 Solution

Writing $y_i$ for the observed counts, independence of the Poisson distributions gives the log-likelihoodIts score function and negative second derivative areUnless all $x_i$ vanish, $J(\beta )>0$, so the log-likelihood is strictly concave. Use Newton method, starting from a finite ${\beta }_0$, withStop when the score function or the change in $\beta$ is small; a reduced step can ensure that the log-likelihood increases. Any finite zero of the score function is the unique maximum-likelihood estimator.

Existence needs a qualification. If the nonzero $x_i$ include both signs, $U$ decreases from $+\infty$ to $-\infty$, so a finite root exists. If all nonzero $x_i$ are positive, a finite root exists exactly when ${\sum }_i{x}_i{y}_i>0$; otherwise the supremum is approached as $\beta \to -\infty$. The corresponding condition for all negative $x_i$ is ${\sum }_i{x}_i{y}_i<0$, with a supremum at $+\infty$ otherwise. When all $x_i=0$, the parameter is unidentifiable. These cases explain when the iterative algorithm actually has a finite estimate to find.