Restricted estimation of the cumulative incidence functions of two competing risks

Abstract

The cumulative incidence function (CIF) plays an important role in the comparison of competing risks in a competing risks model. Its value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. In this paper we consider the estimation of two CIFs, F1 and F2, corresponding to two competing risks when the ratio R(t)≡F1(t)/F2(t) is nondecreasing in t>0. First, we derive their nonparametric maximum likelihood estimators (NPMLE) of these CIFs in the continuous case under this order constraint and show that they are inconsistent. We then develop projection-type estimators that are uniformly strongly consistent and study the weak convergence of the resulting processes. Through simulations, we compare the finite sample performance of the NPMLEs and our estimators and show that our estimators outperform them in general in terms of mean square error at all the scenarios that we consider. We also develop a test for the presence of this order constraint and extend all these results to the censored case. To illustrate the applicability of the theory we develop, we provide a real life example.