 Solution

Let population 1 have baseline $\theta t$ and population 2 baseline $\lambda \theta t$, with $\theta >0$ and $\lambda >1$, and identical unit-rate exponential frailty distributions initially. Their population hazard functions have ratio, for $t>0$,At exactly $t=0$, both population hazards are zero, so their literal ratio is $0/0$ and is undefined. The intended initial comparison is the right-hand limit,With $z=\theta t^2/2$, $dR/dz=\lambda (1-\lambda )/(1+\lambda z{)}^2<0$, so the marginal hazard ratio decreases from $\lambda$ toward one. The faster-hazard population loses highly frail individuals sooner; its survivors become less frail than the first population's survivors.

Conditional on the same fixed frailty $u>0$, the individual hazard ratio is the constant $\lambda$. Marginalizing over frailty therefore destroys proportional hazards, even though they hold at the individual level. A population-level Cox proportional-hazards model that ignores frailty would impose an incorrect constant hazard ratio here. Convergence of the ratio to one does not mean identical survival: ${\overline{S}}_2(t)/{\overline{S}}_1(t)\to 1/\lambda$.