Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 5

Conditioning on $T_1=t$ and using independent random variables giveswhere the notation in the paper uses $F_2$ for the survival function.

With common fixed censoring time $c$, the ordering is known exactly when the earlier event occurs by $c$; continuity makes ties have probability zero. Thus the informative event is $\{min(T_1,T_2)\le c\}$ and

For independent exponential distributions, the numerator from part a(ii) iswhile the denominator is the expression in braces. Their ratio is

Condition on the independently generated random censoring information. For every realized common censoring horizon, part b(i) gives the same conditional probability ${\lambda }_1/({\lambda }_1+{\lambda }_2)$. The law of total probability therefore gives that ratio after averaging over the censoring distribution as well.

Since $T_i$ has cumulative hazard function ${\varphi }_i{H}_0(t)$,Thus the cumulative-hazard time change $U_i=H_0(T_i)$ has an exponential distribution of rate ${\varphi }_i$.

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 givesThis is the pairwise race probability for a proportional hazards family.

In a competing risks model, the latent times to different event types form such a race: only the smallest time and its cause are observed. With proportional cause-specific hazards ${\varphi }_i{h}_0(t)$, the probability that cause $i$ wins is ${\varphi }_i/{\sum }_j{\varphi }_j$, independently of the baseline hazard and under independent censoring. This is the same cancellation derived above.