Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 4 / a

Keep the counting process and at-risk process notation from the introductory derivation. Because the individual hazards add, the aggregate conditional intensity isThe known individual contributions must be averaged over the current risk set, rather than over the original cohort. DefineThen $dN(t)/Y(t)$ estimates $\{{\overline{h}}_0(t)+h_1(t)\}dt$. Subtract the known average contribution to obtain the offset-adjusted cumulative hazard estimatorTake the quotient to be zero when $Y=0$. If $M(t)=N(t)-{\int }_0^t\lambda (u)du$, direct substitution givesThis establishes the target and the mean-zero estimation noise, under the usual integrability conditions. It estimates $H_1(t)$ while there is a nonempty risk set; no data identify its continuation after follow-up has ended. With common known $h_{0i}=h_0$ and individuals at risk throughout $[0,t]$, it reduces to ${\hat{H}}_1(t)=\hat{H}(t)-H_0(t)$.

The estimate may decrease between events because the known offset is subtracted continuously. That is not an algebraic error: it is an unconstrained estimating-equation estimator, not automatically a nonnegative monotone cumulative hazard estimate. If such shape constraints are imposed, the fitting procedure must explicitly account for them. The working hazard model itself must satisfy $h_{0i}+h_1\ge 0$.