## 2 MLEs of Parameters in Generalized Poisson with Under-Dispersion and Its Mean Regression Model
### MLEs of \(\{\beta,\alpha\}\) in the Mean Regression Model
#### 2.2.1 MLE of \(\beta\) Given \(\{\mathbf{\beta}^{(t)},\alpha\}\)  
Since \(\partial\mu_{i}/\partial\beta=\mu_{i}\mathbf{w}_{i}\), we have  
\[\frac{\partial\log(y_{i}^{-1}\mu_{i}+\alpha-1)}{\partial\beta} = \frac{y_{i}^{-1}\mu_{i}}{y_{i}^{-1}\mu_{i}+\alpha-1}\mathbf{w}_{i} \quad\text{and}\] \[\frac{\partial^{2}\log(y_{i}^{-1}\mu_{i}+\alpha-1)}{\partial \beta\partial\beta^{\top}} = \frac{(\alpha-1)y_{i}^{-1}\mu_{i}}{(y_{i}^{-1}\mu_{i}+\alpha-1)^ {2}}\mathbf{w}_{i}\mathbf{w}_{i}^{\top}. \tag{9}\]  
According to (8), we know that \(y_{i}^{-1}\mu_{i}+\alpha-1>0\), thus \(0<1-\alpha<y_{i}^{-1}\mu_{i}\). Given \(\mathbf{\beta}^{(t)}\) and \(\alpha\), to calculate the \((t+1)\)-th approximation of \(\mathbf{\beta}\), we first restrict \(\mathbf{\beta}\) in the following convex set  
\[\mathbb{C}^{(t)}=\bigg{\{}\mathbf{\beta}\colon y_{i}^{-1}\mu_{i}\geqslant T_{i}^{ (t)}(\alpha)\triangleq\frac{1}{2}\Big{[}1-\alpha+y_{i}^{-1}\mu_{i}^{(t)}\Big{]},\quad\forall\,i\in\mathbb{I}_{2}\bigg{\}}, \tag{10}\]  
where \(T_{i}^{(t)}(\alpha)\) is the midpoint of the two endpoints of the open interval \((1-\alpha,\ y_{i}^{-1}\mu_{i}^{(t)})\) and \(\mu_{i}^{(t)}\triangleq\exp(\mathbf{w}_{i}^{\top}\mathbf{\beta}^{(t)})\). Then, for any \(i\in\mathbb{I}_{2}\), since \(\alpha-1<0\), we have  
\[\frac{(\alpha-1)y_{i}^{-1}}{(y_{i}^{-1}\mu_{i}+\alpha-1)^{2}} \stackrel{{(\ref{eq:1})}}{{\geqslant}}\frac{(\alpha-1)y_{i}^{-1}} {[T_{i}^{(t)}(\alpha)+\alpha-1]^{2}}\triangleq\,b_{i3}^{(t)}(\alpha). \tag{11}\]  
On the other hand, we define  
\[h_{i}^{(t)}(\mathbf{\beta}|\alpha)=\log[\mu_{i}+(\alpha-1)y_{i}]-b_{i3}^{(t)}( \alpha)\mu_{i},\quad\forall\,i\in\mathbb{I}_{2}. \tag{12}\]  
By combining (9) with (11), we have  
\[\frac{\partial^{2}h_{i}^{(t)}(\mathbf{\beta}|\alpha)}{\partial\beta\partial\mathbf{ \beta}^{\top}}\geqslant 0;\]  
i.e., \(\partial^{2}h_{i}^{(t)}(\mathbf{\beta}|\alpha)/\partial\beta\partial\mathbf{\beta}^{\top}\) is a positive semi-definite matrix. By applying the second-order Taylor expansion of \(h_{i}^{(t)}(\mathbf{\beta}|\alpha)\) around \(\mathbf{\beta}^{(t)}\), we have  
\[h_{i}^{(t)}(\mathbf{\beta}|\alpha)\geqslant h_{i}^{(t)}(\mathbf{\beta}^{(t)}|\alpha)+ b_{i4}^{(t)}(\alpha)\times(\mathbf{\beta}-\mathbf{\beta}^{(t)})^{\top}\mathbf{w}_{i}, \tag{13}\]  
where the equality holds iff \(\mathbf{\beta}=\mathbf{\beta}^{(t)}\), and \(b_{i4}^{(t)}(\alpha)\triangleq\{[\mu_{i}^{(t)}+(\alpha-1)y_{i}]^{-1}-b_{i3}^{( t)}(\alpha)\}\mu_{i}^{(t)}\). Let \(\ell_{1}(\mathbf{\beta}|\alpha)\) denote the conditional log-likelihood function of \(\mathbf{\beta}\) given \(\alpha\), we have\[\ell_{1}(\beta|\alpha)\ \stackrel{{(\ref{eq:2.2.2})}}{{=}}\ \ell(\beta,\alpha)\] \[\stackrel{{(\ref{eq:2.2.3})}}{{\geqslant}} \sum_{i=1}^{n}\Bigl{\{}\Bigl{[}b_{i1}+b_{i2}b_{i4}^{(t)}(\alpha) \Bigr{]}\mathbf{w}_{i}^{\top}\beta-\Bigl{[}\alpha^{-1}-b_{i2}b_{i3}^{(t)}(\alpha) \Bigr{]}\exp(\mathbf{w}_{i}^{\top}\beta)\Bigr{\}}+c_{5}^{(t)}\] \[\triangleq Q_{1}(\beta|\beta^{(t)},\alpha),\]  
which minorizes \(\ell_{1}(\beta|\alpha)\) at \(\beta=\beta^{(t)}\), where \(c_{5}^{(t)}\) is a constant free from \(\beta\). Note that the \(Q_{1}(\beta|\beta^{(t)},\alpha)\) is a weighted log-likelihood function of \(\beta\) for the Poisson regression model with weight vector \((\alpha^{-1}-b_{12}b_{13}^{(t)}(\alpha),\ldots,\alpha^{-1}-b_{n2}b_{n3}^{(t)} (\alpha))^{\top}\) and observations \(Y_{\text{obs}}^{*}=\{y_{i}^{*}\}_{i=1}^{n}\) with  
\[\frac{\partial^{2}\log(y_{i}^{-1}\mu_{i}+\alpha-1)}{\partial \beta\partial\beta^{\top}} = \frac{(\alpha-1)y_{i}^{-1}\mu_{i}}{(y_{i}^{-1}\mu_{i}+\alpha-1)^ {2}}\mathbf{w}_{i}\mathbf{w}_{i}^{\top}. \tag{9}\]0  
We can calculate the MLEs of \(\beta\), denoted by \(\beta_{*}^{(t+1)}\), of the weighted Poisson regression model directly through the built-in 'glm' function in the VGAM R package.