The <cause-specific hazard> for cause $k$ is
$$
h_k(t)=\lim_{\Delta t\downarrow0}
\frac{\mathbb P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}.
$$
It is the instantaneous rate of cause $k$ among individuals still free of every competing event.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / a / ii / Solution

{title2=$h_k(t)$}

The cause-specific hazard $h_k(t)$ is the instantaneous rate of event type $k$ among individuals who have not yet experienced any competing event.


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