Poisson log-rate ratio ($\hat{\nu }=\log {\overline{Y}}_B-\log {\overline{Y}}_A$)

For independent samples of $n$ observations from Poisson distributions with positive means $a$ and $b$, the maximum-likelihood estimator of $\nu =\log (b/a)$ is the displayed difference when both totals are positive. Its asymptotic variance is $(a^{-1}+b^{-1})/n$, obtained by the delta method or by inverting the two-parameter Fisher information matrix. A zero group total makes its log-mean estimate a boundary value; finite normal intervals require nonzero totals.