Estimation of cumulative incidence functions when the lifetime distributions are uniformly stochastically ordered

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. Assume that the lifetime distributions of two populations are uniformly stochastically ordered. Since this ordering may not hold for the empiricals due to sampling variability, it is natural to estimate these distributions under this constraint. This will in turn affect the estimation of the CIFs. This article considers this estimation problem. We do not assume that the risk sets in the two populations are related, give consistent estimators of all the CIFs and study the weak convergence of the resulting processes. We also report the results of a simulation study that show that our restricted estimators outperform the unrestricted ones in terms of mean square error. A real life example is used to illustrate our theoretical results.