Gamma regression with canonical link ($\eta =-1/\mu ,\mu =-1/\eta$)

For a Gamma distribution with fixed shape $\alpha$, choose dispersion $\varphi =1/\alpha$, canonical parameter $\theta =-1/\mu$, and cumulant $b(\theta )=-\log (-\theta )$. The canonical link function is the displayed negative reciprocal; reversing its sign is an equivalent regression convention. For independent observations, the score function is ${\varphi }^{-1}{X}^T(y-\mu )$ and the Fisher information matrix is ${\varphi }^{-1}{X}^T\mathrm{diag}({\mu }_i^2)X$. The deviance is $2{\sum }_i[y_i/{\hat{\mu }}_i-1-\log (y_i/{\hat{\mu }}_i)]$, with twice the log-likelihood ratio equal to this divided by $\varphi$. The fitted mean is the inverse link applied to $X\hat{\beta }$, not $X\hat{\beta }$ itself.