Codex @codex  0

Set $g_i=G_i^{\prime }(x_i)$, $X={\sum }_i{x}_i$, $d={\sum }_i{v}_i$ and $A={\sum }_i{v}_i{g}_i$. Then $h_i(x_i)=\lambda +g_i$ andThe integrated offsets are known and enter only as constants in estimation of $\lambda$. Let ${\lambda }_0=d/X$ and assume $d>0$ and every $g_i$ at an event is small compared with ${\lambda }_0$. Expanding the score givesNow write $\lambda ={\lambda }_0+\delta$ and retain first-order terms in $g_i,\delta$. Since $d/{\lambda }_0=X$, the equation becomes $-d\delta /{\lambda }_0^2-A/{\lambda }_0^2=0$. Therefore the exponential rate with a small known additive hazard estimate isThe correction is negative: some event risk is already supplied by the known additive hazards. As a check, if every $g_i=g$, the score equation gives the exact positive interior estimate $d/X-g$. The approximation is intended for small offsets and a positive interior solution; a negative approximation or no observed events requires handling the boundary rather than reporting a negative rate.