Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 6 / Unconditional hazard in a frailty model / Solution

A frailty model introduces an unobserved positive random effect $U$ that multiplies an individual's hazard. With $H_0(t)={\int }_0^t{h}_0(s)ds$, marginal survival isDifferentiation givesConsequently $\bar{h}(0)=h_0(0)$ exactly when $\mathbb{E}U=1$.