Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 7 / b

In the counting-process intensity in survival analysis, the counting process $N_i(t)$ jumps once if an observed event occurs; the at-risk process $Y_i(t)$ is one immediately before $t$ only while individual $i$ is still observed and event-free. Conditional on that information, the chance of an event in a short interval is $h_i(t)dt+o(dt)$ for an at-risk individual and zero otherwise. This is the meaning ofThe history ${\mathcal{H}}_{t-}$ is the filtration generated by the observed event and censoring histories, entry information, and available covariates strictly before $t$. In particular, it determines the current risk set but does not reveal future event times. The right-hand side is the infinitesimal compensator of a counting process; the equality is interpreted as an intensity statement to first order in $dt$.

Without censoring, $Y_i(t)=1_{T_i\ge t}$, so its unconditional expectation is survival:The distinction between $>$ and $\ge$ is immaterial for a continuous event time.

With independent censoring, the at-risk process is $Y_i(t)=1_{T_i\ge t,C_i\ge t}$. Independence and the common censoring survivor function $G$ give

Apply the tower property of conditional expectation to the intensity equation. Since $h_i(t)$ is a specified individual hazard function,As in the conditional equation, this describes the first-order expected increment of the observed counting process.