$$
\widehat{\eta}=\tilde{x}^{\prime} \hat{\boldsymbol{\theta}}
$$
Denoting the estimated variance of the estimates as $\widehat{\Sigma}{\theta}=\widehat{V}(\widehat{\theta})=I(\widehat{\theta})^{-1}$, then the estimated variance of the linear predictor is $$ \widehat{V}(\widehat{\eta})=\tilde{\boldsymbol{x}}^{\prime} \widehat{\boldsymbol{\Sigma}}{\theta} \tilde{\boldsymbol{x}}=\widehat{\sigma}_{\hat{\eta}}^{2}
$$
Therefore, the $1-\alpha$ level confidence limits on $\eta$ are
$$
\left(\widehat{\eta}{\ell{1}}, \widehat{\eta}{u}\right)=\widehat{\eta} \pm Z{1-\alpha / 2} \widehat{\sigma}{\widehat{\eta}} $$
MATH0057 COURSE NOTES ：
jth coefficient $\beta_{j}$ can be expressed as
$$
U(\theta){\beta{j}}=\sum_{i} w_{i j}\left(y_{i}-\pi_{i}\right)
$$
with weights
$$
w_{i j}=\left(\frac{1}{\pi_{i}\left(1-\pi_{i}\right)}\right) \frac{\partial \pi_{i}}{\partial \beta_{j}} .
$$