= Poisson score linearization under proportional variance
{c}
{title2=$\operatorname{Var}(\widehat\beta)\approx\phi I^{-1}$}
With $\mu_i=e^{\beta x_i}$, define $U=\sum_i x_i(Y_i-\mu_i)$ and $I=\sum_i x_i^2\mu_i$. The estimating equation and a <Taylor expansion> give $\widehat\beta-\beta_0\approx I(\beta_0)^{-1}U(\beta_0)$. If responses are independent with correct means and $\operatorname{Var}(Y_i)=\phi\mu_i$, then $EU=0$ and $\operatorname{Var}U=\phi I$. Consequently the first-order estimator <variance> is $\phi/I$, even if the working <Poisson distribution> is wrong. Regular asymptotics require growing information and control of leverage and the Taylor remainder.
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