 Profile likelihood for a common Poisson rate ratio with unequal exposures

Suppose independent counts have Poisson distributions with means $p_{i1}{\lambda }_i$ and $p_{i2}{\lambda }_{i}r$, where the known exposures are positive and $r>0$ is a shared rate ratio. Put $t_i=y_{i1}+y_{i2}$. Maximizing over each baseline givesZero-total strata contribute no information about $r$; their baseline maximum is at zero in the parameter-space closure. For every positive-total stratum its fitted second-count fraction increases with $r$, so its contribution to the score function decreases. Explicitly,whenever some $t_i>0$. The limiting score function values are ${\sum }_i{y}_{i2}$ and $-{\sum }_i{y}_{i1}$. If both totals are positive, there is therefore exactly one finite positive maximum-likelihood estimator. A zero total in one condition instead gives a boundary supremum at $r=0$ or $r=\infty$; two zero totals leave $r$ unidentifiable.