Nonidentifiability of competing risks

Frequently one is interested in examining the survival experience of a set of individuals exposed to k risks. If W is the time of failure for the individual and J is the indicator variable for cause of death, then the competing risk framework assumes that W = min X_i and J = j if X_j is the minimum, where the X_i's are the survival lifetimes when only risk R_i is operating on the population. To examine the underlying structure of the survival experience one has to know the joint distribution of the $\text{X}{}_{\mathrm{i}}\text{(F}{}_{\mathrm{<mtext>x</mtext>}}\text{)}$. It is shown here that if only W and J are observed, the joint distribution of the $\text{X}{}_{\mathrm{i}}\text{(F}{}_{\mathrm{<mtext>x</mtext>}}\text{)}$ is nonidentifiable.