Since

T_{i}

ϕ_{i} H_{0} (t)

P {H_{0} (T_{i}) > u} = P {T_{i} > H_{0}^{- 1} (u)} = e^{- ϕ_{i} u} .

Thus the cumulative-hazard time change

U_{i} = H_{0} (T_{i})

ϕ_{i}

ii

Solution

The increasing time change

H_{0}

preserves the ordering of event times and transforms each censoring time by the same rule. Applying part b to

U_{1}, U_{2}

therefore gives

P (T_{1} < T_{2} ∣ informative) = \frac{ϕ _{1}}{ϕ _{1} + ϕ _{2}} .

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