Consider the Poisson log-linear model [y_{i} mid mathbf{x}_{i} sim text { Poisson }left(mu_{i} ight), quad log left(mu_{i} ight)=mathbf{x}_{i}^{top} boldsymbol{beta}, quad 1 leq i leq n]

Consider the Poisson log-linear model

\[y_{i} \mid \mathbf{x}_{i} \sim \text { Poisson }\left(\mu_{i}\right), \quad \log \left(\mu_{i}\right)=\mathbf{x}_{i}^{\top} \boldsymbol{\beta}, \quad 1 \leq i \leq n\]

Show that the score equations have the form (6.9), with \(D_{i}=\frac{\partial \mu_{i}}{\partial \beta}\) and \(V_{i}=\mu_{i}\).

n i=1 DV (y)=0, - (6.9)