Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 37 / 5 / b

Let $f_j(t)=f(t\mid I=j)$ and $S_j(t)=1-F(t\mid I=j)$. A recorded death observes both $T=t$ and $I=1$, so its likelihood contribution, omitting factors from the independent observation mechanism, isIts integration over $t$ gives the eventual death probability $\theta$.

A recorded discharge observes $T=t$ and $I=2$. Its likelihood contribution isThe mixing probability distinguishes the joint outcome-and-time density from the conditional recovery-time density.

For a patient still in hospital, sum over the unobserved eventual outcome. Its likelihood contribution is the mixture survival functionThe probability of eventual death among those still hospitalized is generally different from $\theta$, because the two conditional time distributions can have different tails.