 Solution

A frailty model represents unobserved heterogeneity using a nonnegative frailty random variable $U$ that multiplies an individual's baseline hazard. With $H_0(t)={\int }_0^t{h}_0(s)ds$, the individual and population survivor functions areThus the population survivor function is the Laplace transform of the frailty distribution at $H_0(t)$; at points where differentiation is justified, $\overline{h}(t)=h_0(t)\mathbb{E}[U\mid T>t]$. The last identity follows by differentiating the integral and dividing $-{\overline{S}}^{\prime }$ by $\overline{S}$.

Here $g(u)=e^{-u}$ for $u\ge 0$, $\theta >0$, and $H_0(t)=\theta t^2/2$. Direct integration givesThis is population hazard under exponential frailty. For every fixed $u>0$ the individual hazard function $u\theta t$ increases linearly. In contrast,so the population hazard increases up to $t=\sqrt{2/\theta }$, then decreases, tending to zero like $2/t$. More vulnerable individuals leave the risk set earlier, leaving survivors with smaller frailties. This population decline is selection, not a decline in any fixed individual's hazard. If $\theta =0$, survival is identically one and both hazards vanish; the positive-rate comparison is then inapplicable.