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

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- Solution 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) d s

, marginal survival is

\overset{ˉ}{S} (t) = \int_{0}^{\infty} e^{- u H_{0} (t)} g (u) d u .

Differentiation gives

\overset{ˉ}{h} (t) = h_{0} (t) \frac{\int _{0}^{\infty} u e ^{- u H_{0} (t)} g ( u ) d u}{\int _{0}^{\infty} e ^{- u H_{0} (t)} g ( u ) d u} = h_{0} (t) E [U ∣ T \geq t] .

Consequently

\overset{ˉ}{h} (0) = h_{0} (0)

exactly when

E U = 1

.

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