= Exponential rate with a small known additive hazard
{title2=$\widehat\lambda\simeq d/X-d^{-1}\sum_i v_iG_i'(x_i)$}
Under <independent censoring>, a model $H_i(t)=\lambda t+G_i(t)$ with known $G_i$ has score $\sum_i v_i/(\lambda+G_i'(x_i))-X$, where $X=\sum_i x_i$ and $d=\sum_i v_i$. Expanding about the ordinary exponential estimate $d/X$ gives the displayed first-order correction when every added event-time hazard is small compared with $d/X$. The integrated offsets contribute constants to the <log-likelihood>. If all $G_i'(x_i)=g$, the positive interior estimate is exactly $d/X-g$.
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