partial^{2}\ell_{i}(\boldsymbol{\theta })}{\partial\lambda_{i}^{2}}\bigg{)}=-\mathbb{E}\bigg{(}\frac{v_{i}}{\lambda_ {i}^{2}}-\frac{v_{i}(v_{i}+1)}{\lambda_{i}^{2}}\bigg{[}\frac{t_{i}}{\lambda_{i }}\bigg{]}^{v_{i}}\bigg{)}=\frac{v_{i}^{2}}{\lambda_{i}^{2}},\] \[V_{\
u v}^{(i)} =-\mathbb{E}\bigg{(}\frac{\partial^{2}\ell_{i}(\boldsymbol{\theta })}{\partial v_{i}^{2}}\bigg{)}=-\mathbb{E}\bigg{(}-\frac{1}{v_{i}^{2}}-\left[ \frac{t_{i}}{\lambda_{i}}\right]^{v_{i}}[\log t_{i}-\log\lambda_{i}]\bigg{)}\] \[=\frac{1}{v_{i}^{2}}+\frac{1}{v_{i}}\Big{[}\psi^{\prime}(1)+\psi ^{2}(1)+2v_{i}\log\lambda_{i}\psi(1)+v_{i}^{2}\log^{2}\lambda_{i}-\psi(2)-v_{ i}\log\lambda_{i}+v_{i}\log^{2}\lambda_{i}\Big{]},\] \[V_{\lambda\
u}^{(i)} =-\mathbb{E}\bigg{(}\frac{\partial^{2}\ell_{i}(\boldsymbol{\theta })}{\partial\lambda_{i}\partial v_{i}}\bigg{)}=-\mathbb{E}\bigg{(}-\frac{1}{ \lambda_{i}^{2}}+\frac{1}{\lambda_{i}}\bigg{[}\frac{t_{i}}{\lambda_{i}}\bigg{]} ^{v_{i}}+\frac{v_{i}}{\lambda_{i}}\bigg{[}\frac{t_{i}}{\lambda_{i}}\bigg{]}^{v _{i}}[\log t_{i}-\log\lambda_{i}]\bigg{)}=-\frac{\psi(2)}{\lambda_{i}}.\]  
An alternative way to obtain the ML estimates of \(\boldsymbol{\theta}\) is by using the [PERSON] scoring iterative procedure, providing the following estimation algorithm:  
\[\widehat{\boldsymbol{\theta}}^{(k+1)} =\widehat{\boldsymbol{\theta}}^{(k)}+\left[\mathbf{I}\big{(} \widehat{\boldsymbol{\theta}}^{(k)}\big{)}\right]^{-1}\times\mathbf{U}\Big{(} \widehat{\boldsymbol{\theta}}^{(k)}\Big{)}\] \[=\widehat{\boldsymbol{\theta}}^{(k)}+\left[\widetilde{\mathbf{X}} ^{\top}\mathbf{W}\Big{(}\widehat{\boldsymbol{\theta}}^{(k)}\Big{)} \widetilde{\mathbf{X}}\right]^{-1}\widetilde{\mathbf{X}}^{\top}\mathbf{W}_{1} \Big{(}\widehat{\boldsymbol{\theta}}^{(k)}\Big{)},\quad k=0,1,2,\ldots,\]  
where \(\mathbf{W}_{1}(\boldsymbol{\theta})=(\mathbf{A}_{\mathbf{i}}\mathbf{C}_{ \lambda}\mathbf{D}_{\lambda}.\mathbf{A}_{\mathbf{\
u}}\mathbf{D}_{\
u})^{\top}\). We recommend initializing the algorithm by using, as an initial guess for \(\boldsymbol{\beta}\), the ordinary least squares estimates obtained from the linear regression of the transformed responses: \(g(y_{1}),\ldots,g(y_{n})\) on \(\mathbf{X}\), i.e., \(\left(\mathbf{X}^{\top}\mathbf{X}\right)^{-1}\mathbf{X}^{\top}\mathbf{Z}\), where \(\mathbf{Z}=(g(y_{1}),\ldots,g(y_{n}))^{\top}\). In order to approach the interval estimation and hypothesis testing on the model parameters \(\boldsymbol{\beta}\) and \(\delta\), normal approximation for the ML estimators can be applied. Note that under certain conditions for the parameters, the asymptotic distribution of \(\sqrt{n}(\widehat{\boldsymbol{\theta}}-\boldsymbol{\theta})\) is multivariate normal \(N_{q+r}(\boldsymbol{\theta}_{q+r},\mathbf{I}^{-1}  
Let the \(\theta_{j}\) be the \(j\)th component of \(\theta\). The asymptotic \(100(1-\gamma)\%\) confidence interval for \(\theta_{j}\) is given by  
\[\widehat{\theta}_{j}\pm z_{1-\gamma/2}\;\mathrm{se}(\widehat{\theta}_{j}),\qquad j =1,\ldots,q+r,\]  
where \(\mathrm{se}(\widehat{\theta}_{r})\) is the asymptotic standard error of \(\widehat{\theta}_{j}\), which is the square root of the \(j\)th diagonal element of the inverse of \(\mathbf{I}(\widehat{\theta})\).