Conditional Failure Time Distributions under Competing Risk Theory with Dependent Failure times and Proportional Hazard Rates* Conditional Failure Time Distributions under Competing Risk Theory with Dependent Failure times and Proportional Iv\zard Rates

Suppose that death (or any non repetitive event) can occur due to various causes, each having its own failure time. Assuming independence of failure times and proportional hazard rates over the whole range of time, some authors shown that the single cause failure time distributions conditional on the cause of death, each in presence of the remaining causes, are the same as the distribution of observable failure time, regardless of the cause. It has been shown in the present article, that this result is also valid without the assumption of independence (Section 3). It has also been suggested (Section 5), that in the case of dependent failure time, a conditional limiting distribution (as ex could represent the failure time distribution when cause C ex is eliminated. Three examples (trivariate exponential, bivariate Compertz, and U.S. Life Table 1959-61 data) are given as illustrations.