 Concavity of the Poisson regression likelihood

For a full-column-rank design $X$ and positive fitted means ${\mu }_i=\exp (x_i^T\beta )$, the log-likelihood has Hessian $-X^T\mathrm{diag}({\mu }_i)X$, which is negative definite. Thus any finite stationary point is the unique maximum-likelihood estimator. Full rank does not guarantee existence: with an intercept and all counts zero, the supremum is approached as the intercept tends to $-\infty$. If every observed count is positive, a full-rank design does give a finite maximum: each function $y_i{\eta }_i-e^{{\eta }_i}$ tends to $-\infty$ in either tail and is bounded above, and $\Vert X\beta \Vert \to \infty$ as $\Vert \beta \Vert \to \infty$, so the negative log-likelihood is coercive.