Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 6

A proportional frailty model specifiesIts population survival function is the Laplace transform of the frailty density,If $m=\mathbb{E}U<\infty$, replacing $U$ by $U/m$ and $h_0$ by $m{h}_0$ leaves their product, and hence the model, unchanged. The normalization $\mathbb{E}U=1$ identifies this otherwise arbitrary scale and makes $h_0(0)$ the initial population hazard when $H_0(0)=0$.

For $g(u)=e^{-u}$,Therefore the population hazard function is

By Bayes theorem, the frailty distribution among survivors has densityIt is an exponential distribution with rate $1+H_0(t)$.

The conditional mean frailty among survivors isBecause the cumulative hazard function is nondecreasing, this mean is nonincreasing: high-frailty individuals tend to experience the event earlier, leaving a progressively more robust surviving population.

Conditionally on survival, average the individual hazard $U{h}_0(t)$:which agrees with part a.