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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.

In Poisson regression with log means $a+\beta x_i$, the Fisher information matrix is $\left(\begin{array}{cc}{S}_0& S_1\\ S_1& S_2\end{array}\right)$, where $S_j={\sum }_i{x}_i^j{\mu }_i$. Inverting this matrix gives slope variance $S_0/(S_0{S}_2-S_1^2)$. Writing ${\bar{x}}_{\mu }=S_1/S_0$ reduces it to the displayed reciprocal weighted sum of squares. Thus a common intercept consumes information about the overall rate, while slope information comes from the spread of the covariates under the mean-count weights. Constant covariates make the two parameters unidentifiable.

In a regular common-support statistical model with probability density function $a(\lambda ,y)e^{\lambda t(y;\psi )}$, the score function in $\psi$ is $\lambda t_{\psi }$. Its zero expectation gives $\mathbb{E}{t}_{\psi }=0$ for $\lambda \ne 0$. The mixed log-likelihood derivative is $t_{\psi }$, so the cross Fisher information matrix block vanishes. At zero the same conclusion follows by continuity when the regular derivatives extend there. This proves parameter orthogonality; it does not establish exact independence of fitted parameters.