= Profile likelihood for a common Poisson rate ratio with unequal exposures
{title2=$\widehat\lambda_i(r)=t_i/(p_{i1}+rp_{i2})$}
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 gives
$$
\widehat\lambda_i(r)=\frac{t_i}{p_{i1}+rp_{i2}},\qquad
s(r)=\frac{d\ell_{\rm prof}}{d\log r}
=\sum_i\left[y_{i2}-\frac{t_i r p_{i2}}{p_{i1}+rp_{i2}}\right].
$$
Zero-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,
$$
\frac{ds}{d\log r}=-\sum_i\frac{t_i p_{i1}r p_{i2}}{(p_{i1}+rp_{i2})^2}<0
$$
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.
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