 Solution

Write $f(y;\theta )=\exp \{y\theta -b(\theta )+c(y)\}$ and use canonical parameter ${\theta }_i=X_i^T\beta$. Thenwhere ${\mu }_i=b^{\prime }(X_i^T\beta )$. Geometrically the residual vector is orthogonal to every design column. For $H_0:{\beta }_1=0$, use a two-sided Wald statistic ${\hat{\beta }}_1/\mathrm{se}({\hat{\beta }}_1)$ (or likelihood-ratio statistic), rejecting at the corresponding normal or chi-square critical value.