 Solution

Let $Y_1,\dots ,Y_n$ be independent. In the general exponential-dispersion formulation, with known weights $w_i>0$ and common dispersion $\varphi >0$, their densities have the formThusThe systematic component is ${\eta }_i=x_i^T\beta$, and a specified one-to-one differentiable link relates it to the mean byThe unknown model parameters are the regression coefficient $\beta$ and, when it is not fixed by the family, the dispersion $\varphi$.

The log-likelihood isup to terms independent of the parameters, where ${\theta }_i$ is determined from $g^{-1}(x_i^T\beta )=b^{\prime }({\theta }_i)$.

For a Poisson response,so the natural parameter is $\theta =\log \mu$ and $b(\theta )=e^{\theta }$. The canonical link sets the linear predictor equal to the natural parameter and is therefore