Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 32 / 7 / a

Write $r={\theta }_A+{\theta }_B$, with ${\theta }_A\ge 0$, ${\theta }_B>0$ and a finite $\tau \ge 0$. The competing risks model with transient surgical mortality has survival functionIntegrating the disease cause-specific hazard against this survival function givesThe first term accounts for disease deaths while both causes act; the second includes survivors of that period who then face only disease mortality. The probability of eventually dying from surgery isTaking the limit in $F_B$ gives $F_B(\infty )={\theta }_B(1-e^{-r\tau })/r+e^{-r\tau }$, and thereforeThe ultimate-death conclusion uses the positive continuing disease hazard. If ${\theta }_B=0$, the surviving fraction $e^{-{\theta }_A\tau }$ would instead live indefinitely in this model; the asserted conclusion is not valid in that boundary case.