Exponential rate with a small known additive hazard ($\hat{\lambda }\simeq d/X-d^{-1}{\sum }_i{v}_i{G}_i^{\prime }(x_i)$)

Under independent censoring, a model $H_i(t)=\lambda t+G_i(t)$ with known $G_i$ has score ${\sum }_i{v}_i/(\lambda +G_i^{\prime }(x_i))-X$, 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 $G_i^{\prime }(x_i)=g$, the positive interior estimate is exactly $d/X-g$.